Introduction Dispersion 1 Central Tendency alone does not explain the observations fully as it does reveal the degree of spread or variability of individual observations in a series. Measures of dispersion help us understanding the variability of items. The term dispersion is used in two senses- 1.Firstly dispersion refers to variation of items amongst themselves. e.g. if the value of all the items are same, it will have zero dispersion. 2.Secondly, dispersion refers to variation of items around an average. If the difference between the value of items and average is large, dispersion will be high and if the difference is small, dispersion will be low. “Dispersion is a measure of variation of items.”-Bowley “The degree to which numerical data tend to spread about average is called variation or dispersion of data.”-Spiegel Meaning Definitions

Objectives Dispersion 2 1.To determine the reliability of average- Dispersion helps in determining reliability of average by pointing how far is single average figure is representative of data. If the dispersion is small, the average is a reliable indicator of data. 2.To compare the variability of two or more series- It helps comparison of two or more series of data. The high degree of variability means less consistent data. 3.Facilitates use of other statistical measures-Dispersion serves the basis of other statistical measures like correlation, regression, testing of hypothesis etc. 4.Statistical quality control– Identifies whether the variation in quality of a product is due to random factors or is there some other defect in the manufacturing process.

Properties & types Dispersion 3 Good Average Absolute or Relative Properties of good dispersion are- 1.It should be simple to compute. 2.Should be easy to understand. 3.Should be uniquely defined. 4.Should be based on all observations without unduly affected by extreme observations. 5.Should be capable for further algebraic treatment. Dispersion Absolute Measure of dispersion expressed in the same unit in which data of series is given Relative Measure of dispersion expressed in the percentage or ratio. It is also called coefficient of dispersion.

Methods of Measurement Dispersion 4 Individual series There are 3 main methods of dispersion- 1.Range 2.Interquartile range and quartile deviation 3.Mean Deviation Range-It is defined by difference between maximum and minimum value of a series or a data set. Coefficient of Range- It is relative measure of dispersion and is also called range coefficient of dispersion. Coefficient of range= (x max -x min )/(x max +x min ) Interquartile Range- The difference between the upper quartile (Q 3 ) and the lower quartile (Q 1 ) is called interquartile range. Quartile Deviation-It is half the difference between upper and lower quartile i.e. (Q 3 -Q 1 )/2. The relative measure of quartile deviation is called coefficient of quartile deviation and is defined as- =(Q 3 -Q 1 )/(Q 3 +Q 1 ) Definition

Mean Deviation Dispersion 5 Coefficient of mean deviation Mean Deviation Mean deviation is another measure of dispersion and is also known as average deviation. It is defined as arithmatic average of deviation of various items of a series computed from some measures of central tendency say median or mean. However median is preferred because the sum of deviations of item taken from Median is minimum when signs are ignored. The formulae for calculating mean deviation are- For calculating coefficient of mean deviation- For continuous series, the mid points of various classes and deviations from these values are used to calculate mean deviation and coefficient of mean deviation.

Standard Deviation Dispersion 6 Introduction It is most widely used measure of dispersion and is also called root mean square deviation. It is calculated as square root of arithmatic mean of the squares of the deviation of the values taken from the mean. It is calculated by- Where X is mean value with mean value µ meaning E[X]= µ where as is sigma (standard deviation) Coefficient of SD- Coefficient of SD (relative measure) is obtained by dividing standard deviation by the arithmatic average. The formula is __ Where X is arithmatic average.

Mean Deviation vs SD Dispersion 7 Both are measures of dispersion but they are different in some ways 1.Algebraic signs of deviations are ignored while calculating mean deviation where as in the calculation of standard deviation, the signs of deviations are taken into account. 2.Mean deviation can be computed from either of mean, median or mode where as SD is always computed from mean because sum of squares of deviations taken from mean is minimum. In case of individual series, SD can be calculated by 1.Actual mean method 2.Assumed mean method (in case actual mean is not whole no.) 3.Method based on actual data (when observations are less) In case of discrete series SD can be calculated by 1.Actual mean method 2.Assumed mean method (or short cut method) 3.Step deviation method (used to simplify calculations by dividing deviations by a common factor) In case of continuous series- Under this all the methods used in discrete series can be used as classes are represented by mid values. Differences Calculation of SD

Variance Dispersion 8 Variance is another measure of dispersion. Variance is square of Standard Deviation. Variance= (SD) 2 Variance is calculated by 3 methods as under The standard deviation of two groups can be calculated by Where 12 =Combines standard deviation 1 =SD of first group 2 = SD of second group d 1 =X 1 -X 12 d 2 =X 2 -X 12 Introduction Combined standard deviation

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