Question

Find the mean deviation about the mean for the data
$38,70,48,40,42,55,63,46,54,44$

Answer 1

According to given in the question, we have to Find the mean deviation about the mean for the data $38,70,48,40,42,55,63,46,54,44$ so, first of all we have to find the mean of the given data with the help of the formula as given below:

Formula used: Mean $\overline x = \dfrac{{\sum {{x_i}} }}{n}.................(a)$
Where, $\sum {{x_i}}$is the sum of all the data given and n the total number of given data.
Now, we have to obtain the deviation of the respective observations from the mean obtained with the help of the formula (a) which can be obtained by ${x_i} - \overline x$
Now, we have to obtain the absolute values of the deviations which can be obtained by finding the mode of ${x_i} - \overline x$that is $\left| {{x_i} - \overline x } \right|$
So, to find the mean deviation about the mean for the data we have to use the formula as given below:
Formula used:
Mean deviation$= \dfrac{{\sum\limits_{i = 1}^{i = n} {\left| {{x_i} - \overline x } \right|} }}{n}....................(b)$
Hence, with the help of the formula above we can obtain the mean for the data.

Complete step-by-step solution:
Step 1: First of all we have to find the value of $\sum {{x_i}}$ which can be obtain by adding all the data given hence,
$\sum {{x_i}} = 38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44$
$\sum {{x_i}} = 500$
Step 2: Now, we have to find the value of mean with the help of the formula (a) as mentioned in the solution hint. Hence, on substituting all the values in the formula (a) where, n = 10
$\Rightarrow \overline x = \dfrac{{500}}{{10}} \\\ \Rightarrow \overline x = 50$
Hence, we have obtained the mean for the given data is $\overline x = 50$
Step 3: Now, we have to find the deviations of the respective observations with the help of the mean obtained and with the help of finding ${x_i} - \overline x$ as mentioned in the solution hint.

Data	${x_i} - \overline x$
38	$38 - 50 = - 12$
70	$70 - 50 = 20$
48	$48 - 50 = - 2$
40	$40 - 50 = - 10$
42	$42 - 50 = - 8$
55	$55 - 50 = 5$
63	$63 - 50 = 13$
46	$46 - 50 = - 4$
54	$54 - 50 = 4$
44	$44 - 5 = - 6$

Step 4: Now, we have to obtain the absolute values of the deviations which can be obtained by finding the mode of ${x_i} - \overline x$ that is $\left| {{x_i} - \overline x } \right|$ as mentioned in the solution hint.

Data | $\left| {{x_i} - \overline x } \right|$
---|---
38| $\left| { - 12} \right| = 12$
70| $\left| {20} \right| = 20$
48| $\left| { - 2} \right| = 2$
40| $\left| { - 10} \right| = 10$
42| $\left| { - 8} \right| = 8$
55| $\left| 5 \right| = 5$
63| $\left| {13} \right| = 13$
46| $\left| { - 4} \right| = 4$
54| $\left| 4 \right| = 4$
44| $\left| { - 6} \right| = 6$

Step 5: Now, with the help of the formula (b) as mentioned in the solution hint we can obtain the mean deviation about the mean for the data.
$\Rightarrow (\overline x ) = \dfrac{{\sum\limits_{i = 1}^{i = 10} {\left| {{x_i} - \overline x } \right|} }}{{10}}$
Hence, on substituting all the values,
$\Rightarrow (\overline x ) = \dfrac{{12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6}}{{10}} \\\ \Rightarrow (\overline x ) = \dfrac{{84}}{{10}} \\\ \Rightarrow (\overline x ) = 8.4$

Hence, with the help of formula (a) and formula (b) we have obtain the mean deviation about the mean for the data which is $(\overline x ) = 8.4$

Note: The mean of the absolute values of the numerical differences between the numbers of a set such as, a static data and their mean and median.
The mean deviation is defined as a statistical measure which is used to calculate the average deviation from the mean for the given data set.