Question

Calculate the mean deviation from the mean for the following data :

(i) 4, 7, 8, 9, 10, 12, 13, 17

(ii) 13,17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

(iii) 38, 70, 48, 40, 42, 55, 63, 46, 54, 44

(iv) 36, 72, 46, 42, 60, 45, 53, 46, 51, 49

(v) 57, 64, 43, 67, 49, 59, 44, 47, 61, 59

Answer 1

(i) $M.D=\frac{1}{n}{\sum }_{i=1}^n|d_i|,$ where $|d_i|=|x_i-\overline{x}|$

Let $\overline{x}$ be the mean of the given data.

$\overline{x}=\frac{4+7+8+9+10+12+13+17}{8}=10$

$\begin{array}{cc}{x}_i& |d_i|=|x_i-M|\\ 4& 6\\ 7& 3\\ 8& 2\\ 9& 1\\ 10& 0\\ 12& 2\\ 13& 3\\ 17& 7\\ \text{Total}& 24\end{array}M.D=\frac{1}{n}{\sum }_{i=1}^n|d_i|M.D=\frac{1}{8}\times 24=3$

(ii) Let $\overline{x}$ be the mean of the given data

$\overline{x}=\frac{13+17+16+14+11+13+10+16+11+13+12+17}{12}=14$

$\begin{array}{cc}{x}_i& |d_i|=|x_i-\overline{x}|\\ 13& 1\\ 17& 3\\ 16& 2\\ 14& 0\\ 11& 3\\ 13& 1\\ 10& 4\\ 16& 2\\ 18& 3\\ 12& 2\\ 17& 3\\ \text{Total}& 28\end{array}$

$M.D=\frac{1}{12}\times 28=2.33$

(iii) Let $\overline{x}$ be the mean of the given data

$\overline{x}=\frac{38+70+48+40+42+55+63+45+54+44}{10}=50$

$\begin{array}{cc}{x}_i& |d_i|=|x_i-\overline{x}|\\ 38& 12\\ 70& 20\\ 48& 2\\ 40& 10\\ 42& 8\\ 55& 5\\ 63& 13\\ 46& 4\\ 54& 4\\ 44& 6\\ \text{Total}& 84\end{array}$

$M.D=\frac{1}{10}\times 84=8.4$

(iv) Let $\overline{x}$ be the mean of the given data.

$\overline{x}=\frac{36+746+42+60+45+53+46+51+59}{10}=50$

$\begin{array}{cc}{x}_i& |d_i|=|x_i-\overline{x}|\\ 36& 14\\ 72& 22\\ 46& 4\\ 42& 8\\ 60& 10\\ 45& 5\\ 53& 3\\ 46& 4\\ 51& 1\\ 49& 1\\ \text{Total}& 72\end{array}$

$M.D=\frac{1}{10}\times 72=7.2$

(v) First arrange the given numbers in ascending order write these numbers in ascending order write these numbers in ascending order

57, 64, 43, 67, 49, 59, 44, 47, 61, 59

We get 43, 44, 47, 49, 57, 59, 61, 64, 67

Let X be the mean of given data, we get

$X=\frac{43+44+47+49+57+59+59+61+64+67}{10}=55$

Calculation of Mean Deviations from mean

$\begin{array}{cc}{x}_i& |d_i|=|x_i-\overline{x}|\\ 43& 12\\ 44& 11\\ 47& 8\\ 49& 6\\ 57& 2\\ 59& 4\\ 61& 6\\ 64& 9\\ 67& 12\\ \text{Total}& 74\end{array}M.D.=\frac{\sum d_i}{n}=\frac{74}{10}=7.4$