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.
A.Median
B.Mode
C.Variance
D.Mean

Answer 1

Hint: Here check all properties of median, mode, variance and mean. We should know the proper meaning of all four statistical measures.

Complete step-by-step answer:
We are given that all the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of $10$ to each of the students.
The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should first be arranged in order from least to greatest. In simple form the median is the middle number.
So, after grace marks are added the median will change.
The mode is the value which is repeatedly occurring in a given set. We can also say that the value or number in a data set, which has a high frequency or appears more frequently is called mode. It is one of the methods of measure of central tendency. The value occurring most frequently in a set of observations is its mode. In other words, the mode of data is the observation having the highest frequency in a set of data.
Here, also after adding grace marks mode will be changed.
Variance is the expected value of the squared variation of a random variable from its mean value, in probability and statistics. Informally, it estimates how far a set of numbers (random) are spread out from their mean value. Variance meaning – It is a measure of how data points differ from the mean. According to layman’s terms, it is a measure of how far a set of data (numbers) are spread out from their mean (average) value.
Here, \[Var\left( X\text{ }+\text{ }C \right)\text{ }=\text{ }Var\left( X \right),\text{ }\]where $c$ is constant.
We can say that, by adding grace marks, variance remains the same.
A mean is defined as the average of the given set of numbers. The mean is also considered as one of the measures of central tendencies in Statistics. It gives the central value of the set of values. A mean/arithmetic mean is defined as the average of the given numbers. It is the sum of all the given data values divided by the total number of data values given in the set.
Here, adding grace marks will give different meanings.
The correct answer is option(C).

Note: Here use properties of all four statistical measures. Important thing is about \[Var\left( X\text{ }+\text{ }C \right)\text{ }=\text{ }Var\left( X \right),\text{ }\]where after adding grace marks variance do not change.