Two distributions may have identical averages yet differ completely in the way their observations are spread out. For example, two classes may both have a mean of 60 marks, but in one class all students score between 55 and 65, while in the other scores range from 20 to 100. Averages alone, therefore, give an incomplete picture of the data. To describe a distribution fully, we must also know the extent to which the observations scatter or spread around the average. This scatter or spread is measured by measures of dispersion.
Dispersion is the degree of scatteredness or variability of the observations of a distribution. A measure of dispersion tells us how far, on an average, the observations deviate from a central value. Small dispersion means the observations are close to the average and the average is representative; large dispersion means the observations are spread widely and the average is less representative.
The common measures of dispersion are the range, quartile deviation, mean deviation and standard deviation. Among these, the standard deviation is the most important and widely used measure. This chapter deals with the meaning of dispersion, each measure with its formula and computation for different series, the coefficient of variation, and the relative merits and demerits of the various measures.
Measures of dispersion may be of two types:
Absolute measures: These express the dispersion in the same units as the original data. For example, the range of heights may be 20 cm or the standard deviation of income Rs 500. Absolute measures are useful for comparing the dispersion of two distributions measured in the same units and of similar magnitudes.
Relative measures: These express dispersion as a ratio or percentage, independent of the units of measurement. They are used to compare the variability of two distributions that differ in units or magnitudes. Examples are the coefficient of range, coefficient of quartile deviation, coefficient of mean deviation and the coefficient of variation.
The important absolute measures are:
Range is the simplest measure of dispersion. It is the difference between the largest and the smallest observations:
$$\text{Range} = \text{Maximum value} - \text{Minimum value}$$
The coefficient of range is given by:
$$\text{Coefficient of Range} = \frac{\text{Max} - \text{Min}}{\text{Max} + \text{Min}}$$
The range is easy to compute and understand, and is used in quality control and daily price quotations, but it is based only on two extreme values and is affected greatly by sampling fluctuations.
Quartile deviation is based on the quartiles. Quartiles divide the distribution into four equal parts: Q1 is the value below which 25 percent of the observations lie, Q2 is the median, and Q3 is the value below which 75 percent of the observations lie.
$$\text{Quartile deviation} = \frac{Q3 - Q1}{2}$$
The coefficient of quartile deviation is:
$$\text{Coefficient of QD} = \frac{Q3 - Q1}{Q3 + Q1}$$
The quartile deviation is not affected by extreme values and is easy to compute, but it ignores half of the observations and is not amenable to algebraic treatment.
The mean deviation is the arithmetic mean of the absolute deviations of all observations from a measure of central tendency, usually the mean or the median. Since the algebraic sum of deviations from the mean is zero, we take the absolute values (ignoring signs):
$$\text{Mean deviation about the mean} = \frac{\sum |X - \bar{X}|}{N}$$
For a discrete series:
$$\text{MD} = \frac{\sum f|X - A|}{N}$$
For a continuous series, X is replaced by the mid-values M of the classes.
The coefficient of mean deviation is:
$$\text{Coefficient of MD} = \frac{\text{Mean deviation}}{\text{Average used}}$$
The mean deviation is based on all observations and is easy to understand, but the use of absolute values makes it mathematically inconvenient, and it gives the minimum value when deviations are taken from the median.
The standard deviation is the most important and widely used measure of dispersion. It is the positive square root of the mean of the squared deviations of the observations from their arithmetic mean. Squaring the deviations removes the negative signs and gives greater weight to large deviations.
$$\sigma = \sqrt{\frac{\sum (X - \bar{X})^2}{N}}$$
For a discrete series:
$$\sigma = \sqrt{\frac{\sum f(X - \bar{X})^2}{N}}$$
For a continuous series, the mid-values M of the classes are used:
$$\sigma = \sqrt{\frac{\sum f(M - \bar{X})^2}{N}}$$
Short-cut formula: Using the assumed mean A and deviations d = X - A:
$$\sigma = \sqrt{\frac{\sum fd^2}{N} - \left(\frac{\sum fd}{N}\right)^2}$$
Step-deviation method: When classes are equal, using d' = (X - A)/h:
$$\sigma = \sqrt{\frac{\sum fd'^2}{N} - \left(\frac{\sum fd'}{N}\right)^2} \times h$$
The square of the standard deviation is called the variance:
$$\text{Variance} = \sigma^2 = \frac{\sum (X - \bar{X})^2}{N}$$
The standard deviation is an absolute measure and cannot be used directly to compare the variability of two distributions with different units or very different means. For this purpose we use the relative measure called the coefficient of variation (CV):
$$\text{CV} = \frac{\sigma}{\bar{X}} \times 100$$
The coefficient of variation expresses the standard deviation as a percentage of the mean. A distribution with a smaller CV is more consistent, more stable and less variable. For example, if one company has a CV of profit of 15 percent and another 25 percent, the first is more consistent in its profits. In comparing two series, the one with the smaller coefficient of variation is considered to be more consistent or uniform.
Range: Merits - simplest to compute and understand, useful in quality control and price quotations. Demerits - based only on two extreme values, greatly affected by sampling fluctuations, ignores the distribution of intermediate values.
Quartile deviation: Merits - not affected by extreme values, easy to compute. Demerits - ignores half the observations, not amenable to algebraic treatment, not based on all values.
Mean deviation: Merits - based on all observations, easy to understand, less affected by extreme values. Demerits - ignores signs, difficult to handle algebraically, minimum only about the median.
Standard deviation: Merits - based on all observations, rigidly defined, has well-defined mathematical properties, is the basis of many statistical techniques such as correlation and variance analysis. Demerits - gives greater weight to extreme values, is more difficult to compute and understand.
For a complete description, the standard deviation is generally considered the best measure of dispersion because it is based on all observations, gives the minimum sum of squared deviations, and can be used in further statistical work. Its relative counterpart, the coefficient of variation, is the best measure for comparing the variability of two or more distributions. When the data contain extreme values, the quartile deviation or mean deviation about the median may be preferred because they are less affected by outliers.
| Measure | Formula | Coefficient |
|---|---|---|
| Range | Max - Min | (Max - Min)/(Max + Min) |
| Quartile deviation | (Q3 - Q1)/2 | (Q3 - Q1)/(Q3 + Q1) |
| Mean deviation | Sum | X - A |
| Variance | Sum(X - Mean)^2/N | - |
| Standard deviation | sqrt(Sum(X - Mean)^2/N) | sqrt(Variance) |
| Coefficient of variation | sigma / Mean x 100 | - |
| Measure | Based on | Affected by extremes | Algebraic treatment |
|---|---|---|---|
| Range | 2 values | Yes, highly | Not amenable |
| Quartile deviation | 2 quartiles | No | Not amenable |
| Mean deviation | All values | Slightly | Difficult |
| Standard deviation | All values | Yes | Amenable |
This chapter examined the measures of dispersion, which complete the description of a distribution begun with measures of central tendency. We defined dispersion as the scatter of observations around the average and distinguished absolute measures, expressed in the units of the data, from relative measures, expressed as ratios or percentages. We studied the range, the quartile deviation based on the quartiles, the mean deviation based on absolute deviations from an average, and the standard deviation, the most important measure, together with its variance and the short-cut methods of computation. We saw how the coefficient of variation enables comparison of variability across series with different units and means, a smaller CV indicating greater consistency. With both central tendency and dispersion in hand, we can now turn to the study of the relationship between two variables - correlation - in the next chapter.