Chapter-2-MEASURES OF CENTRAL TENDENCY -01-THEORY

2.1.1 Introduction. An average or a central value of a statistical series in the value of the variable which describes the characteristics of the entire distribution. The following are the five measures of central tendency. (1) Arithmetic mean (2) Geometric mean (3) Harmonic mean (4) Median (5) Mode 2.1.2 Arithmetic Mean. Arithmetic mean is the most important among the mathematical mean. According to Horace Secrist, “The arithmetic mean is the amount secured by dividing the sum of values of the items in a series by their number.” (1) Simple arithmetic mean in individual series (Ungrouped data) (i) Direct method : If the series in this case be then the arithmetic mean is given by , i.e., (ii) Short cut method Arithmetic mean , where, A = assumed mean, d = deviation from assumed mean = x – A, where x is the individual item, d = sum of deviations and n = number of items. (2) Simple arithmetic mean in continuous series (Grouped data) (i) Direct method : If the terms of the given series be and the corresponding frequencies be , then the arithmetic mean is given by, . (ii) Short cut method : Arithmetic mean Where A = assumed mean, f = frequency and x – A = deviation of each item from the assumed mean. (3) Properties of arithmetic mean (i) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero. If , i = 1, 2, …, n is the frequency distribution, then , being the mean of the distribution. (ii) The sum of the squares of the deviations of a set of values is minimum when taken about mean. (iii) Mean of the composite series : If are the means of k-component series of sizes respectively, then the mean of the composite series obtained on combining the component series is given by the formula . 2.1.3 Geometric Mean. If are n values of a variate x, none of them being zero, then geometric mean (G.M.) is given by  . In case of frequency distribution, G.M. of n values of a variate x occurring with frequency is given by , where . 2.1.4 Harmonic Mean. The harmonic mean of n items is defined as . If the frequency distribution is respectively, then Note :  A.M. gives more weightage to larger values whereas G.M. and H.M. give more weightage to smaller values. Example: 1 If the mean of the distribution is 2.6, then the value of y is Variate x 1 2 3 4 5 Frequency f of x 4 5 y 1 2 (a) 24 (b) 13 (c) 8 (d) 3 Solution: (c) We know that, Mean i.e. or or  Example: 2 In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class are 72, then what are the average marks of the girls (a) 73 (b) 65 (c) 68 (d) 74 Solution: (b) Let the average marks of the girls students be x, then (Number of girls = 100 – 70 = 30) i.e., ,  x = 65. Example: 3 If the mean of the set of numbers is , then the mean of the numbers , is (a) (b) (c) (d) Solution: (b) We know that i.e.,  Example: 4 The harmonic mean of 4, 8, 16 (a) 6.4 (b) 6.7 (c) 6.85 (d) 7.8 Solution: (c) H.M. of 4, 8, 16 Example: 5 The average of n numbers is M. If is replaced by , then new average is (a) (b) (c) (d) Solution: (b) i.e.  New average Example: 6 Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is (a) 48 (b) (c) 50 (d) 80 Solution: (c) Sum of 100 items Sum of items added Sum of items replaced New sum  Correct mean 2.1.5 Median. Median is defined as the value of an item or observation above or below which lies on an equal number of observations i.e., the median is the central value of the set of observations provided all the observations are arranged in the ascending or descending orders. (1) Calculation of median (i) Individual series : If the data is raw, arrange in ascending or descending order. Let n be the number of observations. If n is odd, Median = value of item. If n is even, Median = (ii) Discrete series : In this case, we first find the cumulative frequencies of the variables arranged in ascending or descending order and the median is given by Median = observation, where n is the cumulative frequency. (iii) For grouped or continuous distributions : In this case, following formula can be used (a) For series in ascending order, Median = Where l = Lower limit of the median class f = Frequency of the median class N = The sum of all frequencies i = The width of the median class C = The cumulative frequency of the class preceding to median class. (b) For series in descending order Median = , where u = upper limit of the median class As median divides a distribution into two equal parts, similarly the quartiles, quantiles, deciles and percentiles divide the distribution respectively into 4, 5, 10 and 100 equal parts. The jth quartile is given by . is the lower quartile, is the median and is called the upper quartile. (2) Lower quartile (i) Discrete series : (ii) Continuous series : (3) Upper quartile (i) Discrete series : (ii) Continuous series : (4) Decile : Decile divide total frequencies N into ten equal parts. [j = 1, 2, 3, 4, 5, 6, 7, 8, 9] If j = 5, then . Hence is also known as median. (5) Percentile : Percentile divide total frequencies N into hundred equal parts where k = 1, 2, 3, 4, 5,.......,99. Example: 7 The following data gives the distribution of height of students Height (in cm) 160 150 152 161 156 154 155 Number of students 12 8 4 4 3 3 7 The median of the distribution is (a) 154 (b) 155 (c) 160 (d) 161 Solution: (b) Arranging the data in ascending order of magnitude, we obtain Height (in cm) 150 152 154 155 156 160 161 Number of students 8 4 3 7 3 12 4 Cumulative frequency 8 12 15 22 25 37 41 Here, total number of items is 41, i.e. an odd number. Hence, the median is th i.e. 21st item. From cumulative frequency table, we find that median i.e. 21st item is 155. (All items from 16 to 22nd are equal, each = 155) Example: 8 The median of a set of 9 distinct observation is 20.5. If each of the largest 4 observation of the set is increased by 2, then the median of the new set (a) Is increased by 2 (b) Is decreased by 2 (c) Is two times the original median (d) Remains the same as that of the original set Solution: (d) n = 9, then median term . Since last four observation are increased by 2. ∵ The median is 5th observation which is remaining unchanged.  There will be no change in median. Example: 9 Compute the median from the following table Marks obtained No. of students 0-10 2 10-20 18 20-30 30 30-40 45 40-50 35 50-60 20 60-70 6 70-80 3 (a) 36.55 (b) 35.55 (c) 40.05 (d) None of these Solution: (a) Marks obtained No. of students Cumulative frequency 0-10 2 2 10-20 18 20 20-30 30 50 30-40 45 95 40-50 35 130 50-60 20 150 60-70 6 156 70-80 3 159 Here n = 159, which is odd. Median number , which is in the class 30-40 (see the row of cumulative frequency 95, which contains 80). Hence median class is 30-40.  We have l = Lower limit of median class = 30 f = Frequency of median class = 45 C = Total of all frequencies preceding median class = 50 i = Width of class interval of median class = 10  Required median . 2.1.6 Mode. Mode : The mode or model value of a distribution is that value of the variable for which the frequency is maximum. For continuous series, mode is calculated as, Mode Where, = The lower limit of the model class = The frequency of the model class = The frequency of the class preceding the model class = The frequency of the class succeeding the model class i = The size of the model class. Symmetric distribution : A symmetric is a symmetric distribution if the values of mean, mode and median coincide. In a symmetric distribution frequencies are symmetrically distributed on both sides of the centre point of the frequency curve. A distribution which is not symmetric is called a skewed-distribution. In a moderately asymmetric the interval between the mean and the median is approximately one-third of the interval between the mean and the mode i.e. we have the following empirical relation between them Mean – Mode = 3(Mean – Median)  Mode = 3 Median – 2 Mean. It is known as Empirical relation. Example: 10 The mode of the distribut Marks 4 5 6 7 8 No. of students 6 7 10 8 3 (a) 5 (b) 6 (c) 8 (d) 10 Solution: (b) Since frequency is maximum for 6  Mode = 6 Example: 11 Consider the following statements (1) Mode can be computed from histogram (2) Median is not independent of change of scale (3) Variance is independent of change of origin and scale Which of these is/are correct (a) (1), (2) and (3) (b) Only (2) (c) Only (1) and (2) (d) Only (1) Solution: (d) It is obvious. Important Tips  Some points about arithmetic mean • Of all types of averages the arithmetic mean is most commonly used average. • It is based upon all observations. • If the number of observations is very large, it is more accurate and more reliable basis for comparison.  Some points about geometric mean • It is based on all items of the series. • It is most suitable for constructing index number, average ratios, percentages etc. • G.M. cannot be calculated if the size of any of the items is zero or negative.  Some points about H.M. • It is based on all item of the series. • This is useful in problems related with rates, ratios, time etc. • A.M.  G.M.  H.M. and also  Some points about median • It is an appropriate average in dealing with qualitative data, like intelligence, wealth etc. • The sum of the deviations of the items from median, ignoring algebraic signs, is less than the sum from any other point.  Some points about mode • It is not based on all items of the series. • As compared to other averages mode is affected to a large extent by fluctuations of sampling,. • It is not suitable in a case where the relative importance of items have to be considered. 2.1.7 Pie Chart (Pie Diagram). Here a circle is divided into a number of segments equal to the number of components in the corresponding table. Here the entire diagram looks like a pie and the components appear like slices cut from the pie. In this diagram each item has a sector whose area has the same percentage of the total area of the circle as this item has of the total of such items. For example if N be the total and is one of the components of the figure corresponding to a particular item, then the angle of the sector for this item , as the total number of degree in the angle subtended by the whole circular arc at its centre is 360°. Example: 12 If for a slightly assymetric distribution, mean and median are 5 and 6 respectively. What is its mod (a) 5 (b) 6 (c) 7 (d) 8 Solution: (d) We know that Mode = 3Median – 2Mean = 3(6) – 2(5) = 8 Example: 13 A pie chart is to be drawn for representing the following data Items of expenditure Number of families Education 150 Food and clothing 400 House rent 40 Electricity 250 Miscellaneous 160 The value of the central angle for food and clothing would be [ (a) 90° (b) 2.8° (c) 150° (d) 144° Solution: (d) Required angle for food and clothing 2.1.8 Measure of Dispersion. The degree to which numerical data tend to spread about an average value is called the dispersion of the data. The four measure of dispersion are (1) Range (2) Mean deviation (3) Standard deviation (4) Square deviation (1) Range : It is the difference between the values of extreme items in a series. Range = Xmax – Xmin The coefficient of range (scatter) . Range is not the measure of central tendency. Range is widely used in statistical series relating to quality control in production. (i) Inter-quartile range : We know that quartiles are the magnitudes of the items which divide the distribution into four equal parts. The inter-quartile range is found by taking the difference between third and first quartiles and is given by the formula Inter-quartile range Where Q1 = First quartile or lower quartile and Q3 = Third quartile or upper quartile. (ii) Percentile range : This is measured by the following formula Percentile range Where P90 = 90th percentile and P10 = 10th percentile. Percentile range is considered better than range as well as inter-quartile range. (iii) Quartile deviation or semi inter-quartile range : It is one-half of the difference between the third quartile and first quartile i.e., and coefficient of quartile deviation . Where, Q3 is the third or upper quartile and Q1 is the lowest or first quartile. (2) Mean deviation : The arithmetic average of the deviations (all taking positive) from the mean, median or mode is known as mean deviation. (i) Mean deviation from ungrouped data (or individual series) Mean deviation Where |x – M| means the modulus of the deviation of the variate from the mean (mean, median or mode). M and n is the number of terms. (ii) Mean deviation from continuous series : Here first of all we find the mean from which deviation is to be taken. Then we find the deviation of each variate from the mean M so obtained. Next we multiply these deviations by the corresponding frequency and find the product f.dM and then the sum of these products. Lastly we use the formula, mean deviation , where n = f. Important Tips  Mean coefficient of dispersion  Median coefficient of dispersion  Mode coefficient of dispersion  In general, mean deviation (M.D.) always stands for mean deviation about median. (3) Standard deviation : Standard deviation (or S.D.) is the square root of the arithmetic mean of the square of deviations of various values from their arithmetic mean and is generally denoted by  read as sigma. (i) Coefficient of standard deviation : To compare the dispersion of two frequency distributions the relative measure of standard deviation is computed which is known as coefficient of standard deviation and is given by Coefficient of S.D. , where is the A.M. (ii) Standard deviation from individual series where, = The arithmetic mean of series N = The total frequency. (iii) Standard deviation from continuous series where, = Arithmetic mean of series = Mid value of the class = Frequency of the corresponding N = f = The total frequency Short cut method (i) (ii) where, d = x – A = Deviation from the assumed mean A f = Frequency of the item N = f = Sum of frequencies (4) Square deviation (i) Root mean square deviation where A is any arbitrary number and S is called mean square deviation. (ii) Relation between S.D. and root mean square deviation : If  be the standard deviation and S be the root mean square deviation. Then . Obviously, will be least when d = 0 i.e. Hence, mean square deviation and consequently root mean square deviation is least, if the deviations are taken from the mean. 2.1.9 Variance. The square of standard deviation is called the variance. Coefficient of standard deviation and variance : The coefficient of standard deviation is the ratio of the S.D. to A.M. i.e., . Coefficient of variance = coefficient of S.D. . Variance of the combined series : If are the sizes, the means and the standard deviation of two series, then Where, , and . Important Tips  Range is widely used in statistical series relating to quality control in production.  Standard deviation ≤ Range i.e., variance ≤ (Range)2.  Empirical relations between measures of dispersion • Mean deviation (standard deviation) • Semi interquartile range (standard deviation)  Semi interquartile range (mean deviation)  For a symmetrical distribution, the following area relationship holds good covers 68.27% items covers 95.45% items covers 99.74% items  S.D. of first n natural numbers is .  Range is not the measure of central tendency. 2.1.10 Skewness. “Skewness” measures the lack of symmetry. It is measured by and is denoted by . The distribution is skewed if, (i) Mean  Median  Mode (ii) Quartiles are not equidistant from the median and (iii) The frequency curve is stretched more to one side than to the other. (1) Distribution : There are three types of distributions (i) Normal distribution : When , the distribution is said to be normal. In this case Mean = Median = Mode (ii) Positively skewed distribution : When , the distribution is said to be positively skewed. In this case Mean > Median > Mode (iii) Negative skewed distribution : When , the distribution is said to be negatively skewed. In this case Mean < Median < Mode (2) Measures of skewness (i) Absolute measures of skewness : Various measures of skewness are (a) (b) (c) where, = median, = mode, M = mean Absolute measures of skewness are not useful to compare two series, therefore relative measure of dispersion are used, as they are pure numbers. (3) Relative measures of skewness (i) Karl Pearson’s coefficient of skewness : , , where  is standard deviation. (ii) Bowley’s coefficient of skewness : Bowley’s coefficient of skewness lies between –1 and 1. (iii) Kelly’s coefficient of skewness : Example: 14 A batsman scores runs in 10 innings 38, 70, 48, 34, 42, 55, 63, 46, 54, 44, then the mean deviation is (a) 8.6 (b) 6.4 (c) 10.6 (d) 9.6 Solution: (a) Arranging the given data in ascending order, we have 34, 38, 42, 44, 46, 48, 54, 55, 63, 70, Here median M = median is the mean of 5th and 6th items) Mean deviation Example: 15 S.D. of data is 6 when each observation is increased by 1, then the S.D. of new data is (a) 5 (b) 7 (c) 6 (d) 8 Solution: (c) S.D. and variance of data is not changed, when each observation is increased (OR decreased) by the same constant. Example: 16 In a series of 2n observations, half of them equal a and remaining half equal –a. If the standard deviation of the observations is 2, then |a| equ (a) (b) (c) 2 (d) Solution: (c) Let a, a, ..........n times – a, – a, – a, – a, ----- n time i.e. mean = 0 and S.D. . Hence Example: 17 If  is the mean of distribution , then (a) M.D. (b) S.D. (c) 0 (d) Relative frequency Solution: (c) We have, Example: 18 What is the standard deviation of the following s Measurements 0-10 10-20 20-30 30-40 Frequency 1 3 4 2 (a) 81 (b) 7.6 (c) 9 (d) 2.26 Solution: (c) Class Frequency yi , A = 25 fiui fiui2 0-10 1 5 – 2 – 2 4 10-20 3 15 – 1 – 3 3 20-30 4 25 0 0 0 30-40 2 35 1 2 2 10 – 3 9   = 9 Example: 19 In an experiment with 15 observations on x, the following results were available , . On observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is (a) 78.00 (b) 188.66 (c) 177.33 (d) 8.33 Solution: (a) , Increase in , then Increase in , then Variance Example: 20 The quartile deviation of daily wages (in Rs.) of 7 persons given below 12, 7, 15, 10, 17, 19, 25 is (a) 14.5 (b) 5 (c) 9 (d) 4.5 Solution: (d) The given data in ascending order of magnitude is 7, 10, 12, 15, 17, 19, 25 Here item = size of 2nd item = 10 item = size of 6th item = 19 Then Q.D. Example: 21 Karl-Pearson’s coefficient of skewness of a distribution is 0.32. Its S.D. is 6.5 and mean 39.6. Then the median of the distribution is given b (a) 28.61 (b) 38.81 (c) 29.13 (d) 28.31 Solution: (b) We know that , Where M = Mean, = Mode,  = S.D. i.e.  and also know that, 3median – 2mean 3(Median) – 2(39.6) Median = 38.81 (approx.) Example: 22 The S.D. of a variate x is . The S.D. of the variate where a, b, c are constant, is (a) (b) (c) (d) None of these Solution: (b) Let i.e., i.e. , where ,         Thus, new S.D. .