Measures of Central Dispersion
The various Measures of Central Dispersion gives us one single value that represents the entire data. But average alone cannot adequately describe a set of observation, unless all the observation are alike. It’s therefore necessary to describe the variations or dispersions of the observations.
Dispersion of data is the degree to which numerical data tend to spread about an average value. It measures how much an average data tends to spread around an average value or measure of central tendency. It is the extent of scattered items around a measure of central tendency.
Significance of Measures of Central Dispersion
Measures of Central Dispersion are needed for four basic functions:
- To determine the reliability of average.
- To serve as basis for the control of variability.
- To compare two or more series with regard to their variability.
- To facilitate the use of other statistical measures.
Measures of Central Dispersion
The main methods of measuring Measures of Central Dispersion are;
- Range
- Quartile deviation or inter quartile range
- Mean deviation or average deviation
- Standard deviation
- Lorenz curve
The first two (Range and Quartile deviation or inter quartile range) are positional measures because they depend on the values at a particular position in the distribution. Mean deviation or average deviation and Standard deviation are calculated by employing all measures in calculation. The last one is a graphical method.
Range in Measures of Central Dispersion
Range in Measures of Central Dispersion is defined as the difference between the smallest and the largest value of a series.
For grouped data, the range is equal to the difference between the upper class boundary of the highest class and the lower class boundary of the lowest class.
Range (absolute value) = L – S
Coefficient of range =
Where:
L = the largest value of distribution
S = smallest value of distribution
Example:
The following data represents sales of news papers during a week by a vendor:
Day | Monday | Wednesday | Tuesday | Thursday | Friday | Saturday | Sunday |
Sales (Kshs) | 1200 | 600 | 2000 | 1500 | 1800 | 3600 | 4800 |
Find the range and coefficient of the range:
Solution:
Range = L – S where L = 4800 and S = 600
Range = 4800 – 600 = 4200
Coefficient range =
Example:
Calculate the range and coefficient of range from the following data:
Marks | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |
No. of students | 8 | 10 | 12 | 8 | 4 |
Range = L –S = 60.5 – 9.5 = 51
Coefficient of range
Quartile Deviation in Measures of Central Dispersion
Quartile Deviation in Measures of Central Dispersion is also known as semi interquartile range. Inter quartile range is the difference between upper quartile (Q_{3}) and lower quartile (Q_{1}).Quartile deviation is half of interquartile range.
i.e.
Interquartile range = Q_{3} – Q_{1}
Quartile deviation (Q.D) =
Coefficient of Quartile deviation (Q.D)
Where:
Q_{3 }= Upper quartile
Q_{1 }= Lower quartile
Q.D is an absolute measure of dispersion where coefficient of Q.D is a relative measure.
When Q.D is very small it describes high uniformity or small variation of the central 50% of the items and a high Q.D means that the variation among the central items is large.
Example 1
Find out the value of quartile deviation and its coefficient from the data of marks obtained in a class.
Marks 20, 28, 40, 12, 30, 15, 50
Q_{1} = size of
Q_{1} = 15
Coefficient of Q.D =
Example 2
Compute Q.D and coefficient of Q.D from the following data:
Marks | 10 | 20 | 30 | 40 | 50 | 60 |
No. of students | 4 | 7 | 15 | 8 | 7 | 2 |
Solution:
Marks Frequency Cumulative frequency
(x) (f) (C.F)
10 4 4
20 7 11
30 15 26
40 8 34
50 7 41
60 2 43
Q1 = size of
Q1 = 20
Coefficient of Q.D =
Example 3:
Compute Q.D and its coefficient from the following data
Classes | 16-20 | 21-35 | 26-30 | 31-35 | 36-40 |
Frequency | 3 | 6 | 9 | 7 | 3 |
C.F | 3 | 9 | 18 | 25 | 28 |
A quartile is a value below which a certain proportion of data will lie
1^{st} quartile = Q1 = size of n/4^{th} item = 28/4^{th} item which is the 7^{th} item
7^{th} item lies in 21 – 25^{th} class
Q3 = size of 3n/4^{th} item =
21^{st} item lies in 31 – 35 class
3^{rd} quartile class = 31-35
Inter quartile range = Q3 – Q1
= 32.64-23.83
=8.81
Coefficient of Q.D
Variance and Standard Deviation in Measures of Central Dispersion
Variance and Standard Deviation in Measures of Central Dispersion. Variance is the arithmetic mean of squared deviation from the mean. Standard deviation is the square root of the variance. If all the numbers in the sample are very close to each other, the standard deviation is close to zero. If the numbers are well dispersed the standard deviation will tend to be large. A small standard deviation means a high degree of uniformity of the observations as well as homogeneity of series and vice versa.
Example1
Calculate the standard deviation from the following data:
192 288 236 184 260 354 291 530 and 242
S. NO | X | ||
1 | 192 | -68 | 4624 |
2 | 288 | 28 | 784 |
3 | 236 | -24 | 576 |
4 | 229 | -31 | 961 |
5 | 184 | -76 | 5776 |
6 | 260 | 0 | 0 |
Example:
Compute the standard deviation and coefficient of variation from the following data:
Marks 0-10 10-20 20-30 30-40 40-50
No. of students 7 6 15 12 10
Solution:
Marks
x | f | Fx | dx | dx^{2} | f.dx^{2} | Dx(P.M = 27) | f.Dx | f.D^{2}x | |
0-10 | 5 | 7 | 35 | -22.4 | 501.76 | 3512.32 | -22 | -154 | 3388 |
10-20 | 15 | 6 | 90 | -12.4 | 153.76 | 922.56 | -12 | -72 | |
20-30 | 25 | 15 | 375 | -2.4 | 5.76 | 86.40 | -2 | -30 | |
30-40 | 35 | 12 | 420 | 7.6 | 56.76 | 693.12 | 8 | 96 | 768 |
40-50 | 45 | 10 | 450 | 17.6 | 309.7 | 3097.60 | 18 | 180 | 3240 |
50 | 1370 | 8312 | 20 | 8320 |
Lorenz Curve of Measures of Central Dispersion
Lorenz Curve of Measures of Central Dispersion.It shows graphical dispersion of the distribution of data.
Procedure
- Cumulative frequencies of the items of the series.
- Percentages of the various cumulated values taking the total as hundred.
- Making the horizontal scale from right to the left and the vertical scale from left to upward.
- Plot the points taking percentages for the various cumulated values along the y-axis and the items corresponding to them in the x-axis.
- Join the points by a free hand curve.
- Join the zero point on horizontal scale with 100 points on the vertical scale by straight line. This line is known as “the line of equal distribution”. This line indicates that there is no dispersion.
- The concavity of the curve away from the line of equal distribution is a measure of dispersion.
- Variability of the two distributions can be compared from the concavities of their curves.
Example 1
Draw a Lorenz curve for comparison of profits of two groups A and B in business.
Profit (shs million) Number of companies
A B
6 6 1
25 11 19
60 13 26
84 14 14
105 15 14
150 17 13
170 10 6
400 14 7
Profit
Group A | Group B | |||||||
Profit in (million) | Cumulative Profit | % of total | No. of companies | Cum. No. of companies | % no. of total | No. of Co | Cum. No. of co | % no of the total. |
6 | 6 | 0.6 | 6 | 6 | 6 | 1 | 1 | 1 |
25 | 31 | 3.1 | 11 | 17 | 17 | 19 | 20 | 20 |
60 | 91 | 9.1 | 13 | 30 | 30 | 26 | 46 | 46 |
84 | 175 | 17.5 | 14 | 44 | 44 | 14 | 60 | 60 |
105 | 280 | 28.0 | 15 | 59 | 59 | 14 | 74 | 74 |
150 | 430 | 43.0 | 17 | 76 | 76 | 13 | 87 | 87 |
170 | 600 | 60.0 | 16 | 86 | 86 | 6 | 93 | 93 |
400 | 1000 | 100 | 14 | 100 | 100 | 7 | 100 | 100 |
Taking the percentage of companies along x axis and percentage of profit along y axis Plot the points for the group A and the group B. Jon these by free hand curve. Since the curve of group B is further from the line of equal distribution, it represents greater inequality or dispersion.
Read more in Business Training in Kenya.
Conclusion of Measures of Central Dispersion
The Measures of Central Dispersion are also called measures of variation or measures of spread. When a measure of dispersion is expressed in the units of variables, it is called absolute measure of dispersion. If it’s expressed in the form of coefficient, ratio or percentage then is called relative measure of dispersion.Thus Measures of Central Dispersion.