Question

The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from the mean is: -
(a) 2
(b) 2.57
(c) 3
(d) 3.75

Answer 1

Hint: Apply the formula for mean deviation given as: - Mean deviation = \[\dfrac{1}{n}\sum\limits_{i=1}^{n}{\left| \overline{x}-{{x}_{i}} \right|}\], where ‘n’ is the number of observations, \[\overline{x}\] is the mean of the given data and \[{{x}_{i}}\] are the given observations where i = 1, 2, ……, n. To find the value of mean use the formula \[\overline{x}=\dfrac{\sum{{{x}_{i}}}}{n}\]. Substitute all the values in the above formula with n = 7 to get the required mean deviation.

Complete step-by-step answer:
Here, we have been provided with the data 3, 10, 10, 4, 7, 10, 5 and we are asked to determine the mean deviation of these data from its mean.
Now, we know that the formula used to calculate the mean deviation is given as: - Mean deviation = \[\dfrac{1}{n}\sum\limits_{i=1}^{n}{\left| \overline{x}-{{x}_{i}} \right|}\], where \[\overline{x}\] is the mean of the given data, ‘n’ is the number of observations and \[{{x}_{i}}\] are the given data where i = 1, 2, .….,n. So, to find the mean deviation we need to find the mean of the given data first. The mean is given by the formula: - \[\overline{x}=\dfrac{\sum{{{x}_{i}}}}{n}\]. On counting the number of data given to us, we conclude that n = 7. So, we get,
\[\begin{align}
  & \Rightarrow \overline{x}=\dfrac{3+10+10+4+7+10+5}{7} \\
 & \Rightarrow \overline{x}=\dfrac{49}{7} \\
 & \Rightarrow \overline{x}=7 \\
\end{align}\]
Therefore, the mean of the given data is 7. Now, substituting the value of mean in the formula for mean deviation, we get,
\[\Rightarrow \] Mean deviation = \[\dfrac{1}{7}\sum\limits_{i=1}^{7}{\left| 7-{{x}_{i}} \right|}\]
\[\Rightarrow \] Mean deviation = \[\dfrac{1}{7}\left[ \left| 7-3 \right|+\left| 7-10 \right|+\left| 7-10 \right|+\left| 7-4 \right|+\left| 7-7 \right|+\left| 7-10 \right|+\left| 7-5 \right| \right]\]
\[\Rightarrow \] Mean deviation = \[\dfrac{1}{7}\left[ \left| 4 \right|+\left| -3 \right|+\left| -3 \right|+\left| 3 \right|+\left| 0 \right|+\left| -3 \right|+\left| 2 \right| \right]\]
\[\Rightarrow \] Mean deviation = \[\dfrac{1}{7}\left[ 4+3+3+3+0+3+2 \right]\]
\[\Rightarrow \] Mean deviation = \[\dfrac{1}{7}\left[ 18 \right]\]
\[\Rightarrow \] Mean deviation = 2.57
So, the correct answer is “Option b”.

Note: One must remember the formulas of mean and mean deviation to solve the above question. Do not forget to consider the modulus sign otherwise you will get the value of mean deviation equal to 0 and it will be considered as incorrect. Remember that the modulus of any number, whether negative or positive, is always positive. Count the number of observations carefully.