Question

All the students of a class performed poorly in mathematics. The teacher decided to give grace marks of $10$ to each of the students. Which of the following statistical measures will not change even after the grace marks were given?
1). \[Mean\]
2). \[Median\]
3). \[Mode\]
4). \[Variance\]

Answer 1

Hint: We have to find the statistical measure whose value will not change even after the addition of \[10\] marks to each of the students . We solve this question using the concept of statistics . We should have the knowledge of the formulas of each of the quantities I.e. the mean , median , mode , variance etc . We can find the change In the old values by making changes in the actual condition of the various formulas .

Complete step-by-step solution:
Given :
Grace marks of \[10\] are added to each of the students .
Let us consider that there are \[N\] number of students in the class . Also , let the marks of the students in the class be \[{x_1}{\text{ }},{\text{ }}{x_2},{\text{ }}{x_3}{\text{ }},{\text{ }}{x_4}{\text{ }} \ldots \ldots \ldots \ldots ..{\text{ }},{\text{ }}{x_n}{\text{ }}.\]
 Now , starting with the statistical measures :
\[\left( 1 \right){\text{ }}Mean\;\;\]
We know that mean is defined as the average of the observations . It is the ratio of the total sum of the values of the given observations to the total number of observations . This gives us an average value for the whole set of observations .
We know the formula for mean of data , using the formula we can get the change in the mean value of the marks of the class .
Mean of marks of students \[\left( {{\text{ }}{M_1}{\text{ }}} \right){\text{ }} = {\text{ }}\dfrac{{\left[ {{\text{ }}{x_1}{\text{ }} + {\text{ }}{x_2}{\text{ }} + {\text{ }}{x_3}{\text{ }} + {\text{ }}{x_4}{\text{ }} \ldots \ldots \ldots \ldots ..{\text{ }} + {\text{ }}{x_n}{\text{ }}} \right]}}{n}\]
After adding ten grace marks , the new mean is :
New mean \[\left( {{\text{ }}{M_2}{\text{ }}} \right){\text{ }} = {\text{ }}\dfrac{{\left[ {{\text{ }}\left( {{x_1}{\text{ }} + {\text{ }}10} \right){\text{ }} + {\text{ }}\left( {{x_2}{\text{ }} + {\text{ }}10{\text{ }}} \right){\text{ }} + {\text{ }} \ldots \ldots \ldots \ldots {\text{ }} + {\text{ }}\left( {{\text{ }}{x_n}{\text{ }} + {\text{ }}10} \right){\text{ }}} \right]{\text{ }}}}{n}\]
On solving , we get
\[{M_2}{\text{ }} = {\text{ }}{M_1}{\text{ }} + {\text{ }}10\]i.e. the new mean changes .
\[\left( 2 \right)\]\[Mode\]
Mode is defined as the value which occurred for the maximum number of times in an observation . So , if \[\;X\] marks were the mode of the data , then the mode value of marks would also increase by $10$ . As the original marks increased by a value of $10$ for every student .
\[\left( 3 \right)\]\[Median\]
Median in the middle most value of the marks arranged in an ascending or descending order . So if the value of marks increases then the value of median marks also increases .
Also , there is a relation between mode median and mode I.e. \[3{\text{ }}median{\text{ }} = {\text{ }}mode{\text{ }} + {\text{ }}2{\text{ }}mean{\text{ }}.\]
Hence , if any one of these quantities change then the other two also change .
\[\left( 4 \right)\]\[Variance\]
Variance is stated as the sum of squares of the deviations from mean . In simple terms it is the sum of squares of the actual value subtracted from the mean value of the observations .
Thus , we conclude that the statistical term which will not change even after the grace marks is added to every student is variance of the data
Hence , the correct option is \[\left( 4 \right)\] .

Note: The value of mean , median and mode changes for every addition or subtraction of a value in the actual observation . It is also affected by the multiplication and division of terms by any of the constants . The variance and standard deviation of the data remains unaffected when a value is added or subtracted from the actual observations . But these get affected when terms are multiplied or divided by any of the constant terms .