Problem 45 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately [2005] (A) \(22.0\) (B) \(20.5\) (C) \(25.5\) (D) \(24.0\)

Step-by-Step Solution

Answer

The mode is approximately 24.0, matching option (D).

Step 1: Understand the given information

We are given a mean of 21 and a median of 22. The task is to estimate the mode of the distribution using an empirical relationship between mean, median, and mode.

Step 2: Recall the empirical relationship

The empirical relationship between the mean, median, and mode is given by the equation:\[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]

Step 3: Substitute the values into the relationship

Substitute the given values of the mean (21) and median (22) into the formula:\[ \text{Mode} = 3 \times 22 - 2 \times 21 \]

Step 4: Calculate the mode

Perform the arithmetic calculations:\[ \text{Mode} = 66 - 42 = 24 \]

Step 5: Verify with options provided

The calculated mode is 24, which corresponds to option (D), 24.0.

Key Concepts

Frequency Distribution

A **frequency distribution** is a powerful statistical tool that provides a visual representation of data to help us understand its structure. It organizes raw data into a comprehensible format by displaying the frequencies — or counts — of different values in a dataset. This is often done using histograms or frequency tables.

• A frequency distribution gives insights into the shape of the data, such as whether it is symmetrical or skewed.
• It helps in identifying patterns, clusters, and outliers in the data.
• For example, if most values cluster around a particular point, the distribution is centered there.

Learning to compute and understand frequency distributions is foundational in statistics, providing a base for more complex analyses.

Mean Median Mode Relationship

In statistics, understanding the relationship between the mean, median, and mode is crucial. These are the **measures of central tendency** that summarize a data set with a single value that is representative of the dataset.

• The **mean** is the average of all data points, computed by dividing the sum of the data points by the number of points.
• The **median** is the middle value when the data points are arranged in ascending order. If there are an even number of observations, the median is the average of the two central numbers.
• The **mode** is the most frequently occurring value in a dataset.

These measures provide insight into the distribution of data. They can be interconnected using an empirical relationship which relates the three measures in a linear equation. For a moderately skewed distribution, the relationship is:

Empirical Formula

The formula, which applies to many datasets, is: \[\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}\] This relationship helps us to approximate one of the measures if the other two are known. For instance, the exercise uses this formula to estimate the mode of a frequency distribution based on given mean and median values.

Empirical Relationship in Statistics

The **empirical relationship in statistics** is a useful formula that connects the mean, median, and mode of a dataset. While this relationship is not a strict mathematical law, it is a widely-used heuristic that provides a quick estimation in many real-world situations.

• This empirical formula is especially useful when the precise mode is difficult to determine or when a distribution is skewed.
• It assumes a valid approximation when the distribution is moderately asymmetrical, which often occurs in natural data sets.
• The formula helps in estimating unknown measures that require less computation.

In our exercise, where calculating the exact mode is challenging or cumbersome, this empirical method shines by providing an efficient solution based on the values of mean and median. However, it is worth noting that this method works best for unimodal distributions that do not display extreme skewness.