MEASURES OF DISPERSION OR SCATTERDNESS

Question 1

What is a measure of variability

Answer 1

a number that describes the dispersion or variation in a set of observations.

Question 2

What is are absolute and relative measures of dispersion

Answer 2

A measure of dispersion may be either absolute or relative. Absolute measures of dispersion are expressed in the same statistical unit in which the original data are given. e.g km, kg, dollars etc. A measure of relative dispersion is the ratio of a measure of absolute dispersion to an appropriate average. It is sometimes called the coefficient of dispersion, because it is independent of the unit of measurement (it is a pure number).Each absolute measure of dispersion can be converted into its relative measure

Question 3

Give examples of absolute and relative measures of dispersion

Answer 3

Absolute measures include
• The range
• The interquartile range and quartile deviation
• The mean deviation or average deviation
• The standard deviation

Relative measures include:
• Co-efficient of Range
• Co-efficient of Quartile Deviation
• Co-efficient of mean Deviation
• Co-efficient of Variation.

Question 4

What are the four basic purposes of measures of dispersion

Answer 4

1. To determine the reliability of an average
2. To serve as a basis for control of variability
3. To compare two or more sets of observations with regard to their variability
4. To facilitate the use of other statistical measures

Question 5

What are the properties of a good measure of variation

Answer 5

1. Be simple to understand
2. Be easy to compute
3. Be rigidly defined
4. Be based on each and every item of the distribution
5. Be amenable to further algebraic treatment
6. Have sampling stability
7. Not be unduly affected by extreme items

Question 6

What is range and how is it calculated

Answer 6

This the simplest measure of variability. The range is the difference between the value of the largest item and the smallest item in the set of observations.
Range=Largest item-Smallest item.=L-S

For grouped data the range can be found using two methods.
The first is to find the difference between the upper limit of the highest class and the lower limit of the lowest class. The second method is to find the difference between the midpoint of the highest class and the midpoint of the lowest class.

Question 7

What is the coefficient of range

Answer 7

The relative measure corresponding to range , called the coefficient of range is obtained by applying the formula *see notes

Question 8

What are the merits, limitations and uses of range

Answer 8

Merits of the range:
1. It is the simplest to understand and compute compared to other measures of dispersion.

Limitations:

1. It is not based on every item of the distribution.
2. It is subject to fluctuations of considerable magnitude from sample to sample.
3. It cannot tell us anything about the distribution within the two extreme observations.
4. It cannot be computed for open ended distributions

Uses of the range

1. Quality control to ensure the difference between the largest and smallest of mass produced items does not exceed a certain value
2. Fluctuations in share prices
3. Weather forecasts e.g difference between maximum and minimum temperature

Question 9

Briefly describe interquartile range and quartile deviation

Answer 9

The interquartile range=Q3 − Q1
It is reduced to the form of the semi-interquartile range or quartile deviation by dividing it by 2
* See notes for formula .The quartile deviation gives the the average amount by which
by which the two quartiles differ from the median. In a symmetrical distribution the two quartiles
are equidistant from the median. A very small value of quartile deviation indicates that the variation of the middle 50% items is small, likewise a high quartile deviation means variation of
the middle 50% items is large.
The relative measure corresponding to quartile deviation is the coefficient of quartile deviation.

Question 10

What is the coefficient of Quartile Deviation

Answer 10

*See notes for formula

Coefficient of quartile deviation can be used to compare the degree of variation of different distributions.

Question 11

What are the merits and demerits of quartile deviation

Answer 11

Merits of Quartile Deviation

1. In certain respects it is superior to the range as a measure of dispersion.
2. Can be used in open ended distributions.
3. Useful in erratic or badly skewed data

Demerits of Quartile deviation

1. It does not depend on each and every item in the data set. |It ignores the top 25% and the bottom 25% of items in a distribution.
2. It is not capable of mathematical manipulation
3. Its value is affected by sampling fluctuations
4. It is more of a positional average rather than a measure of dispersion as it does not show scatter around an average.

Question 12

Briefly describe average deviation or mean deviation

Answer 12

This is the average of the absolute values of deviations from the mean. *See notes for formula

Question 13

What is the coefficient of mean deviation

Answer 13

The relative measure corresponding to mean deviation is the coefficient of mean deviation
Coefficient of Mean deviation= mean deviation/mean

Question 14

What are the merits and demerits of average deviation or mean deviation

Answer 14

Merits
1. It is simple to understand and easy to compute
2. It is based on each and every item of the data
3. It is less affected by extreme values
Limitations
1. Ignoring of the signs makes the method non algebraic
2. It is not capable of further algebraic treatment.
3. It may not give very accurate results because mean deviation gives best results when deviations are taken from the median, but the median is not a satisfactory measure when variability is high in the data set.

Question 15

What is variance and what are the types

Answer 15

Variance is the arithmetic mean of the square of the deviations from the mean. Population and sample variance
*See notes for formula

Question 16

What is the primary use of sample statistics

Answer 16

The primary use of sample statistics like Sample variance is to estimate population parameters like Population variance

Question 17

Why is n-1 used in the denominator for sample variance.

Answer 17

using n would under estimate the population variance. Dividing by n-1 gives a slightly larger value and an unbiased estimate of the population variance.

Question 18

What is standard deviation

Answer 18

The standard deviation is the square root of variance

Question 19

What is coefficient of variation and what is it used for

Answer 19

Standard deviation is an absolute measure of variation. The corresponding relative measure of variation is the coefficient of variation. It is used when making comparisons of variability of two or more data sets. The data set with a higher coefficient of variation has more variation and that with a lower value has less variation or is more consistent/uniform/homogeneous.
*See notes for formula

Question 20

What are the merits and demerits of SD

Answer 20

1. It is the best measure of variation. It is based on every item of the data set
2. It is amenable to further algebraic treatment. It is possible to calculate the combined standard deviation of two or more groups.
3. It is used in further statistical work e.g in computing skewness, correlation etc

Limitations of Standard deviation

1. It is difficult to compute
2. It gives more weight to extreme items and less to those which are near the mean.