E-LESSON-8 , 9 , 10

Merits and Demerits of 51 Range MERITS DEMERITS  Simple to understand  Can’t be calculated in open  Easy to calculate ended distributions  Widely used in  Not based on all the statistical quality control observations  Affected by sampling fluctuations  Affected by extreme values www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

INTERQUARTILE RANGE 52 & QUARTILE DEVIATION  Interquartile Range is the difference between the upper quartile (Q3) and the lower quartile (Q1).  It covers dispersion of middle 50% of the items of the series.  Symbolically, Interquartile Range = Q3 – Q1  Quartile Deviation is half of the interquartile range. It is also called Semi Interquartile Range.  Symbolically, Quartile Deviation = ������3 −������1 2  Coefficient of Quartile Deviation: It is the relative measure of quartile deviation.  Coefficient of Q.D. = ������3 −������1 ������3+������1 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

STANDARD DEVIATION 53  Most important & widely used measure of dispersion.  First used by Karl Pearson in 1893.  Also called root mean square deviations.  It is defined as the square root of the arithmetic mean of the squares of the deviation of the values taken from the mean σ = Σ ������ − ������ 2 or Σ������2 where x = ������−������ ������ ������  Denoted by σ (sigma) Coefficient of S.D. = σ ������ www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

INDIVIDUAL SERIES 54 ACTUAL MEAN METHOD σ = Σ ������ − ������ 2 or Σ������2 where x = ������−������ ������ ������ Ques: Calculate the SD of the following data: 16, 20, 18, 19, 20, 20, 28, 17, 22, 20 Ans: 3.13 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

DISCRETE SERIES 55 ACTUAL MEAN METHOD σ= Σ������ ������ − ������ 2 or Σ������������2 where x = ������−������ ������ ������ Ques: Calculate the SD of the following data: X: 3 4 5 6 7 8 9 F: 7 8 10 12 4 3 2 Ans: 1.602 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

CONTINUOUS SERIES 56 STEP DEVIATION METHOD σ= Σ������ ������ ′ 2− Σ������������ ������ 2 x i where d’ = ������ −������ ������ ′ Ques Calculate the Mean & SD of the following data: 60-70 70-80 10 4 X 0-10 10-20 20-30 30-40 40-50 50-60 F 5 10 20 40 30 20 Ans: 39.38, 15.69 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

VARIANCE 57  It is another measure of dispersion  It is the square of the Standard Deviation  Variance = (SD)2 = σ2 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Coefficient Of Variation 58 (C.V.)  It was developed by Karl Pearson.  It is an important relative measure of dispersion.  It is used in comparing the variability, homogeneity, stability, uniformity & consistency of two or more series.  Higher the CV, lesser the consistency. Formula: Coefficient of Variation = (Standard Deviation / Mean) * 100. In symbols: CV = (SD/ ) * 100. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Summary 59  Statistics: According to Prof. Horace Secrist, Statistics is the aggregate of facts, affected to a marked extent by multiplicity of causes, numerically expressed, enumerated or estimated according to reasonable standards of accuracy, collected in a systematic manner for a pre-determined purpose, and placed in relation to each other.  Classification of data-The method of arranging data into homogeneous classes according to some common features present in the data is called classification.  Central tendency - A central tendency is a single figure that represents the whole mass of data.  Arithmetic mean- Arithmetic mean is defined as the sum of the values of all observations divided by the number of observations. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

60  Median- Median is the central value of the distribution in the sense that the number of values less than the median is equal to the number greater than the median.  Quartiles- Quartiles are the measures which divide the data into four equal parts, each portion contains equal number of observation. There are three quartiles: If a statistical series is divided into four equal parts, the end value of each part is called a quartile and denoted by ‘Q’.  The lower half of a data set is the set of all values that are to the left of the median value when the data has been put into increasing order.  The upper half of a data set is the set of all values that are to the right of the median value when the data has been put into increasing order. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

 The first quartile, denoted by Q1 , is the median of the lower half of the data 61 set. This means that about 25% of the numbers in the data set lie below Q1 and about 75% lie above Q1.  The second quartile also called median and denoted by Q2, has 50% of the items below it and 50% of the items above it.  The third quartile, denoted by Q3, is the median of the upper half of the data set. This means that about 75% of the numbers in the data set lie below Q3 and about 25% lie above Q3.  Mode - Mode is the value which occurs most frequently in the series, that modal value has the highest frequency in the series.  Dispersion is a measure of the variation of the items from central value. The measures of dispersion is important to compare uniformity, consistency and reliability amongst variables / series. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

 Range- Range is defined as the difference between two extreme 62 observations i.e. the largest and the smallest value. Symbolically, R = L-S Where: R = Range, L = Largest Value, S = Smallest value  Quartile deviation-Quartile deviation is known as half of difference of third (Q3 ) quartile and first quartile (Q1 ). It is also known as semi inter quartile range. Where: Q.D. = Quartile deviation Q3 = Third quartile or upper quartile. Q1 = First quartile of lower quartile. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

 Standard Deviation- Standard deviation is the best and widely used 63 measure of dispersion. Standard deviation is the square root of the arithmetic mean of the squares of deviation of its items from their arithmetic mean. Calculation of standard deviation in individual series. Actual mean method. Where: = Standard Deviation = Sum total of square of Deviation taken from Mean N = Number of items www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

64  Coefficient of variation: When two or more groups of similar data are to be compared with respect to stability (or uniformly or consistency or homogeneity). Coefficient of variation is the most appropriate measures. It is the ratio of the standard deviation to the mean. Where: C.V. = Coefficient of variation = Standard deviation x = Arithmetic mean www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Multiple Choice Questions 65 1 What is the median of the data 78, 56, 22, 34, 45, 54, 39, 68, 54, 84? (a) 54 (c) 55 (b) 53 (d) 51 2 In a week the prices of a bag of rice were 350,280,340,290,320, 310,300. The range is (a) 60 (c) 80 (b) 70 (d)100 Answers: 1.(a) 2.(b) www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Frequently Asked Questions 66 Q1 Define Statistics. Ans: Statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data. Q2 State the properties of arithmetic mean. Ans:  Property 1: If x is the arithmetic mean of n observations x1, x2, x3, . . xn; then (x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x) = 0.  Property 2: The mean of n observations x1, x2, . . ., xn is x. If each observation is increased by p, the mean of the new observations is (x + p). www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

67  Property 3: The mean of n observations x1, x2, . . ., xn is x. If each observation is decreased by p, the mean of the new observations is (x - p).  Property 4: The mean of n observations x1, x2, . . .,xn is x. If each observation is multiplied by a nonzero number p, the mean of the new observations is px.  Property 5: The mean of n observations x1, x2, . . ., xn is x. If each observation is divided by a nonzero number p, the mean of the new observations is (x/p). www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

REFRENCES 68 1. Gupta , S.P and Gupta M.P (2017). Business statistics. Sultan Chand 2. Aggarwal, S.C and Jain, T.R (2008). Business statistics. V.K publications. 3. Vohra, N.D (2014). Business statistics. MC Graw Hill education Pvt. Ltd. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

69 THANK YOU For queries Email: [email protected] www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL