Mean Deviation for Grouped Data
Mean Deviation about the Mean: Recall the arithmetic mean (x̄) of a discrete frequency distribution with n data points is obtained using the formula: x̄ = (ⁿ∑i=₁ xi fi)/ ⁿ∑i=₁ fi = (ⁿ∑i=₁ xi fi)/ N.
Where, N = ⁿ∑i=₁ fi
Now the mean deviation about the mean i.e., M.D (x̄) is obtained by finding the absolute value of the derivative of the date points from the mean i.e. |xi – x̄| and using formula
M.D (mean) = (ⁿ∑i=₁ fi |xi – x̄|)/ ⁿ∑i=₁ fi = 1/N ⁿ∑i=₁ fi |xi – x̄|.
Example: Let us find mean deviation about the mean for the following data
xi | 2 | 5 | 7 | 8 | 10 |
35 |
fi | 6 | 8 | 10 | 6 | 8 |
2 |
Solution: We shall now construct following table to enable us to compute the required statistic.
xi |
fi | xi fi | |xi – x̄| | fi |xi – x̄| |
2 | 6 | 12 | 6 |
36 |
5 |
8 | 40 | 3 | 24 |
7 | 10 | 70 | 1 |
10 |
8 |
6 | 48 | 0 | 0 |
10 | 8 | 80 | 2 |
16 |
35 |
2 | 70 | 27 | 54 |
N = ⁿ∑i=₁ fi = 40 | ⁿ∑i=₁ xi fi = 320 |
140 |
A.M = x̄ = (ⁿ∑i=₁ xi fi)/ ⁿ∑i=₁ fi = 320/40 = 8.
M.D (mean) = (ⁿ∑i=₁ fi |xi – x̄|)/ ⁿ∑i=₁ fi = 1/N ⁿ∑i=₁ fi |xi – x̄| = 1/40 x 140 = 3.5.