MEASURES OF DISPERSION
This also known as measures of spread or variation describes how the data given in any distribution are spread about the ‘Mean’, or the overall spread of the data. These measures are the range, mean – deviation, standard deviation, variance, coefficient of variation, etc.
The Range: The range of a data is the difference between the highest and the lowest value in the data. The formula for calculation of range is:
Range = Highest value – Lowest value
The Mean Deviation: This measures the dispersions around the arithmetic mean. It tells us how far, on the average, the individual observations are from the mean. For a grouped frequency distribution
Mean Deviation = ∑f/x- x/
∑f
Variance: This is the average of the square about the deviations of the measurement about their mean.
Variance = ∑f(X – X)2
∑f
Standard Deviation: This is the square root of the average of the square of the deviations of the measurement about their mean.
Standard Deviation =
∑f(X – X )2
Ʃf
Example:
The data below shows the weight of 50 students to the nearest kg.
65 58 51 36 23 40 53 59 70 51 46 59 50 67 46 39 61 62 73 60 71 51 47 32 48 40 40 51 58 67 60 69 43 52 37 26 38 50 59 40 44 54 42 68 74 45 39 48 55.
- Prepare a grouped frequency table
- Calculate:
- The range
- The mean deviation
- The variance
- The standard deviation
NB: Note that the standard Deviation is the positive square root of the variance.
|
Class |
F |
Mid-point X |
FX |
_
/X- X/ |
_
F/X-X/ |
_
(X-X )2 |
_
F(X-X)2
|
| 21-30 | 2 | 25.5 | 51 | 25.5 | 50.4 | 635.04 | 1270.08 |
| 31-40 | 10 | 35.5 | 355 | 15.2 | 152 | 231.04 | 2310.40 |
| 41-40 | 12 | 45.5 | 546 | 5.2 | 62.4 | 27.04 | 324.48 |
| 51-60 | 15 | 55.5 | 832.5 | 4.8 | 72 | 23.04 | 345.60 |
| 61-70 | 8 | 65.5 | 524 | 14.8 | 118.4 | 219.04 | 1752.32 |
| 71-80 | 3 | 75.5 | 226.5 | 24.8 | 74.4 | 615.04 | 1845.12 |
| 50 | 2535 | 529.60 | 7848 |
∑fx = 2535 = 50.7
∑f 50
- Range = Highest score –Lowest score
= 74 – 23
= 51kg
- Mean Deviation
=∑f/ X –X/ = 529.60 = 10.59kg
∑f 50
(iii) Variance = ∑f/X – X /2
∑f
= 7848 = 156.9kg
50
(iv) Standard Deviation
∑f/X – X /2 156.96
∑f = =12.53kg
EVALUATION QUESTION
- How is the mid-point (X) ascertained in a grouped frequency table?
- State two advantages of the mean over other measures of central tendency (i.e median and mode)
- The table below shows the income of forty Workers in a factory in N
61 78 70 83 92 67 66 83
76 68 79 84 82 86 81 60
78 77 86 77 81 92 80 70
70 40 75 60 74 82 77 87
63 94 76 87 81 77 87 84.
Using a class interval of 40-49, 50-59 etc
- Construct a grouped frequency table of the distribution.
- Calculate the mean of the distribution.
GENERAL EVALUATION QUESTIONS
- Give five reasons why government participates in business enterprises.
- Define ageing population.
- Explain the sources of finance available to a public limited liability business.
- Explain any three weapons that can be used by a trade union during trade
- What is occupational mobility?
WEEKEND ASSIGNMET
- In a class of 5 students the following scores were obtained in a mathematics test 10,2,6,7,12. What is the median score (A) 2 (B) 4(C) 6 (D) 7
- Which measure of central tendency can be applied to find the highest goal scorer in a football match (A) mean (b) Mode (C) Median (D) range
- If the following scores 2, 4, 6, and 10 with frequencies 3,5,7, and 10 respectively makes up a distribution, then the mean score is (A) 5.7 (B) 6.7 (C) 7.5 (D) 5.4
- In the formula X = ∑fx
∑f
∑fX stands for (A) total number of observations (B) Sum of frequencies times observations (C) Sum of frequencies (D) number of elements times frequencies
- The most reliable measure of central tendency is (a) Mode (b) median (c) Histogram (d) mean
SECTION B
- Give the definition of the mean deviation and the standard deviation
- Explain the difference between continuous data and discrete data
See also