["r..-----1( HANDLING DATA )'r--- - - -- - -- -- - - - -- - -- -- - -- - - - -- - -- - - -- - AVERAGES An average is a single value that Median of a distribution is used to represent a collection of data*. Averages are sometimes called The middle value of a distribution* that is measures of central tendency arranged in size order. To find the median or measures of average. Three position, use the formula*: commonly used types of averages are mode, median and mean. median = 1-(n + 1) 2 Mode of a distribution The value or values that occur most often in a where n is the number of values. distribution*. For example, the distribution of time in minutes taken by 10 peopi~ to finish a test is: For example, to find the median of the distribution: 30 31 32 32 35~37 40 4 3 1 8 5 2 1 6 12 The value 36 occurs most often in this 1. Arrange the distribution in size order: distribution, so the mode is 36. 1 1 2 3 4 5 6 8 12 Bimodal distribution 2. Calculate the median position using: A distribution* that has two modes. For example, in the following distribution the median = 1-(n + 1) values 32 and 36 both occur twice: 2 30 31 ~ 35 663\\\"\\\"\\\"6) 39 = .l(g + 1) = 1_ x 10 = 5 This means that 32 and 36 ar~odes. 22 3. Find the value that is in the median position, A distribution which has three or more modes in this case, the 5th value in the list: is called a multimodal distribution. 1 1 2 3 0 5 6 8 12 The median of this distribution is 4 . If there is an even number of values, the median cD>position is halfway between two middle values. 1 123 5 6 8 12 To find the median, add the middle values and divide by two: med\\\"1an =4-+- 5 = 4 .5 2 The median of this distribution is 4.5. Mode of a frequency distribution Median of a frequency distribution To find the mode of a frequency distribution*, To find the median of a frequency distribution*., find the value with the highest frequency*. first calculate the cumulative frequency* of the distribution. Calculate the median position and Sock si:ze Frequency This table shows the find the value that is in that position. --~---+----- frequency of the sizes of pairs Smail 98 of socks bought in a store in a month. You can see from If the median position is 0.5 more than a cumulative frequency, Mediu m 429 the frequency table that the add together the corresponding value and the value above it L 342 category with the highest frequency (429) is medium, and divide by two. In the example below, the median position is arge 25.5 (that is, (50 + 7) \u00b7+ 2). This is 0.5 above the cumulative Extra largi:> 13 1 so lhe mode of this frequency 25, so the median value is 7.5 (that is, (1 + 8) + Z). distribulion is medium. Modal group or modal class Va l u e Frequency Cumulat ive frequen 'Y The class interval* of a grouped frequency distribution* that occurs most often. 6 12 12 13 12 + 13 = 25 Time (mi nutes) Freqrnmcy This table shows the 7 14 25 - 14 = 39 8 11 39 ~ 11 - 50 length of time that 50 9 1-5 10 passengers had to wait Median of a grouped frequency distribution 6-1 0 for a bus one day. The To find the median of a grouped frequency 11 -1 5 25 modal group is 6- 10 distribution*, read the value at the median position 16 -.2 0 minutes as it has the on a cumulative frequency diagram (see page 109). 10 highest frequency. s ~W- \u2022-ciass\u00b7 !~terval 99; Continuous data 96; Cumulative frequency (table) 99; Data, Discrete data, Distribution 96; Formula 75; Frequency 96; Frequency distribution 99 (Frequency table); Grouped frequency distribution (table) 99; Mid-interval value 99; Product 14 (Multiplication); Sum 14 (Addition).","~-----------------------------------,( HANDLING DATA )i----- Mean or arithmetic mean of a distribution Mean of a grouped frequency distribution A measure of the general size of the data. To To find the mean of a grouped frequency find the mean, use the rule: d istribution*, find the mid-interva l value* (x) for each class interva l* and mu lti ply it by the 2: values frequency* (f). Then add the products* to find the total sum* of the va lues a nd find the mean =number of values mean using: where the Greek letter sigma, 2:, means \\\"the 2: values total of\\\" or \\\"the sum* of.\\\" mean = number of values For example, to find the mean of the distribution* : This method can be used for finding the mean 0 5 7 6 2 10 of d iscrete* and continuous data*. As the exact values in a grouped frequ e ncy distribution are use _(0_ +_5_ +_7_ +_6_ +_2_ +_ 1_0_) = 30 = 5 not known, the mean is calculat ed using the 66 mid-interva l value, which is itself an a verage. For this reason, the mean of a grouped frequency The mean of this distribution is 5. distribution can be only an approximation. The rule for finding the mean can also be written as: For example, th e grouped frequency dist ributio n table* below shows how many points were x- = -LX scored by 60 contestants in a g e nera l knowledge quiz. n xwhere is the mean, x is the set of values and n is , the number of values in the set. Mean of a frequency distribution nt\u2022ncyPoints I Mid- IFreq1Jei1cy Mid-interval To find the mean of a frequency distribution*, first find the sum* of the values by multiplying each scored interval (f) vafoe x value (x) by its frequency* (f) and adding these val ue products*. Then find the mean using: (x) 0 10 5.0 () 1 = 5.0 2: values mean = number of values 11 20 15.5 5. 5 X 2 = 31 5.5 x 4 - 102 For example, the frequency distribution table 21 - 30 25.5 below shows the number of books read by a 3 5.5 x 3 - 106.5 group of students in one month. To find the 31- 40 35.5 3 mean of this distribution, first find the sum of 75.545.5 x 9 - 409.5 the values, as described above. 55.5 x 14 \\\"' 777 Number of Freque n r;y Frequency x va Iue 45 .5,_, M41-509 65 .5 x 12 = 786 b ooks {x ) (f) (fx) 55.S 14 0 65.5J l -60 \u00b7 ~ 679 5 1x 0=0 61- 70 12 2 2 2 X 1 =2 71 - 80 85.5 >< 5 = 4.\u00a37.S, 3 0 OX 2 = 0 81-90 85. ~ 5 95.5 x 1 = 95.5 4 1X3 = 3 91 - 100 95.5 1 5 1 1 x 4 ='1 6 \\\"'i f - 60 :::.fx = 34 19 .5 2 2 x 5 = 10 7 To calculate an estimate of the mean number of 0 0 X6 = 0 points scored in the quiz, use: l x7=7 2: values 2:.f = 8 r !fx = 26 mean = number of values Then calculate the mean using: 3,419.5 60 2: values 26 mean = number of va lues = 8 = 3 \u00b725 The mean of this frequency distribution is 3.25 = 56.99 (2 d.p.) books. A mean value does not have to be a whole number, even when the data is discrete*. The mean number of points scored in the quiz was approximately 57. Internet links For links to_ useful vyebsites on averages, go t.o vyww, usb orne-quicklinks. com","HANDLING DATA MEASURES OF SPREAD Spread, or dispersion, is a Quartiles measure of how much a collection Lower quartile or first quartile (Q1) of data* is spread out. There are The value that lies one quarter of the way several methods of describing dispersion. These are known as through a distribution* arranged in ascending measures of dispersion and they order. To find the lower quartile position, use: give different types of information Iower qua rt\\\"1Ie pos1\u00b7rion = -(n -+- 1) about the spread of the data. 4 Range where n is the number of values in the distribution, The range of a distribution* is the difference or the cumulative frequency* of a frequency between the highest and lowest values. (This must be a single value.) To find the range of distribution* or grouped frequency distribution*. a distribution, use the rule: Upper quartile or third quartile (Q3) range =highest value - lowest value The value that lies three-quarters of the way For example, to find the range of this distribution, through a distribution* arranged in ascending take the lowest value from the highest: order. To find the upper quartile position, use: 2 10 3 7 11 5 3 9 6 . .. 3(n+1) range = 11 - 2 = 9 upper quartile pos1t1on = 4 where n is the number of values in the distribution, To find the range of a grouped frequency or the cumulative frequency* of a frequency distribution*, take the lowest possible value from the highest possible value. For example, distribution* or grouped frequency distribution*. if the first group were 0-5 and the last group were 46-50, the range would be 50 (SO - 0). To find the quartiles of a distribution if the quartile position is a decimal*, or to find the quartiles of Comparing distributions a frequency distribution or a grouped frequency To learn more about a set of distributions, it distribution, draw a cumulative frequency diagram* can be helpful to compare the range and the and read off the values at the quartile positions. mean*. .For example, this table shows the Interquartile range (IQR) ... D Iages of residents in two nursing homes: The range of the middle 50% of a distribution*, which eliminates extreme values at either end of \u2022I the distribution. To find the interquartile range, use: . The mean age at each home is 79 (that is, IQR = upper quartile - lower quartile- 553 + 7). The range of ages at Bluebells is 35 years (102 ~ 67) and the range of ages For example, to find the interquartile range of: at The Elms is 5 years (82 - 77). This shows that, although the mean age is the same, the 2 4 5 7 7 9 12 15 16 16 18 range of ages at Bluebells is much greater. 1. Find the value in the lower quartile position: ~ = ..12\u00b1.l =Jl.= 3 4 44 The 3rd value in the distribution is 5, so the lower quartile is 5. 2. Find the value in the upper quartile position: 3(n + 1) = 3 x (11 + 1) = i\u00a7. = 9 4 4 4 The 9th value in the distribution is 16, so the upper quartile is 16. 3. Take the lower quartile value from the upper quartile value (16 - 5 = 11 ). The interquartile range of this distribution is 11. * Cumulative frequency diagram 109; Cumulative frequency table 99; Data 96; Decimal 19; Distribution 96; Formula 75; Frequency distribution 99 (Frequency table); Grouped frequency distribution (table) 99; Mean 100; Substitution 77; Sum 14 (Addition).","HANDLING DATA Standard deviation Standard deviation calculation method 2 Standard deviation from the mean*, usually J'i,: (:xstandard deviation (er) =2 )2 known as standard deviation, tells you how - spread out the values in a distribution* are from its mean . Unlike range and interquartile range, where x is each value in the distribution*, n is standard deviation takes into account every value the total number of values and 2, means \\\"the of a distribution. A high standard deviation means total of\\\" or \\\"the sum* of.\\\" that the values are very spread out. while a low standard deviation means that the values are For example, the distribution below shows the close together. Standard deviation is given in the number of years for which eight employees same units as the original data. It is represented have worked for a company: by the lower-case Greek letter sigma, u . 1 5 6 3 2 10 7 6 Standard deviation calculation method 1 To find the standard deviation : ~standard deviation (u) = 1. Place the values xwhere x is each value in the distribution, is the in a table and calculate the mean of the distribution, n is the total number of value of x2 values and~ means \\\"the total of\\\" or \\\"the sum* of.\\\" for each value of x. For example, the distribution below shows the 2. Find the mean number of years for which eight employees have worked for a company: of the squares 1 5 6 3 2 10 7 6 'Zxof the distribution:2 To find the standard deviation: = 260 = 325 1. Calculate the mean (x) of the distribution : n 8 2: values 3. Calculate the mean of the distribution and mean = number of values sq uare it: )2( ~x = ( ~o )2 = 52 = 25 = (1 + 5+6+3 + 2+10+7 + 6) 4 . Find the standard deviation by su bstituting* the values into the formula *: 8 = 40 = 5 8 2. Find the difference between each value in the = \\\\\/32.5 - 25 = V7.5 = 2.74 (3 s.f.) distribution and the mean (x - x): - 4 0 1 -2 -3 5 2 3 . Find the square of each of these values (x - x)2: The standard deviation is 2.74 years (to 3 s.f.). 16 0 1 4 9 25 4 1 Variance The square of the standard deviation. 4. Find the mean of the squares of the differences, The variance is exp ressed by th e formula *:\u00b7 known as the variance ( :S(x; x)2} :L.(x - x)2 or ~x2 - (2,x)2 16 + 0 + 1 + 4 + 9 + 25 + 4 + 1 n nn 8 where x is each value in the distribution*, = 680 = 7 .5 x is the mean of the distribution, n is the (J'Z{x;t~e5. Find square root of the variance 2 total number of va lues and 2, mea ns \\\"the total of\\\" or \\\"the s um* of.\\\" X) ) V7.5 - 2.74 (3 s.f.) The standard deviation is 2.74 years (to 3 s.f.). lnte~net links .For .links to useful websites on measures ofs.?!ead, .90 to www.usbome-quic;idinks.com","HANDLING DATA - -- x --_r_n_,,...., er,., Changes in standard deviation 45b If every va lu e in a distribution* is increased (or decreased) by the same amount, the mean* is increased (or decreased) by that amount, but the sta ndard deviation stays the same. :;tx -~.::x\\\"' n For example, the distribution below has a mean of 6 and a standard deviation of 2 .83 (3 s.f.): l.. 2 3 2 4 6 8 10 [\\\" If every value is decreased by 3 to give: Most scientific calculators have keys that have statistical functions such as mean* and standard deviation'. Your -1 1 3 5 7 calculator instructions will explain how to use them. The new mean is calculated as: To find the standard deviation of a grouped t -:-::1 + 1 + 3 .,.. 5 + 7) = ~ = 3 frequency distribution Take the mid-interval value* of each class interval * 55 Th e new standa rd deviation is calculated as: as the x value and use either of the methods of \/ (1 + 1 + 9 + 25 + 49) - (\u00b7 ~)\u00b7 2 findi ng standard deviation * described on page 1.03 . The standard deviation formulas* need to be c:r\\\" 5 5. altered to take into accountthe fact that each 8 value has to be mu ltiplied by the frequency* (f). =\/ : - = :\\\\117=9 = v8 \\\"\\\" 2.83 (3 s.f:l Fo'r example, the grouped frequency distribution The mean of the new distribution is increased table* below shows the numb er of telephone calls received on one day by workers in an office. To by 3 (3 + 3 \\\"' 6), but the stand a rd deviation is find an estimate for the sta ndard deviation : 1. Calculate values for x and, as it is a grouped the same for both distributions (2.83). frequency distribution, find fx and fx2 . If every value in a distribution is multiplied (o r divided) by the sa me number, the standard Calls f x fx fx2 deviation a nd the mean will both be multiplied 1-5 (or divided) by that amount. 6-10 93 27 27 x 3 = 81 11 - 15 15 8 120 For example, if every value of the original 16-20 13 13 169 120 x 8 = 960 169 x 13 = 2, 197 distribution above is multiplied by 2, to give: 3 18 54 54 x 18 = 972 4 8 12 16 20 \\\"\\\"'f = 40 2,fx = 370 2, fx 2 = 4,21 O The new mean is ca lculated as: 2 . Fiiid the mean * of the squares* of the (4 + 8 + 12 + 16 + 20) grouped frequency distribution: 5 = =IJX.2 4210 105.25 = 60 = 12 n 40 \u00b7 5 3. Find the mean of the grouped frequency The new standard deviation is calculated as : d istribution and square it: ~((16+ 64 + 144 + 256 + 400) - (~)2 (3: y)2( 2,;x = 0 = 9.252 = 85.5625 5 5. 0 rsso- - = \\\\; - 5- - 122 4 . Find the standard de.viation by substituting* the values into the fo rmul a: = \\\\\/176 - 144 = \\\\\/32 = 5.66 (3 sT) '\\\"\\\"'x' ( )2a- = ~ - 2-fx\\\\n = \\\\\/105.25 - 85.5625 The \u00b7mea n of the new distribution is twice that = \\\\\/19.6875 = 4.44 (3 s:f.) of the o rigi nal distributio n (that is, 6 \u00b7x 2 = 1:2) The standard deviation is approximately 4.44 and the standard deviation is .also doub led calls (to 3 s:f.) . (that is; 2.83 x 2 = 5.66) . * Angle 32;: Area 55; Class interval 99; Data, Distribution 96; Formula 75; Frequency 96i Frequency table 99; Gro~ped frequ~n ~y distribution (table) 99; Mean 101; Mid-inte.rval value 99;\u00b7 Protractor 47; Rounding 16; Sector 65; Squaring 8 (Square numbers); Standard deviation 103; Substitution J]; Sum 14 (Addition).","H\/t.Nl)LIN<:; OA~1---- REPRESENTING DATA There are many different types of Pie chart diagrams and charts you can use A dia g.ram 1n which the frequency* of a to illustrate data*.. The methods you choose might 9epend on distribution* Is represented ' hy the angles* {or. :\\\\\/Vhat you would like to show, as some methods emphasize a reas*) 0f the sectors* .cif a d rcle. The title\u00b7 bf:t he slightly different aspects of pie chart tells you viihat\u00b7it is shuwing, a rid labels the information. or,,a:k(>yexplain what ec;ith sector represeAts. How a .company's employees .usually_travel to work Pictogram, pictograph or ideograph ,P,c:hart orrwhich pletures are used to show Jhe: rrequ~ncy\\\" of a distribvtion*.,Apictogram, \u00b7lndvdes ~ titl~ and a gey;\\\\e)(plainingwh ~t the.: picfor,es ~e~n . Pc;i,rt of~ :picture 2an \\\\je, ljs~d t p represeri-t:smalle.r quantities. \u00b7 Number of ice-cream cones.sold in .a week To fin~ the size of the ailgle. that wiH represent ~ad:i f requency, use th.e for~ ula*; U. . 3 60~ angle .=.. f >< \\\\II\/here F is theJrequency. For exa mple, the fr~quency table* belbw s'~bw~ the data that was ;used to create the pfo t-h'ar( above. Jn ihJ$.1example, 2-f = 60, so ..\u00b7 \u00b7 \u00b7\u00b7\u00b7\u00b7 \u00b7 \u00b7\u00b7 '' f x 3 60\u00b0 \\\"\\\" f. ,'X .'\u00b7q,\u00b0..: angle = . 60 Transport Frequency Angle I ~ 2\u00b7 i_t_e-qeam cones Car 20 20 x 6\u00b0 = 120\u00b0 Bus 15 15 x 6\u00b0 = 90\u00b0 If differe nt symbols are Train 10 10 x 6\u00b0 = 60\u00b0 Ca rpool 2 2 x 6\u00b0 12\u00b0 used, they' sh0\\\\ 11'd't:ie the\u00b7 Walk 8 8 x 6\u00b0 = 48\u00b0 Cycle 5 5 x 6\u00b0 = 30\u00b0 .:same width as e ach' 2,f 60 2: angles = 360\u00b0 other and _al ignedof)tbeC = ?tc~~o,es tlia rt,. 'ltis also helpfuJ if - - ~:'.~qtdogs The~~um* of the a ng les must a lwp,ys\u00b7\u00b7be ~ 60\u00b0. It each symbol represents Tfle;e'pictures are tfte. is SOrTtefiine5c neees\u00b7sa\/ y tG .rOUl'i'd * the ,angJes to .somesii;'e mid represen.t tfie same r:iuniber of the same number of items. the nearest deg ree. 'If sb:, for _eve ryq p,gl'eyg\\\\i roun9 '.l1Jfi :You.wflf ,need. t9 round another :hrie items. This avo ids giving\u00b7 ciowr). u~~ a_ptotrac.wr*t~ m(3asure t he cing.ies, aJ<ilisleadln_g i'mp rnssion f!tJbe C:e~te; ofyour pie. chau: Gf.the results. >-\\\"-l_n_te_x_ne_t~l~in_k_s_F_o_r_lni_: k~s._t_o-__u_\u00b7se.f__u1_.w_e_,b_si.t_e.s__\u2022o_n_r_ep~r_.e_se~n-t_iR_g_d_a_t_a.,_!J_o_t_o_w_w._:~V~l-i\u00b7:~_us_b_o_rr~re~\u00b7q~\\\"u_i~ck~.li_nk\\\"'~\\\"-\u00b7c~9~r;i__ ...,.\u00b7...... .il:t~.?I.","- - - - \\\\_ HANDLING DATA ) Bar chart Component bar chart, composite bar chart, A chart that uses vertical* or horizontal* bars sectional bar chart or stacked bar chart of equal width to show the frequency* of a A bar chart that divides each bar into sections distribution* . The title tells you what the bar to illustrate more than one set of data. chart shows, and labels on the axes* explain what they represent and, where appropriate, For example, the component bar chart below give the units that are used. A bar chart showing uses the same data as the compound bar chart discrete data* has gaps between the bars, but opposite, with the bar representing each gender one showing continuous data* has no gaps. being subdivided to show their preferred drink. Favorite drinks of 100 people Favorite drinks of 100 people Compound bar chart or multiple bar chart The component bar chart below shows the A bar chart that uses multiple bars within a same information but this time with the bar category to illustrate more than one set representing each drink being subdivided to of data. show the split between genders. Favorite drinks of 100 people Favorite drinks of 100 people \u2022 Area 55; Axes 31 (Cartesian coordinate system); Class boundary, Class interval, Class width 99; Continuous data, Discrete data, Distribution, Frequency 96; Grouped frequency distribution (table) 99; Horizontal 30; Polygon 34; Proportional 25; Sum.J 4 (Addition); Vertical 30.","----- ----------,c HANDLING DATA Histogram To calculate frequency from a histogram A bar chart in which the area* of each bar is Use the rule below, which is the rearranged rule proportional* to the frequency* of a grouped for calculating frequency density: frequency distribution*. The bars of a histogram are drawn at the class boundaries*. The height frequency = class width x frequency density of each bar is called the frequency density. This is the same as finding the area* of each of To plot a histogram from a grouped frequency the bars. So, to find the total frequency of distribution, first find the class width* for each people who finished the crossword: class interval*, then calculate the frequency density, using the rule: frequency = (4 x 0.25) (2 x 2.5) (2 x 4.5) + . frequency (8 x 0.75) + (4 x 1) frequency density = c1ass w1\\\"dth = 1 5+9+6 ~ 4 = 25 For example, the grouped frequency distribution The total frequency should be the sum* of the table* below shows the time taken by 25 people frequencies in the table. to finish a newspaper crossword. Time is continuous data* so it is measured to the nearest Frequency polygon minute. The 1-5 class interval therefore extends A chart on which the frequency* (or frequency from 0 .5- 5.5 minutes, so its class width is 5. The density) is plotted against the mid-interval other class widths are calculated in the same way. values* of class intervals*. The points are joined by a series of straight lines and extended to the Ti m@ Frequency Class Freq uency horizontal* axis* to form a polygon*. (mlnu t-es) width density A frequency polygon can also be drawn on a bar chart or histogram, by joining the midpoints of 1- 4 5 4 1 + 4 = 0.2:J the tops of the bars. The area* under the 9 2. 5+ 2 = 2.5 polygon is equal to the area under the histogram. 4-6 6 2. 9 -;- 2 = 4. 5 \/- 8 4 8 6 .;. 8 = 0.75 For example, the frequency polygon below is 9-1 6 4 4-;- 4 = 1 drawn from the histogram illustrating the time 17- 2-0 taken to finish a crossword (see left) . Draw the histogram, plotting the frequency density against the class intervals. Label the axes* and give the histogram a title. Time taken to finish crossword Time taken to finish crossword Internet links For link:s to. use.ful websites on representing data , g.o to www.ustJorne-quicklinks.com","\u2022- - --<( HANDLING DATA Stem-and-leaf display or stem-and-leaf diagram Stem-and-leaf display with large sets of data A method of representing data* by splitting the To display a larger distribution as a stem-and-leaf numbers of the distribution* into two parts. It is diagram, the stem can be subdivided into upper most often used to show the range* and spread* and lower parts to make it easier to read. For of small amounts of quantitative data*. example, the stem-and-leaf display below includes a large amount of data* with a small range* so it For example, to arrange the distribution below looks very crowded: in a stem-and-leaf display, write the tens in the \\\"stem\\\" column, in ascending order, and then Stem Leaves write the units in the \\\"leaf\\\" column. 0 1113345~~-=ti 13. TO 14 'T2 14 9 23 T3 H 21 1 11134~i~ Stem Leaves 1 0 2 1~ 0q Key: 216 stands for 26 1 0133344 By using a - sign to represent the lower part of '- 1 3 the stem (0-4) and a + sign to represent the upper Key: 2j3 stands for 23 part (5- 9), the diagram becomes easier to read. The leaves are usually arranged in ascending order too, especially if 1he display is to be used to find Stem Leaves more information, such as the mode*, median* or range. Turned on its side, the pattern of the leaves o- 111334 is like a bar chart*, but it has the advantage of showing individual values within the distribution. o+ 5~~:tg 1- 11134 1-t- ~i ~ 1- 021 1+ ~ Key: 2 +16 stands for 26 The leaves column can only include one digit Back to back stem-and-leaf display from each number, but the stem can contain any A stem-and-leaf display that shows two sets of number of digits. ,For example, the stem-and-leaf data . To construct a back to back stem-and-leaf display below represents the distribution: display, first choose appropriate units to build the stem. Then form the leaves by writing the 2 0 ~ }j(J ,2.~3 239 240 _240 248 remaining digits of one data set to the left of the stem, and those of the other data set to the right. Stem Leaves For exa mple, to represent the two distributions from the table below in a stem-and-leaf display: 2.0 5 21 ~ A 19 20 23 23 27 30 30 12 3g B 8 17 21 27 31 31 40 23 2.4 0 0 12 1. Bu ild the stem of the diagram, incl uding the tens digits from both sets of data. Key: 2418 stands for 248 2. Form the leaves by writing the units of each To find the median value of the distribution, count set on either side of the stem: the leaves from either end of the diagram until you reach the median position. For example, the display Data A Data B above has 7 leaves so 4 is the median position ~ 0~ (7 + 1 .;- 2). The 4th value is 9, so 239 is the median. :t 3 3 0 1 :t To find the mode, look for the number that occurs 00 most freq uently. In this case, the mode is 240. 1 1t Key: 0 The range is 248 - 205, which is 43. 3 11 40 I3 1 stands for 30 and 31 \u2022 Bar chart 106; Cumulative frequency 99; Data, Distribution 96; Graph (algebraic) 80; Grouped frequency distribution table 99; Interquartile range, Lower quartile 102; Median, Mode 100; Quantitative data 96; Random sample 98; Range, Spread 102; Upper class boundary 99 (Class boundary); Upper quartile 102; Vertical 30 .","---~--t~DLING DATA Cumulative frequency diagram Using cumulative frequency diagrams A graph* on which the cumulative frequencies* Cumulative frequency diagrams can be used of a distribution* are plotted and the points are to find out further information about the data*. joined. On a cumulative frequency curve, the For example, to find the number of people who points are joined by a smooth curve, as shown waited 10 minutes or less, find the point at 10 in the example below. This type of diagram is minutes on the x-axis, and draw a vertical* line also sometimes called an ogive, although the up to the graph*. Look on the y-axis to find the term is becoming less common. If the points are joined by a straight line, the diagram is a cumulative frequency* at this point. Where x = 10, cumulative frequency polygon. y = 39 (approximately), so 39 people waited 10 minutes or less. 16 people waited more than 10 minutes (total number of people (55) ~ 39). For \u00b7example, the grouped frequency distribution To find the median* waiting time, first find the table* below shows the length of time (to the median position. If the total cumulative frequency nearest minute) that a random sample* of \u00a5nis greater than or equal to 100, use a.~: If, as in customers had to wait in a line. this case, it is less then 100, use 1}: Time IFr@qu e ncy Cu rru.1,lative .l.(55 + 1). = 1 x 56 = 28 -(m-1ln-utes) frequency 2. 2 Where y = 28, x - 7. 75 (approximately), so the 0-2 5 5 median waiting time is about 7.75 minutes. 3- 5 6- 8 8 5 8 = 13 To find the interquartile range*, subtract the 9-11 lower quartile* from the upper quartile*. 20 u I\u00b7 20 = 33 Up~e_r quartile = 3(n + 1) _ 3(55 + 1) = 42 11 33 + 11 44 position 4 4 12-14 6 44 + 6 - 50 I ~f S-1 \/ I 50 + 4 54 Where y = 42, x = 11 (app roximately), so the s,1 + 1 55 18-20 upper quartile is 11. The cumulative frequency diagram is drawn Lower quartile = .Q....::L.l = 55 + 1 14 by plotting the cumulative frequencies position 44 against .the upper class boundaries*. The line or curve on both types of cumulative Where y = 14, x 5. 75 (approximately), so the frequency diagram starts at zero on the lower quartile is 5.75 . The interquartile _range is cumulative frequency axis. 5.25 minutes (11 - 5.75). Time spent waiting in a line Time spent waiting in a line , ..-!- - + ,+ -+, + tl Internet links For links to useful websites on representing data, go to w w w.usl)orne-quick\/inks.com","_ _ __,Ci!._ANDLING DAT A ) - -- - - - - - - - -- - - -- - - - - - - - - - -- - = - -- -------'! five number summary l,ir1e graph The lowe~l valt1P, lower quartile~, median *, upper (1uc:1rttle* ;;.od highest value of a A graph on which frequencies\\\" of a distrib ution *. These w1IL~ Ps E)nable you t o assess the range* and interquartile range\\\"' distribution* are plotted, and the poin b are of the data* a l'\\\\d :.ee how symmetrically\\\"' joined by ea s eries ofstraight Jines. The Li l le telfs spread 1t is around the me;:;Jian_ yoli what the line graph sl'Jows; The tal:Jels on the axes* explain what they repreS'er:it and, where appropriate, give the..units that are u~ed . Average maximum temperature in Hambun:i. Germany ,, I I I Box plot or box-and~whisker diag\u00b7ram ,-_I+~' I II + l.~J_-c_ A diagram that shows the five number summary of a distribut ion\u00b7\u2022\u00b7 Box plo s ran be a l ,I.. 'I. u eful way of compari ng the ~pr ead \\\" of two or ~n I ore distributions on the same number line... I ~\/ ' I ..\/ \\\\' -, -I-+'' I -\\\\ 1-!4 L__c__ ~. I\/ I II .... 'u I~ 'I I '~ I I Ii Each diagram contains a rectangular bo>1, Its H-- I \\\\ '--- length r\u20aclprescnts the in terc uartile rnnge', but ,.\\\".r:,: il can b~ <i ny height as this i nu l ~ignificant. II \\\"- ( A vertica l\\\"' Ii e divides the box at the median~_ A each end of the bO !<.. ljn~ called whiskers II extend hor.izontally* to the lowl;l~ and 1ghest values to show the range* o l t 1e d4stribut 1on. I II I \\\" -I I I ~- .__ +-r-+.: .-.- I I I ~ ~ -l- -'-- '__L__,.._ -l- 'I ,-,I \\\\. F II r ...Y' I ~~ l-1- I < \\\"' I - -+-A ...\u2022 - \u2022 .n \u2022 II IJ For example, the box plot below illustrates the . --!+I I I following distributions: ' Uimrn dl>'lrev1a II -r-I' I Sample A 3 10 10 12 12 13 t 5 \/'O Scatter gr\u2022ph or scatter diagram Sample 8 , A graph oo whic ~ pornts are plotted -to ~how 68 9 11 1.2 l4 16 the r~l a lion-;h1p betw<?-~n ~~to sets of qud l'llil tive ~~ data* The points aro not Joined, and you caJ'l have ~ \\\"1~ I ,-j-~- -several points with ttm same x. or y va lue. The title I 'I --- I I and .1xi>~ labels ell yo what he graph shovlf~. I II II I ---~- I I I _! 'I , ' \u2022 Marks udtl~Yed by pupil~ in French and Germ1m test )tl n'\u2022 -- --,-, ' I I Ii u.J-rh'f .LI! \\\" I II l-1 I II 1I I I I _I ...- -- A ror!Qrio!'~ on n tiox plo! --I I I I ____\\\",\\\"_' II \\\" inC.11.ld ' pOtrllS (J I ;pll'5P.f1t -i-~ r::~ I ~n 1- I ' rodi p~l! v 1ckiro. so ~nl I II I- I --- I I II - losing torty or !f1t' 0~>1011'. I II --- II I I. I I' ' I I1 ome ti me~ the d~ta* incJudes a value Lhi:ll i I -, i :.,_ 1, m uch hig er or Jower than all the re l, perhaps I -~- I ;~ because of an mror m measur\\\"og, The-;e values, l' . 1-R- C\\\"all@d outli@rs, are repre~en t~-ci by a individual I> '--t-1-.-:.. '' I point or asterisk (~') beyond the w hisk r_ I I I' I I Z'ig-zag 8 ,_:..__ ~ \\\"\\\" I.> A ripple in an axis.* indici.l ing that the scale does r ot apply ru .,'I \\\\. ,, 61J to that sect~o o l l e axi~ I 4 ,., J -le$ mr '\\\"S l tI ''-- I! I; l\u2022 ' A11;es > (C111rteol111n 'tool'dlnate sy\u00b7n m}: O;at11, Distribution 116; frequency 9~; \\\"oriZ0<1tal !O; lr.terqu111rtile range 102, Lowl!r qu\u2022rtlle ' 02, Me;m 01 \u00b7 Median 100; Number llne 7 (Directed numbers); Quantil.ttlve data 9 , Range, Spr.ud H.12; SymmetTic:;aH 2 {lh11oduct1onl; Opper quilrtile 1fl7.: Verlltal ill","t--------~------------------~----------.~Cr1A.','DUNG DATA Corre'latlon L\u00b7ne of best fit or regression line A r~l.,11 ion hip between two sets of v;:i l ~Je.s. A A lined wn on ~ scatter 9.raph to snow the corrnl,ation be \\\"Neen the two sets of \\\\1ill'u :;. scatter graph cc1ir'I show whether th@re IS aoy correlation beh\u2022lfeen the sets of d;:it;a it represen ~. Often, tne line CM be d rawn D\\\\' eye. Howev c, to dr;;iw a 1 ore ar.:curate line. tmd the @an* <'If An upward t relld in the position of pc;>1ots on & each distribu io'\\\"'\\\"\\\" and draw a Ii c at rig ht ang,es (SOe} to the- oixi s* at th is po int. Then draw th~ srnt t i;r gra ph i~ c~lled p ositlve oorrelat ion . A, linl! o\u00b7f b st fit through the pm t where- th two oownwarJ trend is called nl29ative correlation. lines from the means meet. II 1-h II For e~rnplE!, the t.=i\u00b7ble below shvws the nurn b.P.r I II of swimming pool lerVJlhs swum in various tirnQ~- \u00b71 II Time (mins) 5 8 10 13 15 18, 20 23 I\\\"11) scotier- groph Jho1W'. 'I Lengt hs i 11 1~ 23 27 31 40 39 50 o po:sito've C01reiatidtl. l T Il I rI II Tlli~ 14'Dttef91cplt J\/l(}m i:i l'>i,>got\/'.\u00b7\\\"\\\" ((j\/r1tkiuo11 A sct;tt~r grapn 1mth points bat tie on, or dose To show tha COr!\\\"elation between t hes va l u~, lo, a st rnight Im@ has t1 strong <!orrelatlon. plot them O!'I a scatter $1raph . t co draw a line ,o\\\\.$1 r0 ng cor~lotior\u2022 I of best fit t hrough the poin that represen-~ tf e lr.'d\\\\c(I!~ that the I mf>iln of bot\\\" ~ets of data (show n on th~ gmph by t\u2022.<(G seil. oi wiltoeJ! I the $ymbo l @). crre .:lr;sody ref(ited Thl'l ma;in length' oi time is : 5 -d3110+13 15-i1 8 t 20+ 23 _~ 112 =14 tcr e.00; ii>?r. s8 . The mean number of length~ is: II t -- \\\"\\\" 11 15 -'- 2~ + 27 + 3 l -'- 40 39 .,_ so I B -\u00b7\u00b7 --~- rnA scatter graph'l.vith points lying reu9 hly i3 2-836 - 29.5\\\" straig ht line has a rnoder~te correfotloo, So, t he li n~ of best fit sh ould' 1;>\u20ac olra1...\u2022n t hrough \u00b7 Am~te II the point (1 4, 29. 5~.. In tti1s c~sC?-, thP- line st.lrl:. at 'orrelo \u20221 \u2022 I ((),0) (as in no tim e, oo len.gths would be wvm) . 111r1k11tes:.uiw t\/1(! II Number of le.ngfll:i swu1:11 fl<'ll ~,>\/ 5 ffl Lj(AfU!'l 1-1-'-+-1\\\"-~w\u2022\u00b7+-1-r,1-+-+-r-'-r'-+-1-+--'\\\" --+-+-+-+~t-1-+ I ., ., euf\\\"f: \u00b7 1<~(1.ttd 7' - IV ~'Q<.1> ottlir I I I \/ I I I !\/ -~ I\/ I \/ I . . .fT\\\" \u2022 - -~- - :> II IA I\u2022 A sc~~er graph on w,hic,ti t hf' p<i in1s seem unrelated to a ~tia,ight trn~ is s.a1\u20ac.l 10 have no correratlon. Na cvrR!lalio~ \u2022 l il itrdk-0tc:s tiicrt ill~ ni'(() ~s-o.'vo ~ i I\u00b7 \/ I I ort riO't refu ea rt'1 ,,In ~ _o ---~ It I Ufl)I' linear WU}: II I\/ A r q(tlrow;h f)l.1\\\"ler I 'I I I mrr~ianstiips II l\/ mt'91lt w~i.11 II I ,,..,I\/ between lfl .1 II ''\\\"' '~ i t< ![i!il} II I","HANDLING DATA PROBABILITY Probability is a branch of statistics* that allows you to calculate how likely something is to happen, and give this likelihood a numerical value. For example, if you toss a coin, there are two possible results: either it will land showing the head or the tail. The likelihood of it showing the head is 1 of the 2 results. This probability can be expressed as a fraction* (t), a decimal* (0.5), or a percentage* (50%). Event Theoretical probability Something that happens, for example, the The probability of an outcome occurring in tossing of a coin or the throwing of a pair of dice. theory. Theoretical probability is based on equally likely outcomes: that is, no bias or error is involved. Outcome Tn~- result .of an event, for example, a.tossec:I coin The rule for calculating theoretical probability ,is: landing heads up or a thrown die showing a six. =P(success) total successful outcomes Success \\\"The [equired \u00b7result for example, if youv\\\\fant a. total possible outcomes (oin to lciryd heads up and it does, this would be where P stands for \\\"the probability Of.\\\" ac successful outcome. For example, if a bag contains 6 red balls and Lj. Equiprobable _events Events with equally likely outcomes. For blue balls, the theoretical probability of choosing example:,1tyqu.J.oss a i(>ip;;there is an equal a blue ball at random is: of.c:hance' the outcome being: a head or a tail. tP(choosing a blue ball) = \u00b7i% s Probability scale The theoretical probability of-choosing a blue AsC:'*~ measuring the likelihood Qf an outco111e. The probability of an outcome that ball at random is i . This could also he written as DA. .. 5 .. . . .\u00b7. awill certainly happen Is 1. For ex~ rhple, the. a decimal, a or as percentage, 40%. probability that you will J:ie little aide~ by t he Experimental probability or relative time you have read this sentence is 1.An frequency The nuiJiber of times an outcome occurs fn ah outco'me with a probability of i is described as experiment. a certainty. \\\"'ffi~ probabi1ity of ;~rt outcome that The rule for calculating experimental rdr\u2022.will 'Certainly not happetl' ls o, example, the probability 15: P(success) = -total s.1.1ccessful o utcomes .probability .th:at ybli will t4rrt;i'&t\\\\5)\u00b7Melephant \u00b7is total eve nts o.'An'outcome with a p\\\"robability of (J is where ? stands for \\\"t he prowbility ot'fr d~icribed as an impossibility. \u00b7 ,. 0 f6.r example, if a die f~ thrown .100 times <lt\\\"\\\\d it rmpos5ibilily L_ertdint;y land_s on:'6 a .total of 1'2,times,\u2022the experimental probability, or r~lative frequehci oh hrowing rn~\u00b7 The values 0 <:1 nd 1J :r.re J fre extremes of d ie :aAd' getting a 6 is: probability, ;~H<f the prol:iaJJility ofan outcome: P(throwing a 'slXI = -11.. \\\"' 215.. \\\". \u00b7occurring cari b e anywhere between or including ' 100 :fofand 1. The close! aprobability is fo o ~n the \u00b7 T.he.ex.perimen:tal pro:babifity ofthrowit19 the die scale, the less likely .the outcome is~ Jhe-doser:-a and,getting a 6 is 3 This c(),!lt~ ai~R: b:;'.llvritten as probability is to 1,\u00b0the fl;l?f.~\u00b7 likely>the outcome. 25 a decimal, .0, 1 ~[,='. or as a p\\\"ercenfage, 1~%. 1112:1 \\\"D~lmal 19; fr~o;ticm 17;.Percentage 27: Statistics 9(\\\\ (D~.ta).","Types of events Mutually exclusive events Single event Two':or more events that cannot both have a successful outcome :at'thf!;'sametim~~ For An event that involves only one \u00b7item, for e)(ampl~tossing one :C:(lifk example, if A is the event ''choosing a red card from a deck of playing cards\\\" ancf 8 is the event \\\"choosing a' spade from a Compound event or multiple event deck of playing cards,\\\" \u2022\u2022\u20221\u2022\u20220\u2022\u2022\u2022\u2022\u2022\u20220\u2022\/ An event that involves more than on\u00b7e item, events A and B are mutuaUy exclusive. for exa mpfe, tossing two c;oins. ot t ossing a \u2022\u2022 If you pick one playing coin and a diec. \u00b7 \u00b7\u00b7 card from a deck it. cannot be a red card Independent event and also d\\\"sgade, so I An eventthat has an outcome which is not these are mutually affected by ~nyothe(e\\\\.tent. An independent .exc\/usiw events~ \u2022 ~ven t is also called a random event. .!t. Fm example, when a die 1s thrown ! vvice, the chance of throwing a particular numberoti t he The total probability of .:1 complete set o f s~co,nd occas io n is not aff@cted by$tl\u20ac, first c.-\u00b7\u2022 @Ven~. The prob~ bility of gett~ng a \u00b76-is the same mutually exclusi~e events J Iways adds up to 1. no matte r how many t im@$ the die is t hrown. f.or example, one set of mutually exdusrve eve11ts in choosing a card f rom a deck '<'lr~: Dependent e\u00b7vent \u2022 choosing .;:i red carcH;~:r An event that has ar1 o\u00b7utcome which is affoctecl by a nother event. \u2022 ch oo,sing .a spad e(~~y For ex.ample, if a marble is taken at random \u2022 choosing a club(~~) from a bag of blue and g reen marbles. and is :NoneoHttese can6i\u00a5i.Yr at the same Urne. If not put back into the ba g, the <:olo r of th12 :th~~ pro babtrities ~r~ ~\u00b7ddeci' together: second marble to be p icked will be dependent on the fi rst event. ~+.li+n=g=1 52 $] S2 51 If the m are 3 blu e marbte$ and 3 g reen ma rble$: The result is the same tf a'nother complete ,set~f i - 1p (choos.ing a blue marble) = mutually exclus:ive events ~re addedtog.ether, such as choosing a b lack caret, choosing a he9.rt If a blue marble is ta k.e n out an d not replaced, and choosing a d ia mond. t he probability of p icking ano ther blue m<irble is now l. {as there are only 2 blue marble5 aind The )robability of some-thing riot happe ning is 5 rnarbres altog@ther). 1 t ake away the probability t hat it will happen. The. prqbdbiflty uf rollingca etearid, ,getting \u00b5 iix,'is,\u00a5 \/he PfObabiii~~ The prob.'.lb\/lily Of of p1clrJng one rolling cr d,~ (\/()(j gettmg a ,1u111ber that -0f lh~le Mre i\u00b7il fo'Qr o six (tlror fs, I, m<>1\/rles Qt random irG'm 2, 3, -1, Or 5) ~ '~ \/xig IJ 2. This i~ equal ff) t - ~ ! When t he probo:ibility of a n outcome depends on the probabilitY of a previou> outcome, it is called conditiona l probabiJlity. The condi'tiooal lpm~ability t hat t he second marble is blue is","r.- -- -'( H ANDLING DATA );--- - - -- - - -- - - - -- - -- - - - -- - -- - - -- - -- Combini'ng probabilities The multiplication rule or and rule The rule that is used to find the probability* of The addition rule or or rule a combination of outcomes* occurring . The The rule that is used to find the probability* of multiplication rule states that: one of any number of outcomes* occurring. The addition rule states that: P(A and B) = P(A) x P(B) where P stands for \\\"the probability of,\\\" and A P(A or B) = P(A) + P(B) and B are outcomes. where P stands for \\\"the probability .of,\\\" and A The multiplication rule can be used to find the \/ and B are outcomes. probability of a combination of independent* or dependent events*. The addition rule can be applied to any number of events, as long as they are mutually For example, to find the probability of throwing exclusive*. For example: a 4 with a die and choosing a king from a deck of playing cards, use: P(X or Y or Z) = P(X) + P(Y) + P(Z) P(4 and K) = P(4) x P(K) For' example, to find the probability of choosing where P stands for \\\"the probability of,\\\" and K a red card or a spade or the King of Clubs from stands for \\\"king.\\\" a deck of playing cards: \u00b7 P(R or S or KC) = P(R) + P(S) + P(KC) where P stands for \\\"the probability of,\\\" R stands There are 6 numbers on a die and 4 kings in a for \\\"red,\\\" S stands for \\\"spade\\\" and KC stands deck of 52 cards, so: for \\\"King of Clubs.\\\" =iP(4) There are 52 cards in a deck. 26 of these are red P(K) = ~ =.J-x_.1_ = ~ = ....L cards, 13 are spades and there is only one King 6 52 312 78 of Clubs so: 52 P(R) = 26 P(4 and K) 52 The probability of throwing a 4 and choosing a king is ....L. P(S) = .li 78 52 To use the multi plication rule to find the P(KC) = ....L probability of a combination of dependent events, first calculate any changes in 52' probabi_lity following each outcome, then multiply the results. P(R or S) = 26 + .IT+ ....L = 40 = 1Q 52 52 52 52 13 For example, to find the probability of choosing a king from a deck of cards and then choosing The probability of choosing a red card, a another king, having not replaced the first, use: spade, or the King of Clubs is J.Q, P(K and K) = P(first K) x P(second K) 13 where P stands for \\\"the probability of,'.' and K The addition rule can be used to calculate stands for \\\"king.\\\" the probability of choosing either a red card, or a spade, or the King of Clubs. There are 4 kings in a deck of 52 cards, so: 3,r ....---- - \u00b7-\u00b7\u00b7-,.,-- -r P(first K) = _.1_ \\\\2 \u2022 52 \\\\\u2022 P(second K) = -1.. \\\\ 51 \\\\ P(K and K) = _.1_ X .l.: = -1.L = _L \\\\ 52 51 2,652 221 I The probability of choosing a king from a deck I of cards and then choosing another king, having not replaced tlie first, is d,., \\\\ 221 \\\\ \\\\ __----=- \\\" \u2022Compound event, Dependent events 11 3; Event 112; Independent events, Mutually exclusive events 113; Outcome, Probability 112; Ratio 24; Single event 113.",", HAHDLJt ~ :--- Possible outcomes Probability tree diagram A diagram on w hich the possible outcomes* The possil:ile outcomes* of a n experiment of events* are written on the \\\"branches.\\\" Probabi lity trees are particularly useful for depend on the number qf events* that take dependent event~ ~ . , pl~ce \u00b7an<:twhether the e~~nts ate dependent* For example, if a ba ll is cho en at random from \u2022fl\u00b7 bag of 4 red ilnd 5 green balls, th@ pos>ible ,,9r independent*.. outcom es can be hown as: The' possible outcomes of independent events 9;:this tree diagram y~ R~ can be recorded in a\u00b7list. :.~hows _the probability \\\"'For example, if a coiri is tossed, the possible for a single event\\\", outcomes can be list~d as: H; t . $ where H is heads and' 'r.Js tails. 9 If two coins ar.e tossed\u00b7, the number of possible outcomes incre.ases: If the first ball is riot rep~aced, and hen a secondb~il'. is chosen, t he probabitities are HH; HT; TH; TT dependent-Ontne res1ults of the first outcome: And if three coins are toss.ed; the nuJnber of possible outi:omes increases again: This ire(diagrain shows the HHH; HHT; \u00b7 HTH; \u00b7tffT; ,'THH; THJ;: TTH; TIT probabilityfot a compound event~ Possibility space or probability space i.s (,reen A-table th<!t shows the possible outcomes* of a pair of independent events*. y11 \u00b7~ F.or ~xample, the possibility space \u00b7below shows .i p Crl't'rl -,. the possible outcomes :Ofcombined sG'.or.es 'from throwing two dice. . \u00b7\u00b7 _. . \u00b7 .\u00b7. . . Green First die nm&a11 Seeond bail 1 2., 3 4 5 \\\"1 l 3 + s If a 'red .ball' ls chosen fir$l, and nol rep~aced, t he \\\"2., 3 4 5 '7 1- g =r probability'.of choosing a second red ball is now (J)sQj on.ly 18. (as only. 3'red b<11lls. .=.re left out o1 a total of 8 bal,ls). If a green ball i chosen first. the ~ 3 '1- G:i 1- g probability of choosing '1 sec;o nd g reen ball is @4\\\".c0C., 5 now only i (as onfy 4 gr~n bal ls am left out of =r g \\\" 0)5 t gbQj 10 a total of 8' balls). II\\\\ 10 11 @)b =r g 10 11 12. Odds The probability* of an event* occurring o'find the probability of a certa in number expressed as a ratio*\u00b7 of it occurring to it being the total score from ~the two' dic;e, :c;o.tint not occurring . For example, the odds of throwing a six with a die are 1:5 (there is t he number of times tha t core appears i~ the f:one chance of success to five of failure) . table. f or example, the number 9 a ppears four The probability of throwing a six is times out of 36 possibilitie>, so the .pf<iqability of t he tota l core being 9 is~. Which i l, 36 '3 lraternet li nks For links to u~~ful websites on p robability, go to w1.ri1w.usborn.e-qvicklinh.com","A-Z OF MONEY TERMS Here is an alphaheticar Jist of some Currency money-related terms you m_ay meet. The money in current use by a country. For example, the yen is the currency of Japan. Annual' perc;entage rate (APR) The total c~st of any form o1 borrowing, Debit including mortgages, expressed as a percentage* Debit has many meanings. In a debit transaction, of th_e amount borrowed . APR includes any money is taken from an account- If you are in actministration fees as. well as the interest rate*. debit, yo.u owe money. A debit can also mean a sum of money taken froli:i your account. Balance The amou nt of money in an account when all Debit balance mcomings and outgoings have been considered. The amount of money that you owe, for example to a bank or credit card company. Base rate or base charges A fi>H1d fae charged by a utility company for Debit card providing a service to a property. For example, an A card issued by a bank or other financial electricity bill can include a daily base rate plus a institution that can be used to make purchases cost for each unit of electricity used. using the money in your bank account. Commission Deductible A-fee thatj~ earned for giving a service, such as A fixed amount that a person must pay toward selling-~ c~r on behalf 9f so-meone else. Commission an insurance claim. For example,' if you have an is often a percentage* qfthe value of the item sold. insurance policy with $100 deductible, you would have to pay the first $100 of any claim, Credit and the insurance company would pay the rest. Credit has many-meanings. In a credit transaction, money is received \u00b7into an account. If your acc\u00b7ount Discount has a credit, it rne~l:!s that you have money. ff you A deduction from -the price of an item. A 10% have credit, you have the ability to borrow money. discount means that the price i~ reduced by 10%. Credit ha.lance Earnings The amount of money in your bank account. Sala,.Y or wages. Gross earnings. is the amount Credit card earned\u00b7 before any deductions(:such as i_ncome , A ea rd that is used to make purchases bu( pay f9r tax, and pension contributions) are made: Net them later. Payments are.usually made monthly earnings, or take home pay, is the amount left and inte.rest* may be-charged on the amount of after deductions h.ave been made. money borrowed, Ach~rge card or \u00b7stor~ card is Exchange rate The rate at Which a unit of one currency'.l:a'.i:J b@ \u00b7a form of credit card issued by a store. exchanged for another curre ncy. \u00b7c;redit .Jimit Income tax The ma~imum amo4nt that qm be borrowed Tax related to a person's earnings. Everyone can on a credit card. earn a certain amount without paying tax on it (personal tax allowance) and may be able to Credit rating _claim uther allowances. lricome tax is payable on all earnings 9bove th~se combined allowa'rices. on\u00b7 A _points sy.stem, based mai.nly income and Tax is c;ha rged :as\\\"a\\\"percentage* of taxable income. the more you ear{!; the more tax you pay. credit history, Whlc::l1 is\u00b7 used by banks and other f\\\\n~neiaJ lnstitutionstb dedd'e 1-\\\\ciw muth mciney\u00b7a customer caii borrow. 11161 \u2022 J~t~rest lnte~estAver:.ge:\u00b71'Q.O; 28; rat\\\" 28; Percen!age-2?.","NCJNJBER 1.ncome'tax withholding program Satary The money to b(i!ecirhed o'w'.er;ay~~ Th i~ is A program by which income tax is deducted u su ally divided into 1~ equal part~: paid monthly. from a person's-gross earnings, before any Sales and services taxes ismoney p.aid tothe pemin. Taxes thatare added to. the cost :Qt.goods a nd lriflcttion services; for example, rest aurant bills: Th~ amount An al;tera.ge\\\"' increase in the cost of goods of tax is a fixed percentage* of he serling price ~}\\\"a~a \u00b7services over time. There are different ways of the goods, and it is decided by the govern ment. of measuring inflation, such as the Consumer Some cou ntries have a com bi ned t ax for sales Price Index (CPI), which measures the changing and services called Value Added Tax (VAT). cost of a range of commonly used goods and services. Savings account An account that pays inte rest* on the money in it . Insurance premium The amount that you pay to an insurance Statement company. Premiums are usually payable yea rly. A report from a bank or finance compa ny t hat A no claims bonus is a reduction in an insurance shows the incomings (amounts received) and premium after several years when no claims have outgoings (amounts paid out) of an a ccou nt. been made. A credit card statement will also show how much payment is due. Investment income Money received from various forms of investments, Stock market such as a savings account or mutual funds. The market for buying and selling stocks an d shares in companies. If a company is perfo rmi ri~? Mortgage w ell, its share price is high. If a compa ny is A large sum of money loaned by a bank or performing poorly, its share price is low. credit union for buying a house or other property. It is repaid, with interest*, over a Tax number of years. Money collected on behalf of t h_e government so they can provide. services, such as schools, Overtime for the country. There are ma ny differe n1:Jaxes. Money paid for working more than the agreed The act of taxing people is called taxation. number of hours. Overtime is often paid at a Direct taxation takes place before any moi;iey is different rate from normal pay. spent. Income tax is a common form of-<lire_ct taxation. Indirect taxation takes plate when_ \u00b7Pension program money is spent. Sales tax is a n example of A,.ri .arrangement to pay a person a regular indirect taxation . sun{ of money after retirement. A pension Unit trust or mutual fund \u00b7 program: is usually funded by payments made A program run by an investment company t hat . inve~ts people's money in .;:i, range of shares. before ret irement bythat person and\/or Utility bill i helr .employer. A:_b itl fOr an essential servke to a property, suich i>ersonal loan iii \u00a7as; electricity and water services. Utility bill s: As'1:1m ofmoney loaned to a person by a bank for .ahy pu:rpose. The Joanis subject to inter~st*. -can-be\u00b7 paid weekly, monthly or qua rterly. p.j~ee rate Wage$ The money earned over a period of time;, usually A fixed .rat e.of pay that il> calculated p~r item paid weekly or monthly after t he end olthe p~riod. -p r;cid.uced cir.processed, such .as ffie 11uf1J ber;of bricks laid by a brk kiayer. Wo rk. tbat:is~pi:iftj in this.;way is c;alled piece-work. \u00b7\u00b7 . .. . Internet links For links to useful websites on managing money, go to\u00b7 www. usbome-quJc_klfnks,eom \u00b7","INDEX MATH SYMBOLS The following list includes symbols commonly used in math that you need to be able to use and recognize. (The letters n and m are used where appropriate to represent any given values.) + Addition sign \u00b1n Positive or negative ::;; Less than or equal to (see page 90) (see page 14) number (see page 11) ~ Greater than or equal to e.g. 2 + 5 = 7 e.g. v16 = ::'::4 (see page 90) Subtraction sign n Recurring number =f. Not equal to (see page 14) e.g. 23 - 4=19 .(see page 19) (see page 90) e.g. 3 x 2 =F 4 x Multiplication sign e.g . 10 + 3 =3.3 = Is approximately equal to (see page 14) n:m Ratio e.g. 6 x 5 = 30 (see page 72) (see pages 24-26) e.g. 100 + 9 = 11 Division sign e.g. 3 : 2 (see page 15) 2: The sum of e.g. 45 + 9 = 5 ex Proportional to (see pages 25-26) (see pages 14 and 101) Equals sign e.g. L(1, 2, 3) = 6 (see page 79) no Degrees n The mean of e.g. 2 + 3 = 6 - 1 (see pages 32-33) e.g. angles in a circle = (see page 101) n2 Squared number 360\u00b0 {n} Set (see pages 12-13) (see pages 8 and 21) 'IT Pi (see page 66) e.g. 42 = 4 x 4 i.e. 3.141 592 654... e.g. set A = {3, 5, 8} and setB={1,2,3} n3 Cubed number L Angle (see page 32-33) E Is a member of the set (see pages 8 and 21) e.g. a right angle is 90\u00b0 (see page 12) e.g. 33 = 3 x 3 x 3 e.g. 3 E {3, 5, 8} I Right angle Vn Square root g Is not a member of the set (see page 32) (see page 11) (see page 12) a Unknown angle alpha e.g. \\\\\/49 = 7 (see page 60) e.g. 4 t\u00a3 {3, 5, 8} Vn Cube root e Unknown angle theta cg Universal set (see page 11) (see page 60) (see page 12) e.g. \\\"& = {{A}{B}{...}} e.g.~ = 5 - Identity sign (see page 75) 0 or Empty set % Percent e.g. 3x = 5x - 2x {} (see page 12) (see pages 27-28) < Less than e.g . .2l = 50% u Union or cup (see page 90) +n Positive number e.g. 1 < 3 (see page 13) (see page 7) > Greater than e.g. {3,5,8} u {1,2,3} e.g. + 2 x +3 = -+:6 (see page 90) = {1 ,2,3, 5,8} n Negative number e.g. 3 > 1 n Intersection or cap (see page 7) e.g. + 3 x - 4 = - 12 (see page 13) e.g. {3,5,8} n {1,2,3} = {3} ~h1!]","_-;;ii~.- - - - - - , - - - - - - - - - , - - - - - : - - - - - - - , - - - - , - - - - - - - , 7 +i - - - - - - - - - - - - - - - { ( INDD(.. )-~-:\\\"=--- INDEX The page numbers listed in the index are of two types. Those printed in bold type (e.g. 92) show where the main definitions of words can be found. Page numbers in lighter type (e.g. 92) refer to supplementary entries. Singulars, abbreviations and symbols are given in parentheses after indexed words. If a page number is followed by a word in parentheses, it means that the indexed word can be found inside the text of the definition indicated. 12\/24-hour clocks 74 and rule 114 arithmetic 14, 15, 16 angle-angle-side (AAS) A meah see mean congruence 38 acceleration 73, 95 angles 32-33, 48, 53, 64, 105 with decimals 20 accounts (financial) 28, 29, with fractions 18 dihedral 40 with vectors 46 116,117 in circles 70-71 arms of angles 32, 48, 49, acres S5 (introduction) naming 70 50, 70 acute angles 32, 35 of depression 53 arrowheads 39 acute-angled triangles 37 of elevation 53 associative laws 14, 15, 46 addition (1) 14, 15, 16 of polygons 34, 35, 37 asymmetry 42 (introduction) of rotation 43 (rotation) average speed 73 (speed) in algebraic of triangles 37, _60-63 averages 73, 99, 100-101 expressions 76 annual percentage rate (APR) 116 axes (sing . axis) \u2022 of algebraic fractions 77 ante meridiem (am) of ellipses 69 of rotation(al) symmetry of decimals 20 74 (12-hour clock) of exponents 22 apex 41, 42,43 of fractions 18 of vectors 46 of a pyramid 41 x- and y- 31 (Cartesian rule 114 of a triangle 37 (angles adjacent coordinate system), angles 33 in a triangle), 56 80, 111 side of a triangle approx.imately equal to (= ) B to() 60, 61 72 (equivalent units) arcs 47, 48, 49, 50, 65, 70 back-to-back stem-and-leaf algebra 75-95 displays 108 rules 76-79 finding lengths of 66 arccos 61 (finding unknown balance 116 algebraic bar charts 99, 106, 107, angles) equations see equations arcsin 61 (finding unknown 108 expressions 75, 76, 77, base rate 117 angles) base ten 6 (number system) 78, 79, 90 arctan 61 (finding unknown base two 6 (number system) fractions .77 bases graphs 80-84 angles) identities(=) 75 area 55-57, 68, 69 number 6 (number system) alpha (a) 60 of polyhedra 41 alternate of bars in a histogram 107 angles 33 of circles 57 , 66, 105 bearings 73 see also three- segment property of ellipses 69 of polygons 55-57 figure bearings 71 bias 97, 98 segments 71 (alternate parallelograms 57 bimodal distributions 100 rectangles 56 binary numbers 6 (number: segment prop\u00b7erty) squares 56 ambiguous case 49, 63 trapezoids 57 , 94 system) amplitude 64 (variations triangles 56, 63, 77 binomial expressions 75, 78 . of sectors 67 bisectors 48, 51 on graphs) under a graph 94-95_ box plots or box-and-whisker diagrams 110","I I boxes 11 0 (box plot) class conditional braces ({}) 12 boundaries 99, 107, 109 inequality 90 intervals 99, 100, 101 , probability 113 (dependent c 104, 107 event) calculating length see class widths cones 56, 59, 68 frequency from a limits 99 (class interval) congruent histogram 107 sizes see class widths interest 29 widths 99, 107 figures 44 probability 112-115 clockwise 32, 43 triangles 38, 70 standard deviation 103, 104 cluster sampling 98 consecutive numbers 6 clusters 98 (cluster sampling) constants 10, 24, 75, 83, 84, canceling fractions coefficients 75, 81, 85, 86, 88 17 (equivalent fractioni;), collecting data 97 85, 93 18, 27 collinear points 30 of proportionality (k) 25 colum!:Y-'ectors (~) 45 (vector construction cap (n) 13 (intersection notation), 46 of compound loci 51 of sets) combining probabilities \u00b7 114 of geometric figures 47-50 commission 116 capacity 59, 72 arcs 47 Cartesian common bisectors 48 factors 11 , 24 circles 47 coordinates 31 fractions 18 perpendicular bisectors coordinate system 31, 80 multiples 11, 79 categorical data see 48 qualitative data commutative laws perpendicular lines 48 census 97 of addition 14, 15, 46 regular polygons 50 centiliters (cl) 72 (metric units) of multiplication 14, 15, straight-sided solids 50 centimeters (cm) 72 (metric 46, 76 triangles 49 units) consumer price index (CPI) 117 center comparing (inflation) of a circle 70, distributions 102 contingency tables 99 quantities 24, 25 continuous data 96, 101, 106, 65 (introduction) ratios 24 107 of enlargement sets 13 convenience sampling 98 convex 44 (enlargement) . compass 53 polygons 35 of rotation(al) symmetry compasses 47, 48, 49, 50, ,. polyhedra 40 51, 54 coordinates (x, y) 30, 42,43 complementary angles 33 , 37 31 (Cartesian coordinates), certainties 112 (probability complements of sets 13 80, 82, 83, 84, 88, 89, 91 completing the square 86, 89 coplanar points 30 _ scale) complex fractions 18 correlation 111 component or composite bar corresponding angles 33 change in position see charts 106 cosine composite graph or cosine curve 64 displacement functions 92 ratio (cos) 60, 61, 93 changes in standard deviation numbers 7 rule 62, 63 compound counterclockwise 32, 43 104 bar charts 106 counting numbers see charge cards 116 (credit card) events 113 natural numbers Chinese triangle 10 interest 28, 29 \u00b7 credit 116 chords 65, 69, 70, 95 loci 51 balance 116 circle measures 73 card 116, 117 computer databases 99 rating 116 \u00b7 equations 84 (circle concave transaction 116 (credit) graphs) 89 polygons 35, 39 - union 28, 117 polyhedra 40 cross sections 41, 58, 69 functions 93 graphs 84 \u00b7 circles 47, 51, 55, 57, 65-71 circular functions 60-64, 93 graphs 64 circumference 34, 55, 65, 66, 67, 68, 70, 71 . formula 66","cube degrees (0 32 (introduction), drawing (cont'd) l numbers 8 ) straight-sided so lids 50 roots 11, 22 to scale 52- 54 33, 53 cubes 40, 58 E cubic deltas 39 earnings 116 curve 83 denominators 9, 17, 18 \u00b7edges 40 (polyhedron), 41 functions 93 elements (E) 12, 13, 92 graphs 83 density 59, 73, 94, 95 elevation 41 units of measurement 58 elimination cubing 8 (cube number) dependent cumulative frequency 99, 100 of like terms with (cumulative frequency tables) events 113, 114, 117 coefficients not equal curves 109 (cumulative or opposite 88 variables 75 frequency diagram) of terms the same or diagrams 100, 102, 109 diagonals opposite 87 polygons 109 (cumulative of polygons 34, 39 ellipses 69 frequency diagram) empty sets ({} or 0) 12 tables 99 of polyhedra 41 enlargement 44, 52 cup 72 (US customary units) equal cup (U) 13 (union of sets) diameter 55, 65, 66, 69, 70 ratios or equivalent ratios curly brackets see braces diamonds see rhombuses 24 currencies 28, \u00b729, 116 curved surfa'ce area 67, 68 difference 14 (subtraction) vectors 45 equals sign (= ) 79 cy~lic between two squares 78 equations 60, 61, polygons 34, 71 digits 6 79 (introduction), 80, 81, quadrilaterals 71 85, 86, 87, 88, 89, 90 cylinders 58, 67 dihedral angles 40 of straight lines 81 quadratic 85 D dimensions 30, 31 simultaneous 87-89 equiangular porygons 35 data (sing. datum) 96-97 direct equidistant points, lines or handling 96-117 segments 48, 65 lists 96, 99 proportion 25, 26 equilateral log 97 (data logging) polygons 35 logging 97 taxation 117 triangles 36, 37, 40 equiprobable events 112 days 74 directed equivalent fractions 17 debit 116 in algebra 77 (algebraic lines 45 (vector notation) fractions) balance 116 ratios 24 card 116 numbers 7, 76 estimating area 55 transaction 116 (debit) Euler's theorem 40 decagons 34 (polygons) direction 53 even numbers 7 decahedra 40 (polyhedra) events 112, 113, 114, 115 deceleration 73 discontinuities 64 (tangent exchange rates 116 decimal experimental probability 112 fractions (decimals) 6, 9, graph) exponential curve 84 19, 20, 27, 74, 102, 112 discounts 116 functions 93 as percentages 27 graphs 84 places (d .p.) 16, 19, 20 discrete data 96, 101, 106 notation 21 point 6, 19, 20, 23, 27, 74 exponents 16, 21, 22 system 19, 72 dispersion see spread in algebraic expressions 75, 76 decrease proportionately in size see also scale 52 displacement 43 (translation), deductible 116 deductions 116 (earnings) \u00b7. 45, 95 . distance 73 distance-time graph 73 distribution 96, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111 distributive property 78 division(+ ) 14, 15, 16, 76 by a decimal 20 in a given ratio 26 in algebraic fractions 77 of fractions 18 of powers 22 dodecagons 34 (polygons), 36 dodecahedra 40 (polyhedra) domains 92, 93 double inequality 90 drawing compound loci 51 geometric figures see also construction 47-49 graphs see plotting graphs","expressing a quantity as a formulas (cont'd) geometry 30-31, 32-44, I percentage of another 27 averages 100, 101 47-50, 51, 52-54, 55-57, density 59 58-59, 60-64, 65-71 l expressions see algebraic gradient 80 with vectors 46 mean 101 expressions median 100 glide reflection 44 exterior or external angles 34, pie charts 105 grams (g) 72 (metric units) probability 112 37, 71 range 102 per cubic centimeter extremes of probability rearranging 60, 61, 62, 63 (g\/cm3) 59 (density) sequences 10 graphs 112 (probability scale) solving quadratic algebraic 80-84 equations 86 area under 94-95 F speed 73 circle 84 standard deviation 103 cubic 83 faces 40 (polyhedron), 41 substitution into 77 distance-time 73 factoring - volume 58-59, 67, 68, 69 exponential 84 cones 59, 68 functions 92, 93 expressions 78 cylinders 67 general terms 80 quadratic equations 85, prisms 58 inequalities 91 pyramids 59 line 110 86, 89 rectangular prisms 58 quadratic 82 factors 11, 78 spheres 59, 69 reciprocal 25, 84 scatter 110, 111 identifying factors of fractional simultaneous quadratic equations 85 exponents 21, 22 feet (sing. foot) (') scale factors 44 equations 88-89 72 (imperial units) (enlargement) sketch a linear graph 81 Fibonacci sequenc;e 1O transformation 93 finding fractions 6, 9, 17-18, 19, 21, trigonometric 64 area under graphs 94-95 24, 27, 112 greater than (> ) 90 \u2022 expressions for vectors 46 original quantities 27 algebraic 77 (inequality notation) percentages of known expressed as or equal to (? ) 90 quantities 27 percentages 27 (inequality notation) unknown angles of frequency (f) 96 greatest common factors (GCF) a triangle 60-61 density 107 (histogram) 11 (common factor), 24 unknown sides of distribution 99 (frequency gross earnings 116 (earnings), a triangle 60-61 table), 100, 101, 105, 117 finite 106, 110 grouped frequency\u00b7 99 polygons 107 decimals 19 tables 99, 105 (grouped freque~cy sets 12 front elevation 41 (elevation) distribution table) first qu\u00b7artile (Q 1) see lower frustrum 41 (cross section) distribution 99 (grouped quartiles full turns see whole turns frequency distribution five-number summary 11 O function form 80 table) 100, 101, 102, flat angles 32 , functions (f) 80, 82, 84, 104, 107 flow c_harts or diagrams 92 92- 93 finding standard deviation fluid ounces (fl. oz) graphs 93 of 104 72 (US customary units) trigonometric or circular tables 99 foot see feet 60-64, 93 grouping data 99 formal interview 97 (interview) G H formulas acceleration 73 gallons (gal) 72 (US customary handling data 96- 115 algebraic 75 units) hemisp_heres 69 area 56-57 hendecagons 34 (polygons) general form 81 he.ptagons 34 (polygons) of circles 57, 66 heptahedra 40 (polyhedron) of ellipses 69 hexagons 34, 36 (polygons) of sectors 67 of t~iangles 56, (using trigonometry) 63, 77","hexahedra 40 (polyhedron) informal interview 97 like terms 75 (term), 77, 78, 88 histograms 107 (interview) homogeneous population line 30, 31 , 32, 33, 34, 48 information from graphs 94-95 98 (random sampling) insurance premiums 117 graphs 110 horizontal 30, 53, 81, 84, 95, integers 6, 12, 19 interest 28, 29, 116, 117 of best fit 111 106, 107, 110 hours 74\u00b7 rate 28 of sight 53 hyperbolas 84 interior angles 34, 35, 37, hypotenuse 45, 60 (Pythagorean of symmetry see also 50, 71 theorem), 61 interquartile range (IQR) 102, reflection symmetry icosagons 34 (polygons) 109, 110 37, 39,42 icosahedra 40 (polyhedron) intersections icosidodecahedra 40 (semi- segments 30, 31 , 48, 51 of lines 48, 49, 51 regular polyhedron) of sets (n) 13 symmetry 42 identifying factors 85 interviews 97 identity(\\\"\\\") 75 invariance property. 44 linear ideographs 105 inverse illustrating cosine, sine and equations 81 , 8~ 89, 91 data 105-111 tangent ratios 61 functions 93 functions 92, 93 functions 92 \u00b7 image operations 14 (addition), 15 graphs see straight line , of f 92 (result) proportion 25 of object after transformation investment income 117 graphs _ irrational numbers 9, 66 43 (introduction) isometric paper 50 scale factor 44 (enlargement) impossibilities 112 (probability isosceles triangles 37 sequences 10 scale) trapezoids 39 improper fractions 18 liters (I) 59, 72 (metric units) inches (\\\") 72 (US customary K loci (sing. locus) units) kilograms (kg) included angles 37 (angles in a 72 (metric units) 51 (introduction) triangle), 49, 63 per cubic meter (kg\/m3) long income tax 116, 117 59 (density) diagonals 41 (diagonal) withholding program kilometers (km) 116 72 (metric units) division 15 . increase proportionately in size per hour (kph, km\/h, kmh - 1) multiplication 14 73 (speed) see also scale 52 lower kites 39 independent bounds .16 events 113, 114, 115 L variables 75 (dependent class boundaries 99 (class variable) labeling polygons 35 lateral faces 41 (prism) boundary) indirect taxation 117 laws inequalities 90-91 class limits 99 (class interva l) of arithmetic 15 graphs of 91 of exponents 22, \u00b775 quartiles (Q 1) 102, 109, 110 solving 90 least common multiples (LCM) lowest solving double 91 11 (common multiples) inequality notation 90 length 72, 96 common denominators infinite less than (<) decimals 19 17 (equivalent fractions), sets 12 90 (inequality notation) infinity (oo) . 12 or equal to( ,;;) 24 inflation 117 90 (inequality notation) common multiples (LCM) 11 (common multiples), 88 possible terms 17 (equivalent fractions), 27 M magnitude 45, 46 major arcs 65 (arc) axes 69 (ellipses) sectors 65 (sector) segments 65 (segment) mapping functions 92 transformation 43 (introduction) maps 53","mass :23, 59, 72, 73, 94, mode 100, 108 nominal data 96 95, 96 moderate correlation nonagons 34 (polygons) nonahedra 40 (polyhedron) mean 100, 101 , 102, 103, 111 (correlation) non-periodic or non-repeating 104, 111 money 28, 29, 116-117 mortgages 11 6, 117 decimals 19 measurement 72-73 motion 73 non-right-angled -triangles measures multimodal distribution 62-63 of central tendency or 100 (bimodal distribution) non-terminating decimals 19 average see averages multinomial expressions noon 74 (12-hour clock) not equal to (\u00a5') 90 of spread or 75 (algebraic expression) notation dispersion 102-103 multiple mapping 92 measuring bar charts 106 of inequalities 90 angles 47 . events 113 ofsets 12,13,92 temperature 7 multiples 11 of vectors 45 multiplication (X ) 14, 15, 16, 21 null median 100, 108, 109,J10 by a decimal 20 angles 32 in algebraic -expressions 76 sets({} or 0) 12 members of sets see elements in algebraic fractions 77 numb_er 6-31 of a vector by a scalar 46 bases 6 (number system) meters (m) 72 (metric units) of fractions 18 lines 7 (directed numbers), per second (m\/s, ms - 1) of powers 22 73 (speed) rule 114 90, 92 per second per second multipliers 28, 29 numerators 9, 17, 18, 62 (m\/s 2, ms - 2) multiplying factor 29 73 (acceleration) (compound interest 0 short method) metric measurement system multi-stage sampling 98 object of transformation 72-73 mutual funds 117 43 (introduction) mutually exclusive events 111id-interval values 99, 101, 13,114 oblique prisms 41 (prism) 104, 107 N oblongs see rectangles midnight 74 (12-hour clock) midpoint N-gons 34 observation 97 naming angles 70 obtuse-angled triangles of a chord 70 natural of a class interval 99 37, 56 of a line segment 48, 54 numbers 6, 12, 78 obtuse angles 32, 35 miles 72 (US customary units) perfect squares octagons 34, 36 (polygons) per hour (mph) octahedra 40 (polyhedron) .78 (perfect square) odd numbers 7 73 (speed) negative odds 115 milligrams (mg) 72 (metric angles 32 ogives see cumulative units) correlation 111 (correlation) milliliters (ml) 59, 72 (metric gradients 80 frequency curves numbers (-) 6, 7 one-dimensional 30, units} parabolas 82 millimeters (mm) 72 (metric rotation 43 (rotation) 31 (dimensions) scale factors 44 (enlargement) opposite side of a triangle units) square roots (-Vn) 11 milliseconds (ms or msec) 74 net earnings 116 (earnings) to fJ 60, 61 minor nets or rule 114 of cylinders 67 order of rotation(al) arcs 65 (arc) of polyhedra 41 axes 69 (ellipses) no-claims bonus symmetry 42 sectors 65 (sector) 117 (insurance premium) segments 65 (segment) orders see exponents minutes (min) 74 ordinal data 96 mirror lines 42, 43, 44, see origin 31 ounces (oz) 72 (US customary also lines of symmetry mixed units) fluid (fl. oz) 72 (US decimals 19 numbers 6, 18 customary units) operations 16 modal group or class 100","outcomes \u00b7112, 113, 114 pictograms or pictographs principal 28 (interest), 29 outliers 11 0 105 prisms 41, 58, 67 overtime 117 probability 10, 112-117 piecharts 105 p piece rates 117 scale 112 piece-work 117 (piece rates) spaces 115 pair of compasses see pilot surveys 97 tree diagrams 115 compasses pints (pt) 72 products 14 (multiplication), place value 6, 9, 16, 19 25, 56, 78, 101 pairs of angles 33 plane 30, 31, 43 proper fractions 18 palindromes 9 properties pandigital numbers 9 figures 30 of angles parabolas 82 of symmetry 42 of angles in circles 7-0-71 parallel 30, 33, 39, 41, 44, sections (of a solid) 41 of angles in cyclic plans 41 45, 50, 51, 57, 81 Platonic solids 40 quadrilaterals 71 parallelograms 39 plotting graphs 80 of circles 66 parentheses 16 cubic 83 of numbers 6-9 functions 93 ofpolygons 34-39 in algebraic expressions 76, inequalities 91 of polyhedra 30, 40-41 78, 79 linear 80, 82 of tangents 71 quadratic 82 proportion (OC) \u00b724, 25, 26, 52, participant observation reci procal 84 98, 107 97 (observation) simultaneous equations protractors 47, 49, 50, 105 pyramids 41, 59, 68 parts of circles 65, 70-71 88-89 Pythagorean theorem 37, 38, Pascal's triangle 10 point of contact 71 (tangents) 45, 60, 68 PEMDAS 16, 76 points see also vertices 30, 31, Pythagorean triples or triads pension plans 117 38 pentagons 34 (polygons), 32, 33,34,42,48, 51 on a compass 53 Q 40, 50 polygons 34-35, 40, 41, 50, pentahedra 40 (polyhedron) 55, 56, 57, 107 quadrant arcs 65 per annum (p.a.) polyhedra .(sing. polyhedron) see quadrants . also three-dimensional objects 28 (interest), 29 and solids 40-41 of circles 65 percent 27 (introduction) polynomial expressions 75 on a plane 31 percentage (algebraic expression) quadratic \u00b7 population 97, 98 equations 85-86, 89 change 28 positive expressions (see also square decrease 28 angles 32 increase 28 correlation 111 (correlation) numbers) 78, 82, 85 rate of interest 29 gradients 80 (introduction) percentages (%) 18, numbers (+) 6, 7 formula 86, 89 27 (introduction), 28, 29, 112, parabolas 82 functions 93 116,117 rotation 43 (rotation) graphs 82 perfect square roots (Vn) 11 sequences 10 numbers 11 possibility space 115 quadrilaterals 34, 39, 71 squares 78, 86 possible outcomes 115 cyclic 71 perigons see whole turns post meridiem (pm) 74 qualitative data 96, 97 perimeter 55 \u00b7 (12-hour clock) quantitative data 96, 97, 108 period (of graph) 64 (sine graph) pounds (lb) 72 \u00b7 quart (qt) 72 (US customary perpendicular powers 6, 16, 19, 21, 22, 76, 84 units) bisectors 43, 48, 51, primary data 96 quartiles 102 prime questionnaires 97 (of a chord) 70 factors 11 quindecagons 34 (polygons) height 41, 56, 57, 68, 75 numbers 7, 11 quota sampling 98 lines 30, 32, 48, 71 quotient 15 (division) personal loans 117 tax allowance 116 (income tax) pi ('IT) 9, 19, 5'5, 66","R reverse percentages 27 (to sectors 67, 68, 105 find an original quantity) area of 67 radius (pl. radii) 51, 57, 65, 66, 67, 68, 69, 70, 71 revolution 32 (whole turn) segments rhombuses 35, 39 of circles 65, 70 random right of lines 30, 31, 48, 51 events 113 sampling 98 angles 30, 32, 41, 48, 50, semi- 51, 56, 57, 64, 70, 71 circular arcs 65, 70 range major axes 69 (ellipses) of a distribution 102, 108, 110 prisms 41 (prism) minor axes 69 (ellipses) set of results 92, 93 pyramids 41 (pyramid) regular polyhedra 40 right-angle-hypotenuse-side regular tessellation 36 rate see speed (RHS) congruence 38 rates of interest 28 (interest), 29 right-angled triangles 37, 38, semicircles 51, 65, 70 ratio (n:m) 24 (introduction), 45, 56, 60'--61, 70 sense 43 (reflection) roots septagons 34 (polygons) 25, 26, 52, 66 cube 11, 22 sequences 10 method 26 of quadratic equations 85 sets . 12-13, 92, 98 rational square 11, 22, 86 number perfectsquares 78 roster notation of sets 12 (set notation 12, 92 numbers 9, 12, 78 notation) \u00b7 shapes see polygons raw data 96 (primary data) rotation 43, 44 short diagonals 41 (diagonal) real numbers 9, 12, 92 rotation symmetry or showing proportion 25 rearranging rotational symmetry 39, 42 side-angle-side (SAS) equations 60, 61, 62, 79, round angles see wh.ole turns rounding congruence 38 87,88, 90 decimals 20, 86 side elevations 41 (elevation) - trigonometric formulas 60, error 20 sides 34, 35, 37, 49, 60,.62 integers 9, 16 side-side-side (SSS) 61, 62, 63 ' rules reciprocal 18, 76, 77 functions 92-93 congruence 38 in sequences 10 significant figures (sig. fig. or curve 84 in sets 12 functions 93 of algebra 76-79 s.f.) 6, 9, 16, 23 graphs 25, 84 similar recording data 99 5 rectangles 35, 39, 56, 67 figures 44 rectangular coordinate system salaries 117 triangles 38 31 (Cartesian coordinate sales and services taxes 117 simple system) samples 97, 98 fractions 18 rectangular prisms 41, 58 sampling 98 interest 28, 29 recurring decimals 9, 19 random sampling 98 reduction 52 error 98 simplification \u00b7reflection 43, 44 savings accounts 117 of expressions and i::quations reflection symmetry or reflective scalar or scalar quantity 46 symmetry 42 scalar multiplication 46 77, 78, 87, 89 reflex angles 32, 35 scale 52 of fractions see canceling regression line 111 regular drawing 52-54 fractions polygons 35, 36, 40, 41, 50 factor 44, 52, 54 of ratios 24 polyhedra 40, 41 scalene triangles 37 simultaneous equations 87-89 prisms 41 (prism) scaling up or down 52 sine pyramids 41 (pyramid) scatter graphs or diagrams graph or sine curve 63, 64 tessellation 36 110, 111 ratio (sin) 60, 61,. 93 relative frequency 112 scientific notation 21 ruJe 62, 63 remainders (r. or rem.) 7, 15 secondary data 96 wave 64 representing data 105-111 seconds (s or sec) 74 single event 113 result (f) 92 sec:tional bar charts 106 resultant vectors 46 size see magnitude \u00b7 slant height 41, 68 slope (m) 25, 80, 81 and tangents 95 of graphs 94, 95","slope\/intercept form 81 subtraction (-) 14, 15, 16 three-dimensional objects see SOHCAHTOA 61 in algebraic expressions 76 also polyhedra and solids solids -see also polyhedra 30, in algebraic fractions 77 30, 31, 40, 50, 58 of a decimal 20 40-41,48 of exponents 22 volume of 58-59 symmetry 42 of a fraction 18 three-figure bearings 53 see transformation 43 of vectors 46 volume 58-59 also bearings solutions 79 (introduction), success 112 time 73, 74 82, 83 successful outcomes tonnes (t) 72 (metric units) solve for___ see rearranging tons 72 (US customary units) - equations 112 (success), 113 top heavy fractions 18 solving total probability 113 (mutually equations 79, 85-86 sum CD 14, 38, 55, 101 inequalities 90, 91 exclusive events) proportion and ratio supplementary angles 33 transformation 43-44 surface area 57 problems 26 of graphs 93 simultaneous equations 87-89 of cones 68 translation 43, 44 speed 46, 73, 74, 96 of cylinders 67 transversals 30, 33 speed-time graphs 94 (area of spheres 69 trapezoid rule 95 (to find the under a graph) surveys 97, 98 spheres 59, 69 symmetry 37, 39, 42, 69, 82 area under a curved graph) spirals 10 systematic trapezoids 39, 57, 94 spread 102, 108 observation 97 (observation) trial and improvement square sampling 98 n~mbers 8, 10, 21, 38, method 79 78, 85 T triangles 30, 34, 37-38, 56, roots (\\\"\\\\I) 9, 11, 22, 78, 86 units of measurement 55 take home pay 116 (earnings) 60-63 squares 36, 39, 56 tallies 99 (tally chart) constructing 49 squaring 8 (square number), 21 tally charts 99 triangular. stacked bar chart 106 tangent standard numbers 8 deviation 103, 104 graph or tangent curve 64 prisms 41 statements 117 ratio (tan) 6-0, 61, 62, 93 trigonometric statistics 96 (introduction), 112 tangents (to a curve) 71, 95 functions 60~64, 93 stem-and-leaf displays or taxable income graphs 64 diagrams 108 116 (income tax) trigonometry 60-64 stock markets 117 taxation 117 (tax) trinomial expressions stones (st) 72 (US customary taxes 11 6, 117 75 (algebraic expression) units) terminating decimals 19 two-dimensional shapes store card 116 (credit card) terms 30 (dimensions), 31, straight in sequences 10 angles 32 money-related 116-117 41, 54 line graphs 80, 81, 82 of algebraic expressions 75, two-way tables 99 strata (sing. stratum) 98 (stratified sampling) 78, 79,88 u stratified sampling 98 of ratios 24 strong correlation to describe constructions 48 unconditional inequality 111 (correlation) to describe graphs 80 90 subsets (C) 12, 13, 92 tessellation 36, 39 substitution 77, 79, 81, 85, 86, tetrahedra 40 (polyhedron) uniform cross-section 58 87,88, 89 theorems (volume formulas) 'subtended angles 70 Euler's 40 - Pythagorean 37, 38, 45, 60, union of sets (U) 13 unit cubes 58 68 unitary theoretical probability 112 theta (e) 60, 61 method 26 third quartile (Q3) see upper ratio 24 units of measurement 24, 55, quartiles 58, 72-73, 74 universal sets (~) 12, 13 unknown angles 60 unlike terms 75 (term), 76","upper vertical 30, 31, 50, 95, 106, x bounds 16 - 109, 110 x-axis 31 (Cartesian coordi\u00b7nate class boundaries vertically opposite ang les system), 45, 80, 81, 93, 94, 95 99 (class boundary) 33 x-coo rd inate 31 (Cartesian class limits 99 vertices (sing. vertex) 91 coord inate system) (class interval) of polygons 33, 34, 35, 36, 37,48, 57, 70, 71 x-intercept 80 \u00b7 quartiles (Q3) 102, of polyhedra 40, 41 109, 110 y volume 58-~9 , 94, 95 US customary units 72-73 cones 59 (volume of a yards (yd) 72 (US _customary utility bills 117 pyramid), 68 units) rectangular prisms 58 v cylinders 67 y-axis 31 (Cartesian prisms 58, 68 coordinate system), 45, 80, Value Added Tax (VAT) pyramids 59, 68 81 , 93, 94, 95 117 (sales and services tax) spheres 59, 69 y-coordinate 31 (Cartesian variables 75, 79, 80, 87, w coordinate system) 88, 90 wages 117 y-inte rcept 80, 81 variance 103 weight 72 (mass) variations on graphs 64, 93 whiskers 110 (box plot) z vector notation 45 whole numbers see integers vectors 43, 45-46 whole turns 32, 71 z-axis 31 (dimensions) velocity 73, 95 zero \u00b7. Venn diagrams 13 vertex see vertices angles 32 exponent rule 22 zig-zags 110 Acknowledgements Website adviser Lisa Watts Photos Page 23 \u00a9UC Regents\/ Lick Observatory; page 35 courtesy of Glo beXplorer.* *Every effort has. been made to trace the copyright holders of the materia l in this book. If any rights have been omitted, the publishers offer to rectify this in any future edition, following notification. The publishers are grateful to th e organizations and individuals concerned for th eir contribution and permission to reproduce material. Trademarks Macintosh and Qu ickTime are trademarks of Apple Computer, Inc., registered in the US and other countries._ RealOne Player is a t rademark or registered trad emark of RealN etworks, Inc., registered in th e US and other countries. Flash and Shockwave are trademarks of Macromedia, Inc., registered in the US and other countries. Usborne Publishing is not res ponsible and does not accept liabi lity for the availability or content of any websi t e other than its own, or for any exposure to harmful, offensive, or inaccurate material which may appear on the Web. Usborne Publishing will have no liabi lity for any damage or 'loss caused by viruses that may be downloaded as a result of browsing the sites it recommends. First published in 2003 by Usborne Publishing Ltd, Usborne House, 83-85 Saffron Hill, London EC1 N 8RT, En gland. AE First published in America, 2004. www.usborne.com Q eCopyright \u00a9 2003 Usborne Publishing Ltd. The name U sb~rne and the devices are Trade Marks of Usborne Publishing Ltd. All rights reserved. No part of thi s publ ication may be reproduced , stored in a retrieval system or transmitted in any form o r by any means, electronic, mechanical, photocopying, recording or otherwise, wi thout the prior permission of the publisher. Printed in Dubai.","ILL STRATED DI CT IONARY OF Everyone studying math needs this book. tts concise explanations , enhanced by examples and diagrams, provide the confidence and understanding of the subject that are the key to exam success . \u00b7Over 500 clear definitions of: all the main mathematical terms and concepts . \u00b7 More than 300 illustrations and diagrams help to interpret, darify and explain each subject. \u00b7 Over 100 worked examples show how to put theory into practice. \u00b7Comprehensive cross-referencing and a detailed index guarantee easy access to information . \u00b7Internet links to recQrnmended websites complement each topic. Other title s in this series: The Us borne Illustrated Dictionary of Physics The Usborne Illustrated Dictionary of Chemistry The Usborne Illustrated Dictionary of Biology The Us borne Illustrated Dictionary of Science For more information on Usborne books, visit www.edcpub.com or www.ubah.com $12.95 ISBN 0-7945-0662-3 Published in the USA by 111111111111111111111111111111 JFM MJJASOND\/ OS EDC PUBLISHING, 10300 E. 55th Place, 9 780794 506629 Tulsa, Oklahoma 74146, USA. NOT FOR SALE IN CANADA Mad e with paper fr-O rn a sust ain able .s ource."]
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