• September 30th, 2016
• Posted by athanne

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 (Q3) and lower quartile (Q1).Quartile deviation is half of interquartile range.

i.e.

Interquartile range = Q3 – Q1

Quartile deviation (Q.D) =

Coefficient of Quartile deviation (Q.D)

Where:

Q3 = Upper quartile

Q1 = 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

Q1 = size of

Q1 = 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

1st quartile =  Q1 = size of n/4th item = 28/4th item which is the 7th item

7th item lies in 21 – 25th class

Q3 = size of 3n/4th item =

21st item lies in 31 – 35 class

3rd 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 dx2 f.dx2 Dx(P.M = 27) f.Dx f.D2x 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.