# Mean Deviation for Grouped Data

### 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.