Question

Find the mean deviation of the data about its mean:
3, 10, 10, 4, 7, 10, 5.

Answer 1

Hint: We will first calculate the number of terms we have with us. Then, we will calculate the mean of the given data. After that we will use the formula of mean deviation about the mean and put in the values to get the required answer.

Complete step-by-step solution:
We have 3, 10, 10, 4, 7, 10, 5 that is in total 7 terms with us.
Therefore, we have n = 7.
Now, let us find the mean.
We know that we have the formula of mean given by:-
Mean of the data is the sum of observations divided by the number of observations given to us.
Therefore, we will get after putting in the values:-
$ \Rightarrow Mean = \dfrac{{3 + 10 + 10 + 4 + 7 + 10 + 5}}{7}$
On solving the RHS of the above expression, we will get:-
$ \Rightarrow Mean = \dfrac{{49}}{7}$
On solving the RHS of the above expression further, we will get:-
$ \Rightarrow Mean = 7$
$\therefore $ we have $Mean = \bar x = 7$.
Now, let us first note the formula of standard deviation which is given by the following expression:-
Mean deviation from the mean is given by $\dfrac{{\sum {|x - \bar x|} }}{n}$.
Now, let us put in the values from the data we have.
We will get the following expression:-
Mean deviation from the mean is given by:
$\dfrac{{|3 - 7| + |10 - 7| + |10 - 7| + |4 - 7| + |7 - 7| + |10 - 7| + |5 - 7|}}{7}$.
On simplifying the values inside the modulus sign in the numerator of the RHS, we will get:-
Mean deviation from the mean is given by $\dfrac{{| - 4| + |3| + |3| + | - 3| + |0| + |3| + | - 2|}}{7}$.
Now, we also know that $|x| = \left\{ {\begin{array}{*{20}{c}}
  {x,x \geqslant 0} \\
  { - x,x < 0}
\end{array}} \right.$
$\therefore $ Mean deviation from the mean is given by $\dfrac{{4 + 3 + 3 + 3 + 0 + 3 + 2}}{7}$.
Now, simplifying the values on the numerator, we will then get:-
$\therefore $ Mean deviation from the mean is given by $\dfrac{{18}}{7}$.

$\therefore $ Mean deviation from the mean is given by 2.57.

Note: The students must know what does the term “Mean deviation” refers to and what does it represent in a given data.
It basically tells us about how far the values are from the middle. Because, we are basically calculating the distances of all the observations from the mean. It is also sometimes known as Absolute Deviation because we are calculating the absolute values of deviation. (Absolute refers to the modulus here).
The students must also know that we can calculate mean deviation from a point always because we need to calculate the distances of observation from some point only.