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Measures of Dispersion Measures of central tendency do not provide complete information about the composition and character of a series. In order to get a comprehensive picture of the series, we use various methods of dispersion. In the words of Spiegel “the degree to which numerical data tend to spread about an average value is called dispersion of the data.” Objectives of measures of dispersion To know about the composition of the series To compare the disparity between two of more series Absolute and relative measure of dispersion Absolute measure - These measures of dispersion can be used for comparison only if all the values are expressed in same units. Relative measure – these are calculated by dividing the absolute measure by a measure of central tendency. These measures are also known as co-efficient of dispersion. These measures helps in comparing values expressed in different units. Absolute measures Relative measures Range Quartile deviation Mean deviation Standard deviation Co-efficient of range Co-efficient of quartile deviation Co-efficient of mean deviation Co-efficient of standard deviation Range- it is the difference between the largest and the smallest value.(L - S) Co- efficient of range = L –S L +S (Note; in discrete series, range is the difference between highest and lowest variables and not frequency. In continuous series, it is difference between highest upper limit and lowest lower limit.) Merits and demerits of range Merits It is simple to calculate and understand It can be used in statistical series relating to quality control in production. control charts are prepared based on the basis of range. Demerits It is an unstable measure of dispersion. It depends upon extreme values of the series It is not based on all values in the series. This cannot be used in the case of open-ended classes. 1 Quartile deviation and its coefficient Quartile deviation is the half of the difference between the third quartile and the first quartile Q.D = (Q3 – Q1) /2 For the calculation of Q3 and Q1 refer previous chapter. Coefficient of quartile deviation = (Q3 – Q1) / (Q3 Q1) Merits and demerits of quartile deviation Merits It is simple to calculate and understand Less affected by extreme values. Demerits It is not based on all the values of the series. Mean deviation Mean deviation is the arithmetic average of the deviations of all the values taken from some average value (mean or median) of the series ignoring signs of the deviations Calculation of mean deviation in individual series Steps Find the average of the series (arithmetic mean or median) Find the absolute value of the deviations of the variables from its average(mean or median) and sum them up. Divide the sum of the deviations by the number of the observations. / I.e. MD = ∑│d│ N d = X – A.M or Median n = number of observations. Calculation of MD in discrete series Steps Find the average of the series (arithmetic mean or median) Find the absolute value of the deviations of the variables from the average.(arithmetic mean or median) Multiply the deviations with the respective frequency and find the sum. Divide the sum by the sum of the frequencies. / 1. e. MD = ∑│fd│ ∑f d = X – A.M or Median ∑f = total frequency 2 Calculation of MD in continuous series Steps Find the mid value of the classes Find the average of the series (arithmetic mean or median) Find the absolute value of the deviations of the variables from the average.(arithmetic mean or median) Multiply the deviations with the respective frequency and find the sum. Divide the sum by the sum of the frequencies. / 1. e. MD = ∑│fd│ ∑f d = X – A.M or Median X = mid value ∑f = total frequency Coefficient of mean deviation In order to find the coefficient of mean deviation, divide the mean deviation by arithmetic mean if deviations are found from AM. If deviations are taken from median, divide MD by median. Coefficient of mean deviation = Mean deviation /AM Or MD / median Merits and demerits of mean deviation Merits It is simple to calculate It is based on all observations Less affected by extreme values Demerits Suffers from inaccuracy since the signs are ignored. Not capable of further algebraic treatment Standard deviation SD is the most scientific method of calculating dispersion. Standard deviation is defined as the square root of the arithmetic mean of the squares of deviations of the items from their mean value. Calculation of standard deviation in individual series Standard deviation in individual series can be calculated using following methods. Direct method Steps Calculate the arithmetic mean of the series. Find the deviations of the given variables from the arithmetic mean of the series(d) 3 Square each deviation and sum them up. (d2) Divide the sum of the deviations by number of items and find the square root of it. SD = d=x – actual arithmetic mean Short cut method Take one of the given items as the assumed mean.(A) Find the deviations of the given variables from the assumed mean of the series(d) Square each deviation and sum them up. (∑d2) Apply the formula SD = √(∑d2)⁄N –(∑d)/N)2 d=x –A(assumed mean) Calculation of standard deviation in discrete series Direct method Steps Calculate the arithmetic mean of the series. Find the deviations of the given variables from the arithmetic mean of the series(d) Square each deviation (d2) Multiply d2 with frequency and sum them up. Apply the formula. SD = = √(∑ fd2)/∑f d=x – actual arithmetic mean Short cut method Take one of the given items as the assumed mean.(A) Find the deviations of the given variables from the assumed mean of the series(d) Square each deviation (d2) Multiply d2 with frequency and sum them up. Apply the formula SD = √ (∑fd2) ∑f – (∑fd/∑f) 2 d=x –A (assumed mean) Step deviation method Take one of the given items as the assumed mean.(A) Find the deviations of the given variables from the assumed mean of the series (d) and divide it by a common factor. Square each deviation (d2) 4 Multiply d2 with frequency and sum them up. Apply the formula SD =√ (∑fd’2) ∑f – (∑f d’/∑f) 2 × c d=x –A (assumed mean) c = common factor Calculation of standard deviation in continuous series Direct method Steps Calculate the arithmetic mean of the series. Find the deviations of the mid points from the arithmetic mean of the series(d) Square each deviation (d2) and denote it by(d’2) Multiply d’2 with corresponding frequency and sum them up. Apply the formula. SD = √ (∑ fd2)/∑f d=x – actual arithmetic mean Short cut method Take one of the given items as the assumed mean.(A) Find the deviations of the mid point from the assumed mean of the series(d) Square each deviation (d2) Multiply d2 with corresponding frequency and sum them up. Apply the formula SD = √ (∑fd2) ∑f – (∑fd/ d=x –A (assumed mean) x = Midvalue Step deviation method Take one of the given items as the assumed mean.(A) Find the deviations of the mid points from the assumed mean of the series (d) and divide it by a common factor and denote it by d’. Square each deviation (d’2) Multiply d’2 with corresponding frequency and sum them up. Apply the formula SD =√ (∑fd’2) ∑f – (∑f d’/∑f) 2 × c d’ = x – A/ c x = Midvalue c = common factor Coefficient of variation 5 This relative measure is used for comparing dispersion of two series. SD/AM x 100 (standard deviation divided by arithmetic mean multiplied by 100) Merits and demerits of standard deviation Merits It is based on all values Capable of further statistical treatment Less affected by changes in samples Demerits Can be used only by experts Extreme values tend to get greater importance. Construction of Lorenz curve Find the cumulative of the given values and convert them into percentage (each cumulative value divided by the total multiplied by 100) Cumulative percentage of frequencies are plotted on X – axis of a graph, while cumulative percentage of variables are plotted on y-axis Draw a line joining (0,0) and (100,100) to get the line of equality. Plot the actual data on the graph and join the points with a smooth hand curve and extend it to the points (0,0) and (100,100). The actual distribution curve is called the Lorenz curve. The gap between the line of equality and the actual curve gives the variation in the distribution. Larger the gap, more is the dispersion and vice versa. Find range from the following 1. 23 56 78 98 ans 91 56 7 2. 23 56 78 98 56 100 Ans = 99 3 . wages 25 50 75 80 85 90 workers 4 6 9 3 2 1 ans 65 4. Income 10-20 20-30 families 46 54 Ans = 40 34 34 30-40 42 90 1 34 34 87 87 34 34 56 40-50 30 6 7 8 9