Problem 70 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Consider the following statements : (A) Mode can be computed from histogram (B) Median is not independent of change of scale (C) Variance is independent of change of origin and scale. Which of these is / are correct? (a) (A), (B) and (C) (b) Only(B) (c) Only (A) and (B) (d) Only (A)

Step-by-Step Solution

Answer

Only statement (A) is correct.

Step 1: Evaluating Statement A

Statement (A) claims that the mode can be computed from a histogram. A histogram provides a visual representation of data distribution, and the mode is the value that appears most frequently. By observing the highest bar in the histogram, the mode can indeed be identified. Therefore, statement (A) is correct.

Step 2: Evaluating Statement B

Statement (B) claims that the median is not independent of change of scale. When data is scaled, i.e., each data point is multiplied by a constant, the percentage relationship between the values remains the same, meaning the median stays proportionally consistent. Thus, the median is actually affected by scaling, making statement (B) incorrect since it claims the opposite.

Step 3: Evaluating Statement C

Statement (C) claims that variance is independent of change of origin and scale. When data is shifted (change of origin), variance remains unchanged. However, when data is scaled (each data point and the mean are multiplied by a factor), the variance is affected as it gets multiplied by the square of the factor. Thus, variance is not independent of change of scale, making statement (C) incorrect.

Key Concepts

Understanding the Mode

The mode is a statistical term that represents the most frequently occurring value in a data set. To grasp this concept, think of it as the number that "pops up" most often.
For example, in the data set \( \{2, 3, 3, 4, 5\} \), the mode is \(3\) because it appears more times than the other numbers.

• Simple: Just count how many times each number appears.
• Visual: In histograms, look for the tallest bar.

A histogram can easily show the mode by highlighting the tallest bar because this bar represents the data point occurring most frequently. So whenever you're asked to find the mode, look for that peak in the histogram. This makes determining the mode quick and often very straightforward, making it a handy tool not just in statistics but in real-world data analysis too.

Grasping the Median

The median is the middle value in a data set when the numbers are ordered from smallest to largest. It provides a way to understand the central tendency, or "middle point," of the data.

• Order: Always arrange the numbers first.
• Middle: If there is an odd number of observations, the median is the middle number; if it's even, it's the average of the two central numbers.

When you multiply or scale all the data points by a constant, all values including the median are affected proportionally. This meaning the median itself is altered by changes in scale. For example, if the original data set is \( \{1, 2, 3\} \) and we multiply each by \(2\), we obtain \( \{2, 4, 6\} \) with the new median being \(4\) instead of \(2\). Thus, scale impacts the median directly, keeping its place as the center point but shifting its value.

Variance Explained

Variance measures how much the values in a data set deviate from the mean, or how "spread out" they are. Understanding variance involves comprehending both consistency and variability in your data.

• Calculation: Compute the average of squared differences from the mean.
• Interpretation: Larger variance means greater spread.

When discussing changes of origin (shifting the data set by a constant), variance remains the same. Why? Because variance looks at differences, or spread, rather than absolute values. However, when you change the scale by multiplying all data by a factor, variance changes by the square of that factor. For example, if every number in your set is doubled, the variance increases by \(4\) (\(2^2\)).
This makes variance independent of changes in the origin but sensitive to changes in scale, affecting how the spread appears in modified data sets. It's a crucial concept in statistics that assists in understanding the underlying spread and consistency of data.