Dispersion

 Dispersion is the average of variation from average. Dispersion word came from the word spread. It shows the spreading of actual data from its average data. 

Come and learn it with a simple example, we have calculated our 3 coaching classes student's average in last 8 years

Years	Students of	Students of	Students of
	Accounts	Maths	Science
1	15	12	1
2	15	14	3
3	15	14	4
4	15	15	15
5	15	18	15
6	15	15	27
7	15	16	26
8	15	16	29
Sum	120	120	120
Average	15	15	15

From our table, we are seeing that our have average students in last 8 years just 15 in all these classes. Which we did not give us the logical information for our decisions. 

Now we find deviation Data

Years	A - Average	M - Average	S - Average
1	0	-3	-14
2	0	-1	-12
3	0	-1	-11
4	0	0	0
5	0	0	0
6	0	0	12
7	0	1	11
8	0	1	14

From above measurement of dispersion, we find level of unity is in Accounting students because deviation is zero and highest differences from mean is of Science students and it help to bring more focus to science students and bring unity of data in next 8 years. 

Above is graphically presentation of dispersion of our 3 coaching classes.  Our science students classes are more dispersed than the our account students class and math student class. Even the average number of students are equal but dispersion are different of each class. 

Benefits of Measuring Dispersion

1. To examine the reliability of average 

Measurement of dispersion helps us to to check the reliability of average. If there is low dispersed from average, we will say, this data has equal level of unity. but if it is more dispersed, its average is not reliable.

2. Helpful to Compare the Series on the basis of Dispersion

In daily life, we have to compare different things for self improvement like our income, like temperature and business loss. We have to compare its measured dispersion. With this, we can find our weakness and take action for better improvements.

3. To Control Undesirable Variations between series of data

When we measures the dispersion of two series of data, we can focus the reasons of its dispersion. For example, we say, we have kept very high fees, so, students of science is very less in beginning when we had decreased the fees, no. of students has increased. That is reason of its dispersion. So, with this cause, we can take better decision for our new fees plan. 

4. Dispersion Measurements helps to Use Other Statistical Tools

If we have measured the dispersion, we can calculate other statistical tools like co-relation, regression, hypothesis on the basis of dispersion

5. Find the Time Series Trend

On the measurement of dispersion, we can established the trend in time series. Whether trend is seasonal, or accidental.

Methods of Measuring Dispersion

(A) Algebraic Method of Measuring Dispersion

1. Method of limit

1. Range

Range is the difference between largest value and smallest value of any series

Range = Largest value - smallest value 

Coefficient of range = Largest value - smallest value  / Largest value + smallest value 

Example 

Years	Students of	Students of	Students of
	Accounts	Maths	Science
1	15	12	1
2	15	14	3
3	15	14	4
4	15	15	15
5	15	18	15
6	15	15	27
7	15	16	26
8	15	16	29

Range of Account students = 15-15 = 0

Range of maths students = 18-12 = 6

Range of science students = 29-1 = 28

2.The inner quartile range

In this, we find 3rd quartile average of data and 1st quartile average of data and then take its difference 

= Q 3 - Q 1

Q3 = 3 ( N +1 ) /4

Q1 = N+1/4

Quartile deviation = Q 3 - Q 1 / 2 

Coefficent of Quartile deviation =  Q 3 - Q 1 /  Q 3 + Q 1

Years	Students of	Students of	Students of
	Accounts	Maths	Science
1	15	12	1
2	15	14	3
3	15	14	4
4	15	15	15
5	15	18	15
6	15	15	27
7	15	16	26
8	15	16	29
Sum	120	120	120
Average	15	15	15

Inner quartile range of Account students =    3(120+1/4) -  (120+1/4) 

= 93.75-31.25 =62.5

Inner quartile range of maths students =    3(120+1/4) -  (120+1/4) 

= 93.75-31.25 =62.5

Inner quartile range of science students =    3(120+1/4) -  (120+1/4) 

= 93.75-31.25 =62.5

3. The percentile range

4. Quartile deviation 

(II) Method of moments

1. Mean deviation

Mean deviation is also called average absolute deviation. It is the average of absolute deviatioin from its orginal data

 

Mean Deviation = Total ( X - Average )  / N

For example

Years	A - Average	M - Average	S - Average
1	0	-3	-14
2	0	-1	-12
3	0	-1	-11
4	0	0	0
5	0	0	0
6	0	0	12
7	0	1	11
8	0	1	14

Mean Deviation of Account Students = 0

Mean Deviation of Account Students = -3/8

Mean Deviation of Account Students = 0

2. Variance 

Variance is the measure of dispersion. It is finding following way

1. first deduct average from data and

2. then square of this data

3. Now adding this square data

4. Now dividing this added square data with total no. of value

Formula

Variance = Total ( X - Average ) ² / N

2. Standard deviation

Square root of variance will be standard deviation. 

Standard deviation = √Variance

3. Third moment of dispersion 

(B) Graphic Method of Measuring Dispersion

1. Lorenz Curve

Lorenz curve is used for graphically presentation of dispersion. It is tool of measurement of dispersion. In this, we show distribution of income and wealth. It is made by Lorenz in 1905 for showing the inequality of wealth distribution. In this x'axis, we show the share of people from lowest to highest income. On y'axis, we show the cumulative share of income earned.