MEASURES OF DISPERSION

Presented by,Shiwani Agrawal & R. Naveena

1.  Range:  The simplest of our methods for measuring dispersion is range.  Range is the difference between the largest value and the smallest value in the data set.  While being simple to compute, the range is often unreliable as a measure of dispersion since it is based on only two values in the set. 

A range of 50 tells us very little about how the values

are dispersed.  Are the values all clustered to one end with the low value (12) or the high value (62) being an outlier?

Or are the values more evenly dispersed among the range?

For Example:-

Population form: Sample form:

The population form is used when the data being analyzed includes the entire set of possible data.  When using this form, divide by n, the number of

values in the data set.

All people living in the US.

The sample form is used when the data is a random sample taken from the entire set of data. 

When using this form, divide by n - 1.(It can be shown that dividing by n - 1

makes S2 for the sample, a better estimate of for the population from which the sample was taken.)

Sam, Pete and Claire who live in the US.

The population form should be used unless you know a random sample is being analyzed.

Before discussing our next methods, let's establish some vocabulary:

Mean Deviation:-Average of the difference of the values of the items from some average of the series(ignoring negative sign) i.e. the arithmetic mean of the absolute differences of the values from their averages.

Note:-• MD uses average but ignores sign, hence appears unmethodical.• MD is calculated from mean as well as from median for both ungrouped data using direct method and for continuous distribution using assumed mean method and short-cut method.•Average used is either arithmetic mean or median.MD=∑ d / N (Individual series)MD = ∑ f d / N (discrete and continuous series) Where, ∑ d = Sum of all deviations ( taken either from mean or median ignoring ± sign)

Standard Deviation:-Standard deviation is used to measure the spread of items in a set of observations. If all the observation values are identical or distribution of items of a set is uniform, then deviation of every value from mean is zero. Where elements of set are moredispersed, then SD becomes larger.

σ = √[1/N {( X1-X)2+.....................+(XN-X)2}]

Standard deviation enables us to determine with a great deal of accuracy, where the values of a frequency distribution are located in relation to the mean.

Variance:-It is the square of the standard deviation.V=σ2

Coefficient of variation:-The ratio of standard deviation to the mean in percentage is known as “Coefficient of variation”C.V.= σ/X*100