Problem 43 - A Complete Resource Book in Mathematics for JEE Main

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) only (A) (B) only (B) (C) only (A) and (B) (D) \((\mathrm{A}),(\mathrm{B})\) and \((\mathrm{C})\)

Step-by-Step Solution

Answer

(C) only (A) and (B) are correct.

Step 1: Understanding Statement (A)

The mode is the value that appears most frequently in a data set. It can indeed be computed from a histogram, as the highest bar in a histogram represents the mode. Thus, statement (A) is correct.

Step 2: Understanding Statement (B)

The median is a measure of central tendency that is affected by the scale of the data but not by changes in origin (like adding a constant to each data point). When we change the scale, such as by multiplying each data point by a constant, the median changes, so it is not independent of scale. Therefore, statement (B) is correct.

Step 3: Understanding Statement (C)

Variance measures how much the data is spread out from the mean. It is not independent of change of scale since multiplying each data value by a constant requires multiplying the variance by the square of that constant. However, variance is independent of changes in origin. So, statement (C) is incorrect as it states independence from both origin and scale changes.

Step 4: Evaluate and Select the Correct Option

Based on the analysis, statements (A) and (B) are correct while statement (C) is incorrect. Therefore, the correct option that reflects this is (C) only (A) and (B).

Key Concepts

Mode and Histogram

The mode of a data set is the value that occurs most frequently. It is a measure of central tendency, much like the mean and median. In the context of a histogram, which is a graphical representation of data distribution, the mode can be easily identified. The tallest bar in the histogram corresponds to the mode, representing the data range with the highest frequency.

Histograms are made up of bars, where each bar height depicts the number of occurrences (frequency) of data within a particular interval. To find the mode using a histogram, simply look for the bar that outmatches others in height. This bar's interval is your mode. It is crucial to remember that a histogram can sometimes be unimodal, bimodal, or multimodal, indicating one mode, two modes, or multiple modes.

• A **unimodal** histogram has one clear peak.
• A **bimodal** histogram shows two distinct peaks, each representing a mode.
• A **multimodal** histogram features more than two peaks.

Identifying the mode from a histogram offers quick insights into the data's most common values without complicated calculations.

Median and Scale

The median of a dataset is another measure of central tendency. It is the middle value in a dataset arranged in ascending or descending order. Unlike the mode, the median is sensitive to scale changes but not to shifts in origin.

A change in **scale** refers to multiplying each value in the dataset by a constant. This transformation will also multiply the median by the same constant. However, if you change the **origin** by adding or subtracting a constant to every data point, it won't affect the relative order of the numbers. Hence, the median remains unchanged with such origin changes but alters with scale changes.

• When data is multiplied by a constant (change of scale), the median is multiplied by the same constant.
• Adding a constant (change of origin) does not affect the median.

Understanding the median's response to scale and origin changes is vital for analyzing how different transformations will affect data interpretation.

Variance and Origin/Scale

Variance measures how much the values in a dataset differ from the mean. It is a pivotal statistic in understanding data dispersion. Importantly, variance is affected differently by changes in origin and scale.

**Origin** changes, such as adding or subtracting a constant to each value in the data set, do not affect variance. This is because variance is a measure of spread and not location. However, a change in **scale**, which involves multiplying all data values by a constant, affects variance significantly. Specifically, if all data values are multiplied by a constant, the variance is multiplied by the square of that constant. This is because variance is based on squared deviations from the mean.

• Changing the origin by adding/subtracting a constant does not impact variance.
• Multiplying data by a constant changes variance, scaling it by the constant squared.

This concept is critical when performing operations on datasets because it illustrates how variance remains invariant to shifts but not to scaling, thus allowing a better grasp of data variability.