MEASURES OF DISPERSION IN STATISTICS FREE NOTES

MEASURES OF DISPERSION IN STATISTICS

ABSOLUTE MEASURES OF DISPERSION

The absolute measures of dispersion can be compared with one another only if the two belong to the same population and are expressed in the same units like inches, kilograms, rupees etc. The various absolute measures of dispersion are:

• Range
• Quartile deviation
• Mean deviation
• Standard deviation
• Variance

The detailed explanation of all the terms is as follows:

RANGE

Range is the simplest measure of dispersion. The difference between the highest value and lowest value of a series is known as range. It is defined as the difference between the two extreme items of the distribution. In other words, range is the difference between the highest and lowest values of the distribution.

Absolute Range = Highest Value-Lowest Value

R=H.V.-L.V.

MERITS OF RANGE: Range can be easily calculated as we require only two extreme values. If one is interested in getting a quick, but not very accurate picture of variability, one can compute range Range is useful for the following reasons:

1. Quality Control: Quality control is done to check the quality of the product without 100% inspection. Range plays an important role, when statistical methods of quality control are used. In preparing R-charts, range plays an important role. Thus range is useful in determining and maintaining the quality of the products.

2. Variations in Stock and Shares Prices: Range is useful in studying the variations in the prices of shares and stock. By computing range, we can get an idea about the range of fluctuation, say, gold, jewellery prices etc..

3. Forecasts: The meteorology does make use of the range in determining the difference between the minimum and maximum temperature. This information is useful for general public.

DEMERITS OF RANGE: Range suffers from the following limitations:

1. Range is not based on all the items of the series: We require only the extreme items largest and smallest. The items in-between are not given any weightage. Under the circumstances, range cannot be representative and hence it is not suitable for mathematical treatment.

According to W.I.King who states “Range is too indefinite to be used as a practical measure of dispersion.”

2. Range is affected by unusual items: Range is affected by unusual items of the series as it takes into account only items-largest and smallest. But what about the items which are very close to the smallest and the largest Under such circumstances, range cannot be considered as a reliable measure of Dispersion.

3. Not fruitful in open end series: In open-end classes, range cannot be determined, because the lower and upper classes are not given.

4. Affected by fluctuations of sample: Range is very much affected by the fluctuations of sampling. The value of range i e is never stable, it varies from sample to sample. As the size of the sample increases, the range tends to increase though not in the same ratio.

5. Do not represent all values of the series: If the two values ie, the smallest and the largest items of a distribution are not changed whereas other items are replaced by a set of observations within these two limits, the range will not change. But the value of range is considerably changed if any item is added or deleted on either side of the extreme value. It means range does not take into account the range composition of the series. Its concern is mainly with smallest and largest items.

QUARTILE DEVIATION or INTER-QUARTILE RANGE

Quartile Deviation or Inter-quartile range is the difference between First quartile and third quartiles. By using quartile deviation, the dependence on extreme items can be avoided.

QD= (Q₃-Q₁ )/2

MERITS OF QUARTILE DEVIATION:

1. Quartile deviation is easy to calculate. It is based on two values only i.e. lower quartile and upper quartile. It does not involve any mathematical complications. Since quartile deviation makes use of 50% of the data, it is a better measure than range which takes into account just two items ie., the largest and the smallest.
2. Quartile deviation is superior to range because quartiles are generally located where there is big concentration of items The values of quartiles will not change much it one more value of any size is added to or taken away from the distribution.
3. The quartile deviation like range is not influenced by the extreme items. It can be used in highly skewed distribution where other measures of dispersion do not give reliable results. The quartile deviation is influenced by the position in an array of each item in the series.
4. In open-end distribution, quartile deviation is suitable. The quartile deviation is also useful where data is ranked but not measured qualitatively, say in case of beauty, intelligence, tastes etc.
5. The quartile deviation provides wonderful short-cut for estimating standard deviation
6. It is quite satisfactory when only the middle half of the group is dealt with as it takes into account 50% of the middle data, i.e., between lower quartile and upper quartile.

DEMERITS OF QUARTILE DEVIATION:

1. Quartile deviation ignores 50% of the items i.c., the first 25% and the last 25% The value of quartile déviation does not depend on each and every item of the series. So it cannot be called a good method of measuring dispersion. An ideal method should be based on each and every item.
2. It is not a measure of dispersion, because it does not show the scatter around an average but rather a distance on a scale. It means quartile deviation is not itself measured from an average. But it is a positional average. Some Statisticians are of the view that quartile deviation is a measure of partition rather than a measure of dispersion.
3. Quartile deviation is not capable of further algebraic treatment because it ignores the first and the last 25% items on both sides.
4. Quartile deviation depends on the values of central items. If these values are irregular, then the results are affected adversely.
5. Quartile deviation is highly influenced by fluctuations of sampling.
6. Quartile deviation is not affected by the distribution of items between lower quartile and upper quartile or by the distribution outside the quartiles. That means the values of the quartiles may be the same for two dissimilar series.

Thus quartile deviation has limited reliability, particularly, in case of those distributions where variation is substantial.