Question

How do you find the mean, median and mode of $ - 500,350,475, - 300, - 500,450,425, - 400$?

Answer 1

Hint: We will apply the various formulas for the central tendencies to get the mean, median and mode for the given distribution. On doing some simplification we get the required answer.

Formula used: ${\text{Mean = }}\dfrac{{{\text{Sum of terms}}}}{{{\text{number of terms}}}}$
\[\begin{array}{*{20}{c}}
  {{\text{Median = }}{{\left[ {\dfrac{{{\text{n + 1}}}}{{\text{2}}}} \right]}^{th}}{\text{ term if n is odd,}}} \\
  {{\text{ = }}\left[ {\dfrac{{{{\left[ {\dfrac{{\text{n}}}{{\text{2}}}} \right]}^{th}}{\text{ term + }}{{\left[ {\dfrac{{\text{n}}}{{\text{2}}} + 1} \right]}^{th}}{\text{ term}}}}{{\text{2}}}} \right]{\text{ if n is even}}{\text{.}}}
\end{array}\]

Complete step-by-step solution:
Now to find the mean we have the formula as:
${\text{Mean = }}\dfrac{{{\text{Sum of terms}}}}{{{\text{number of terms}}}}$
Now the total numbers of terms in the distribution are $8$, therefore on substituting the sum of all the terms and the number of terms in the formula we get:
$\Rightarrow$${\text{Mean = }}\dfrac{{ - 500 + 350 + 475 - 300 - 500 + 450 + 425 - 400}}{8}$
Now on adding all the terms in the numerator we get:
$\Rightarrow$${\text{Mean = }}\dfrac{0}{8}$
Therefore, ${\text{Mean = 0}}$
Now to find the median, we have to first arrange all the terms in ascending order, it can be written as:
$ - 500, - 500, - 400, - 300,350,425,450,475$
Now to find the median the formula is:
\[\begin{array}{*{20}{c}}
$\Rightarrow${{\text{Median = }}{{\left[ {\dfrac{{{\text{n + 1}}}}{{\text{2}}}} \right]}^{th}}{\text{ term if n is odd,}}} \\
{{\text{ = }}\left[ {\dfrac{{{{\left[ {\dfrac{{\text{n}}}{{\text{2}}}} \right]}^{th}}{\text{ term + }}{{\left[ {\dfrac{{\text{n}}}{{\text{2}}} + 1} \right]}^{th}}{\text{ term}}}}{{\text{2}}}} \right]{\text{ if n is even}}{\text{.}}}
\end{array}\]
Since the total numbers of terms in the distribution are $8$, which is an even number, the formula of median is:
$\Rightarrow$\[{\text{Median = }}\left[ {\dfrac{{{{\left[ {\dfrac{{\text{n}}}{{\text{2}}}} \right]}^{th}}{\text{ term + }}{{\left[ {\dfrac{{\text{n}}}{{\text{2}}} + 1} \right]}^{th}}{\text{ term}}}}{{\text{2}}}} \right]\]
On substituting we get:
$\Rightarrow$\[{\text{Median = }}\left[ {\dfrac{{{{\left[ {\dfrac{8}{{\text{2}}}} \right]}^{th}}{\text{ term + }}{{\left[ {\dfrac{8}{{\text{2}}} + 1} \right]}^{th}}{\text{ term}}}}{{\text{2}}}} \right]\]
On simplifying we get:
$\Rightarrow$\[{\text{Median = }}\left[ {\dfrac{{{{\left[ 4 \right]}^{th}}{\text{ term + }}{{\left[ 5 \right]}^{th}}{\text{ term}}}}{{\text{2}}}} \right]\]
On substituting the value of the $4th$ and $5th$ term, we get:
$\Rightarrow$\[{\text{Median = }}\left[ {\dfrac{{{\text{ - 300 + 350}}}}{{\text{2}}}} \right]\]
On simplifying we get:
$\Rightarrow$\[{\text{Median = }}\dfrac{{50}}{2}\]
Therefore, the median is: $25$.
Now mode is the term which is presented the greatest number of times.
In the above distribution $ - 500$ is present $2$ times and all the other elements are present only $1$ time, the mode is $ - 500$.

The mean of the distribution is: $0$
The median of the distribution is: $25$
The mode of the distribution is: $ - 500$.

Note: A place where the student could go wrong is while writing the correct formula of median; the formula of median is dependent on the number of terms in that distribution.
Mode is to be used when the given distribution is nominal, which implies that the class or category is not based on numbers for example: city, age, gender etc.
Mean is the most commonly used measure of central tendency and it is usually considered to be the measure of it. Mean should not be used when the data is non-numeric or when there are extreme values in the distribution.