Provide Information Regarding Statistics & Econometrics : Measure of Dispersion Lecture 11

Measure of Dispersion

Lecture 11

Dispersion

Dispersion means the scattering of values about their central value (average).

Measure of Dispersion

A quantitative measure that measures the amount of variation or scatterness in data about their central value is called a measure of dispersion. Measures of dispersion provide an idea about variation in a data series.

Types

There are two types of measures of dispersion. They are given below:

1. Absolute measure

Absolute measures dispersion measures the amount of dispersion in terms of the same units (or square of units) of the original observation. e.g., if the given data measure is in kilometres, the measure of dispersion will also be in kilometres (or km sqr.). The absolute measures of dispersion are not helpful to compare two or more data sets having different units of measurements.

2. Relative measure

Relative measures of dispersion measure the degree of dispersion and are independent from the units of measurement. If the original data is in kilometers, the relative measures of dispersion will be free from these units. These measures are a kind of ratio and are called coefficients. The relative measures are suitable for comparative studies.

Main measures of dispersion

1. Range

2. Quartiles Deviation (The Semi-Interquartile Range)

3. Mean Deviation

4. Standard Deviation.

The Range

The difference between the largest (or maximum) value and the smallest (or minimum) in the data set is called range. It is the simplest measure of dispersion based on two extreme values of the data set.

The range is an absolute measure of dispersion, while its relative measure of dispersion is known as the coefficient of range.

Characteristics of Range

i. It is the simplest and crude measure of dispersion.

ii. It is not based on all the observations of the given data.

iii. It is affected by extreme values.

iv. It gives an idea of the dispersion very quickly.

Example 4.1: Find range and its relative measure for the following data measure in kilograms.

50, 87, 64, 45, 53, 89, 92, 112, 87, 96, 125.

Solution:

Merits of Range

i. It is defined precisely.

ii. It is very simple to measure.

iii. It is very useful in statistical quality control.

iv. It is useful in studying variation in price and stocks.

Demerits of Range

i. It is not a stable measure of dispersion affected by extreme values.

ii. It is based on only two extreme values.

iii. It is a crude measure of dispersion and not suitable for precise and accurate studies.

The Quartile Deviation

The difference between the upper and lower quartiles is called the interquartile range, and the half of the interquartile range is called the quartile deviation or semi-interquartile.

Quartile deviation is an absolute measure of dispersion. A relative measure of dispersion based on quartile deviation is known as the coefficient of quartile deviation. Given by:

Example 4.2: Find quartile deviation and relative measure based on quartiles of the following data.

497 495 480 465 440 490 443 398 390 365 400 412 432 416 389

Solution: Arrange the data in ascending order of magnitude and assign ranks.

Rank	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Data	365	389	390	398	400	412	416	432	440	443	465	480	480	495	497

The relative measure of quartile deviation is given below:

Example 4.3: Find the quartile deviation and coefficient of quartile deviation for the following frequency distribution.

Class	20- 30	30 – 40	40 – 50	50- 60	60–70	70 - 80	80–90
Frequency	20	17	44	57	68	55	34

Solution:

Class	Frequency	Cumulative frequency
20- 30	20	20
30 – 40	17	37
40 – 50	44	81
50- 60	57	138
60–70	68	206
70–80	55	261
80 - 90	34	295
	295

For Q1, we divided the total observation by 4.

We select the class that lies against 73.75 are not available in the cumulative frequency column. We select the class lies against 81, which is 40–50.