Measures of Dispersion - SS2 Mathematics Lesson Note
More information can be gotten through measures of dispersion, which measure how spread out the data items are from their collective center. They measure variability or deviation from what is assumed to the norm of the data, the central tendencies.
These include the range, mean absolute deviation, variance and standard deviation.
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Range is the difference between the lowest and highest scores in the set of data.
Mean absolute deviation is the arithmetic mean of the absolute values of difference of each score from the mean and dividing by their total frequency. \(MD = \ \frac{\sum_{}^{}{f|d|}}{\sum_{}^{}f}\), where \(|d|\) is the modulus of the deviations, \(\sum_{}^{}{f|d|}\) is the sum of the product of the modulus of the deviations and their frequencies and \(\sum_{}^{}f\) is the total frequency.
Standard deviation is how far or near a score is to the mean score. \(SD\ or\ \sigma = \ \sqrt{\frac{\sum_{}^{}{f{(X - \overline{X})}^{2}}}{\sum_{}^{}f}}\)
Variance is the square of the standard deviation. \(\sigma^{2} = \frac{\sum_{}^{}{f{(X - \overline{X})}^{2}}}{\sum_{}^{}f}\)
| \[f\] | \[cf\] | \[X\] | \[Xf\] | \[X - \overline{X} = d\] | \[d^{2}\] | \[fd^{2}\] | \[|d\text{|}\] | \[f|d|\] | |
|---|---|---|---|---|---|---|---|---|---|
| \[100\ –\ 81\] | \[67\] | \[67\] | \[90.5\] | \[\ = PRODUCT(LEFT)\ 6063.5\] | \[21.5\] | \[462.25\] | \[30,970.75\] | \[21.5\] | \[1,440.5\] |
| \[80\ –\ 61\] | \[34\] | \[67 + 34 = 101\] | \[70.5\] | \[\ = PRODUCT(LEFT)\ 2397.0\] | \[1.5\] | \[2.25\] | \[76.5\] | \[1.5\] | \[51\] |
| \[60\ –\ 41\] | \[56\] | \[101 + 56 = 157\] | \[50.5\] | \[\ = PRODUCT(LEFT)\ 2828.0\] | \[- 18.5\] | \[342.25\] | \[19,166\] | \[18.5\] | \[1,036\] |
| \[40\ –\ 21\] | \[12\] | \[157 + 12 = 169\] | \[30.5\] | \[\ = PRODUCT(LEFT)\ 366.0\] | \[- 38.5\] | \[1,482.25\] | \[17,787\] | \[38.5\] | \[462\] |
From example 1, the mean absolute deviation, \(MD = \frac{\sum_{}^{}{f|d|}}{\sum_{}^{}f} = \frac{2,989.5}{169} = 17.69\)
\[standard\ deviation,\ \sigma = \ \sqrt{\frac{\sum_{}^{}{f{(X - \overline{X})}^{2}}}{\sum_{}^{}f}} = \sqrt{\frac{68,000.25}{169}} = \sqrt{402.36834} = 20.06\]
\[variance,\ \sigma^{2} = {20.06}^{2} = 402.3683\]