Problem 56 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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) mean (b) median (c) mode (d) variance

Step-by-Step Solution

Answer

Variance will not change after adding grace marks.

Step 1: Understanding the Problem

We need to determine which statistical measure remains unchanged when each student receives an additional 10 marks.

Step 1: Calculate Mean With Grace Marks

The mean is calculated by summing all values and dividing by the number of observations. Adding a constant to each value increases the sum by the constant multiplied by the number of observations, thus increasing the mean by the constant. Therefore, adding 10 to each score raises the mean by 10.

Step 2: Examine the Median Effect

The median is the middle value of a data set when arranged in order. Adding a constant to each value shifts the entire distribution by that constant, so the median also increases by that constant. Therefore, the median changes when all scores increase by 10.

Step 3: Analyze Mode Changes

The mode is the most frequently occurring value in the dataset. When 10 is added to each value, the distribution shifts, and the new number appearing most frequently becomes the old mode plus 10. Thus, the mode also changes by 10.

Step 4: Determine Variance

Variance measures the spread of data points around the mean. Adding a constant to each data point shifts all data by that constant, but the relative distance between data points remains unchanged. As variance is a measure of relative distance, adding a constant does not alter variance.

Key Concepts

Mean

The mean, often referred to as the average, is a measure used to describe a set of data by calculating the sum of all values divided by the number of values. It's a useful measure to determine the central tendency of the data. If a teacher decides to give every student in a class an additional 10 marks, this change affects the mean. For example, suppose the mean score initially is 50. When every score is increased by 10, the total sum of scores increases by the number of students times 10. Therefore, the new mean becomes 60. The mean is not resistant to adding a constant to each value, and it changes by the amount of the added constant.

Median

The median is the central value of a dataset when the numbers are arranged in an increasing or decreasing order. It acts as a robust measure of central tendency, especially when dealing with outliers or skewed distributions. When all students receive 10 additional marks, each score shifts up by 10 points, and the overall distribution of scores moves along the scale. However, the relative position of the scores remains unchanged, which means the exact middle point of the sequence also moves up by 10. As a result, the median is affected by the addition of a constant to each data point and increases by that constant.

Mode

The mode of a data set is the most frequently occurring value. It identifies the score that appears most often in the dataset. Similar to the mean and median, when a constant such as 10 is added to each score in a dataset, the value that was originally the mode moves up by the same constant. This means if the mode was originally 45, with the addition of 10 to each score, the new mode becomes 55. Like the mean and median, the mode changes by the added constant because the frequency of the mode simply shifts with the data.

Variance

Variance is a measure of how much the data in a dataset varies or spreads out from the mean. It is calculated by determining the average of the squared differences from the mean. Variance provides insight into the variability within a set of numbers. When you add the same constant to every data point, each point shifts equally, but the overall spread of the data relative to its own mean does not change. Since variance measures how data points differ from each other rather than from a fixed point, adding a constant does not affect these internal differences. Thus, even if the students receive 10 extra marks, the variance remains unchanged, as it does not depend on the specific values but rather on their dispersion or spread.