Measures Of Dispersion SS2 Economics Lesson Note

MEASURES OF DISPERSION  OF A GROUPED FREQUENCY DISTRIBUTION

MEASURES OF DISPERSION:  

They are also known as measures of spread or variation that describe 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.

1. The Range: The range of data is the difference between the highest and the lowest value in the data. The formula for the calculation of the range is:

Range = Highest value – Lowest value

2. The Mean Deviation: This measures the dispersions around the arithmetic mean. It tells us how far, on average, the individual observations are from the mean.  For a grouped frequency distribution 

           Mean Deviation =✓∑f/x- x/  

                                ✓∑f               

3. Variance: This is the average of the square about the deviations of the measurement about their mean.

Variance = ∑f(X – X)2

            ∑f                 

4. Standard Deviation: This is the square root of the average of the squares 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: 

1. The range 
2. The mean deviation 

iii. The variance 

1. The standard deviation 

NB: Note that the standard Deviation is the positive square root of the variance.

∑fx = 2535 = 50.7

∑f        50

Range = Highest score –Lowest score 

= 74 – 23 = 51 kg

 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                                 

✓∑f

✓156.96

✓50                                                    =12.53kg 

ASSIGNMENT 

1. How is the midpoint (X) ascertained in a grouped frequency table? 
2. State two advantages of the mean over other measures of central tendency (i.e median and mode)
3. The table below shows the income of 40 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 

1. Construct a grouped frequency table of the distribution.
2. Calculate the mean of the distribution.