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M.B.A 2 All right are reserved with CU-IDOL Quantitative Techniques for Managers Course Code: MBA602 Semester: First SLM Units: 3-4-5 E-LESSON : 3 www.cuidol.in Unit-3,4 & 5 (MBA602)

Introduction to Statistics 33 OBJECTIVES INTRODUCTION Student will be able to : The business environment of today being Understand the major concepts of descriptive very complex and complicated, the decision analysis . making for business is a very difficult job Know various concepts of Arithmetic Mean Understand the use of data for calculation of centre The statistical data constitutes the basic raw tendency material, for its useful gain in decision making. Analyze the various concepts of measures of In this chapter we will elaborate as to location their applications in research. how the statistical data can be tabulated and www.cuidol.in Unit-3,4 & 5 (MBA602) presented in a form to draw a meaning inference at a glance. . INASllTITriUgThEt aOrFeDreISsTeArNveCdE AwNitDh OCNUL-IIDNOE LLEARNING

TOPICS TO BE COVERED 4 > Introduction to Descriptive Analysis > Measure of Central Tendency and Location > Empirical Relation among Mean, Mode and Median > Measures of Location > Percentiles and Quartiles. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

Descriptive Summary measures 5 www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

INTRODUCTION 6  Measures of central tendency are statistical measures which describe the position of a distribution.  They are also called statistics of location, and are the complement of statistics of dispersion, which provide information concerning the variance or distribution of observations.  Simpson and Kafka defined it as “ A measure of central tendency is a typical value around which other figures congregate”  Waugh has expressed “An average stand for the whole group of which it forms a part yet represents the whole”. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

ARITHMETIC MEAN  Arithmetic mean is frequently referred to as ‘mean’. It is obtained by dividing sum of the values of all 7 observations in a series (ƩX) by the number of items (N) constituting the series.  Thus, mean of a set of numbers X1, X2, X3,………..Xn denoted by x̅ and is defined as Mean = ������������ /N= Sumation of items/no.of items www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

ARITHMETIC MEAN CALCULATED METHODS 8  Direct Method : ������ = ������fm/N  Short cut method : X = A+������fd/N  Step deviation Method : X= A+������fd/N www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

ADVANTAGES/ DISADVANTAGES OF MEAN 9  ADVANTAGES  It is easy to understand & simple calculate.  It is based on all the values.  It is rigidly defined .  It is easy to understand the arithmetic average even if some of the details of the data are lacking.  It is not based on the position in the series.  DISADVANTAGES  It is affected by extreme values.  It cannot be calculated for open end classes.  It cannot be located graphically  It gives misleading conclusions.  It has upward bias. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

ADVANTAGES OF MEDIAN 10  Median can be calculated in all distributions.  Median can be understood even by common people.  Median can be ascertained even with the extreme items.  It can be located graphically.  It is most useful dealing with qualitative data.  It is not based on all the values.  It is not capable of further mathematical treatment.  It is affected fluctuation of sampling.  In case of even no. of values it may not the value from the data. http://makemeanalyst.com/explore-your-data-mode-median-and-mean/ www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

MEDIAN  Median is a central value of the distribution, or the value which divides the distribution in equal parts, each part 1 1 containing equal number of items. https://www.onlinemathlearning.com/median.html www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

MODE  Mode is the most frequent value or score in the distribution. 12  It is defined as that value of the item in a series which occurs most of the time  It is denoted by the capital letter Z.  It is the highest point of the frequencies distribution curve. Croxton and Cowden : defined it as “the mode of a distribution is the value at the point armed with the item tend to most heavily concentrated. It may be regarded as the most typical of a series of value” . The exact value of mode can be obtained by the following formula. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

ADVANTAGES/ DISADVANTAGES OF MODE 13  ADVANTAGES  Mode is readily comprehensible and easily calculated  It is the best representative of data  It is not at all affected by extreme value.  The value of mode can also be determined graphically.  It is usually an actual value of an important part of the series.  DISADVANTAGES  It is not based on all observations.  It is not capable of further mathematical manipulation.  Mode is affected to a great extent by sampling fluctuations.  Choice of grouping has great influence on the value of mode. www.cuidol.in Unit-3,4 & 5 (MBA602) https://www.chilimath.com/lessons/intermediate-algebra/mean-median- mode-and-range/ All right are reserved with CU-IDOL

EMPIRICAL RELATION AMONG MEAN, 14 MODE AND MEDIAN  The following empirical relationship has been developed by Prof. Karl Pearson to connect Mean, Mode and Median. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

APPLICATIONS OF CENTRAL TENDENCY 15  Average provides the overall picture of the series.  Average value provides a clear picture about the field under study for guidance and necessary conclusion.  It gives a concise description of the performance of the group as a whole and it enables us to compare two or more groups in terms of typical performance. Problem : Calculate the average value of age for a class of 10 students with their ages as under : Solution : www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

PERCENTILES  Percentiles are the values which divide the arranged data into hundred equal parts. There are 99 16 percentiles i.e. P1, P2, P3, ……..,P99. The 50th percentile divides the series into two equal parts and P50 = D5 = Median.  Similarly the value of Q1 = P25 and value of Q3 = P75.  The different percentiles can be found using the formula given below: Pi = L1+ L2 –l1/f[iN/100-c] i=1,2,3..... Where, L1 = lower limit of ith percentile class L2 = upper limit of ith percentile class c = cumulative frequency of the class preceding the ith percentile class f = frequency of ith percentile class. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

QUARTILES There are three quartiles, i.e. Q1, Q2 and Q3 which divide the total data into four equal parts when it has been 17 orderly arranged. Q1, Q2 and Q3 are termed as first quartile, second quartile and third quartile or lower quartile, middle quartile and upper quartile, respectively. The first quartile, Q1, separates the first one-fourth of the data from the upper three fourths and is equal to the 25th percentile. The second quartile, Q2, divides the data into two equal parts (like median) and is equal to the 50th percentile. The third quartile, Q3, separates the first three-quarters of the data from the last quarter and is equal to 75th percentile. www.cuidol.in https://www.mathematics-monster.com/glossary/quartile.html All right are reserved with CU-IDOL Unit-3,4 & 5 (MBA602)

CALCULATION OF QUARTILES 18  The calculation of quartiles is done exactly in the same manner as it is in case of the calculation of median.  The different quartiles can be found using the formula given below: Q1 = L1+ L2 –l1/f[iN/4-c] ,i=1,2,3..... Where, L1 = lower limit of ith quartile class L2= upper limit of ith quartile class C = cumulative frequency of the class preceding the ith quartile class F = frequency of ith quartile class. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

SKEWNESS  The term skewness refers to the lack of symmetry. The lack of symmetry in a distribution is always determined 1 9 with reference to a normal or Gaussian distribution. Note that a normal distribution is always symmetrical.  The skewness may be either positive or negative. When the skewness of a distribution is positive (negative), the distribution is called a positively (negatively) skewed distribution. Absence of skewness makes a distribution symmetrical.  It is important to emphasize that skewness of a distribution cannot be determined simply my inspection.  If Mean > Mode, the skewness is positive.  If Mean < Mode, the skewness is negative.  If Mean = Mode, the skewness is zero. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

MEASURES OF SYMMETRY 20  Many distribution are not symmetrical.  They may be tail off to right or to the left and as such said to be skewed.  One measure of absolute skewness is difference between mean and mode. A measure of such would not be true meaningful because it depends of the units of measurement.  The simplest measure of skewness is the Pearson’s coefficient of skewness www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

MEASURES OF PEAKEDNESS 21 Kurtosis: is the degree of peakedness of a distribution, usually taken in relation to a normal distribution.  A curve having relatively higher peak than the normal curve, is known as Leptokurtic.  On the other hand, if the curve is more flat-topped than the normal curve, it is called Platykurtic.  A normal curve itself is called Mesokurtic, which is neither too peaked nor too flat-topped. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

MEASURE OF VARIABILITY 22 Variability: Variability means ‘Scatter’ or ‘Spread’. Thus measures of variability refer to the scatter or spread of scores around their central tendency. The measures of variability indicate how the distribution scatter above and below the central tender. Measures of Variability:  The Range  The Standard Deviation  Range is the difference between in a series. It is the most general measure of spread or scatter. It is a measure of variability of the varieties or observation among themselves and does not given an idea about the spread of the observations around some central value. Range = H—L Here H = Highest score L = Lowest score www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

MERITS/ DEMERITS OF RANGE: 23  Merits of Range: 1. Range is easily calculated and readily understood. 2. It is the simplest measure of variability. 3. It provides a quick estimate of the measure of variability.  Demerits of Range: 1. Range is greatly affected by fluctuation of scores. 2. It is not based on all the observations of the series. It only takes the highest and the lowest scores in to account. 3. In case of open ended distributions range cannot be used. 4. It is affected greatly by fluctuations in sampling. 5. It is affected greatly by extreme scores. 6. The series is not truly represented by range. A symmetrical and A symmetrical distribution may have same range but not the same dispersion. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

USES OF RANGE 24 Range is used as a measure of dispersion when variations in the value of the variable are not much.  Range is the best measure of variability when the data are too scattered or too scant. Range is used when the knowledge of extreme score or total spread is wanted. When a quick estimate of variability is wanted range is used. www.cuidol.in https://www.mathsisfun.com/data/range.html All right are reserved with CU-IDOL Unit-3,4 & 5 (MBA602)

THE STANDARD DEVIATION (SD) A widely used measure of variability, consisting of the square root of the mean of the squared deviations 2 5 of scores from the mean of the distribution.” Standard deviation is the square root of the average value of the squared deviations of the scores from their arithmetical mean. The SD is computed by summing the squared deviation of each measure from the mean, divided by the number of cases and extracting the square root. To be more clear, we should note here that in computing the SD we square all the deviations separately, find their sum, divide the sum by the total number of scores and then find the square root of the mean of the squared deviation. So that it is also called the ‘root mean square deviation’. The square of standard deviation is called as Variance (σ2). It is referred to as the mean square deviation. It is also called as the second moment dispersion. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

THE STANDARD DEVIATION The standard deviation is the square root of the variance. Thus, the standard deviation of a population is:2 6 σ = sqrt [ σ2 ] = sqrt [ Σ ( Xi - μ )2 / N ] where σ is the population standard deviation, μ is the population mean, Xi is the ith element from the population, and N is the number of elements in the population. Statisticians often use simple random samples to estimate the standard deviation of a population, based on sample data. Given a simple random sample, the best estimate of the standard deviation of a population is: s = sqrt [ s2 ] = sqrt [ Σ ( xi - x )2 / ( n - 1 ) ] where s is the sample standard deviation, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

THE VARIANCE  In a population, variance is the average squared deviation from the population mean, as defined by the 2 7 following formula: σ2 = Σ ( Xi - μ )2 / N where σ2 is the population variance, μ is the population mean, Xi is the ith element from the population, and N is the number of elements in the population.  Observations from a simple random sample can be used to estimate the variance of a population. For this purpose, sample variance is defined by slightly different formula, and uses a slightly different notation: s2 = Σ ( xi - x )2 / ( n - 1 ) where s2 is the sample variance, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample. Using this formula, the sample variance can be considered an unbiased estimate of the true population variance. Therefore, if you need to estimate an unknown population variance, based on data from a simple random sample, this is the formula to use. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

COEFFICIENT OF VARIANCE 28  Coefficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another. www.cuidol.in https://www.mathsisfun.com/data/range.html All right are reserved with CU-IDOL Unit-3,4 & 5 (MBA602)

MULTIPLE CHOICE QUESTIONS 1. Change of origin and scale is used for calculation of the 29 a) Arithmetic mean (b) Geometric mean b) Weighted mean (d) Lower and upper quartile 2 Scores that differ greatly from the measures of central tendency are called: a) Raw scores b) The best scores c) Extreme scores d) None of the above 3) Any measure indicating the centre of a set of data, arranged in an increasing or decreasing order of magnitude, is called a measure of: a) Skewness b) Symmetry c) Central tendency d) Dispersion 4) The measure of central tendency listed below is: a) The raw score b) The mean c) The range d) Standard deviation www.cuidol.in Answers: 1.(a) , 2.(c) , 3. (c) , 4. (b) Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

SUMMARY 30  A measure of central tendency is a measure that tells us where the middle of a bunch of data lies.  Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average.  Median is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order.  Mode is the value that occurs most frequently in a set of data.  Standard deviation is the square root of the average value of the squared deviations of the scores from their arithmetical mean.  Range is used as a measure of dispersion when variations in the value of the variable are not much www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

FREQUENTLY ASKED QUESTIONS 31 Q1. What is meant by Measure of Central Tendency? Ans: Averages are the values lying between the largest and the smallest value of the observations and denote the central part of the distribution of the data. These are called ‘Measure of Central Tendency’. For further details Refer to the SLM. Q2. Explain the method of locating mode. Ans: (1) Arrange the terms in ascending or descending order (Preferably Ascending) (2) Note the term occurring maximum times if it is or is a unique one. (3) This term is Mode. (Z). For further details Refer to the SLM. Q3. which average do you consider to be the best and why? Ans. The median is a good measure of the average value when the data include exceptionally high or low values because these have little influence on the outcome. The median is the most suitable measure of average for data classified on an ordinal scale. For further details Refer to the SLM. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

REFERENCES 32  Black, K. (2008). Business statistics for contemporary decision making. New Delhi: Wiley India.  Schiller, J., Srinivasan, R.,Spiegel, Schaum's.M(2012)..Outline Of Probability and Statistics. New Delhi: McGraw-Hill.  Levin, R. I.,Rubin, D. S.(1999). Statistics for management. New Delhi: Prentice Hall of India.  Webster, A. (2006). Applied statistics for business and economics. New Delhi: McGraw Hill. www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL

33 THANK YOU For queries Email: [email protected] www.cuidol.in Unit-3,4 & 5 (MBA602) All right are reserved with CU-IDOL