Answer

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**Hint**: This question is based on statistics. In this question a number of observations are given and we have to find the mean, median and mode for this observation. The mean of a data also known as the average is defined by the ratio of the sum of all the observations to the number of observations. The median of a data is the middle value when the data is arranged in ascending or descending order. For example, if there are $n$ number of terms then the formula for the median is given by –

${\rm{Median = }}\left( {\dfrac{{{\rm{n + 1}}}}{{\rm{2}}}} \right){\text{th observation}}$

When $n$ is an odd number

And,

${\rm{Median = }}\dfrac{{\left( {\dfrac{n}{2}} \right)th + \left( {\dfrac{n}{2} + 1} \right)th}}{2}{\rm{observation}}$

When $n$ is an even number.

The mode is defined as the observation having the highest number of occurrences.

**:**

__Complete step-by-step answer__Given:

The number of students $ = 15$

The marks of the students

$41,39,48,52,46,62,54,40,96,52,98,40,42,52,60$

The mean of the data given can be calculated in the following way –

${\rm{Mean = }}\dfrac{{{\rm{Sum of all the marks of the students}}}}{{{\rm{Number of the students}}}}$

Substituting the values in the formula we get,

$

{\rm{Mean}} = \dfrac{{41 + 39 + 48 + 52 + 46 + 62 + 54 + 40 + 96 + 52 + 98 + 40 + 42 + 52 + 60}}{{15}}\\

= \dfrac{{822}}{{15}}\\

= 54.8

$

Now to calculate the median of the data given, rearranging the data given in the ascending order, we get,

$39,40,40,41,42,46,48,52,52,52,54,60,62,96,98$

So, total number of the data observations $n = 15$

Since the number of observations $n$ is an “odd number” we use the formula for the median of the odd number of observations. The formula is given by –

${\rm{Median = }}\left( {\dfrac{{{\rm{n + 1}}}}{{\rm{2}}}} \right){\text{th observation}}$

Substituting the value of $n$ in the formula we get,

$

{\rm{Median = }}\left( {\dfrac{{15 + 1}}{2}} \right){\text{th observation}}\\

{\text{ = 8th observation}}

$

And we know from the data given in the ascending order that the $8{\rm{th}}$ observation is the number 52

So,

${\rm{Median = 52}}$

Now to calculate the mode of this data we have to find the number with the maximum number of occurrences or the frequency.

By observation we found that -

The number with the highest frequency is 52 (total 3 times).

So, the mode of these observations is –

${\rm{Mode = 52}}$

Therefore, the mean of the data is 54.8, the median for the data is 52 and the mode for the data is 52.

**Note**: It should be noted that depending upon the distribution of the data in the given number of observations the value for the mean, median and mode changes. For example, if the data distribution is symmetrical about the centre then it is a normal distribution and for this distribution the value of mean, median and mode is equal.

So, for normal distribution –

Mean = Median = Mode

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