If M is the median; then what is the value of 3M?

Question

Prepp Team · Accepted Answer

Finding the Median of Grouped Data

The question asks us to first find the median (M) from the given frequency distribution table and then calculate the value of 3M.

To find the median for grouped data, we follow these steps:

Calculate the total frequency ($\Sigma f$), which we denote as N.
Find the position of the median item, which is $\frac{N}{2}$.
Determine the median class. This is the class interval where the cumulative frequency is greater than or equal to $\frac{N}{2}$ for the first time.
Use the formula for the median of grouped data.

Calculating Total Frequency and Median Position

Let's look at the provided data table:

Class	40-50	50-60	60-70	70-80
Frequency (f)	4	3	1	2

The total frequency (N) is the sum of all frequencies:

$N = 4 + 3 + 1 + 2 = 10$

The position of the median item is $\frac{N}{2}$:

$\frac{N}{2} = \frac{10}{2} = 5$

So, the median is the value of the 5th item in the distribution.

Identifying the Median Class

Now we need to find which class interval contains the 5th item. We can do this by looking at the cumulative frequencies.

The 1st to 4th items are in the 40-50 class (CF = 4).
The 5th to (4+3=7th) items are in the 50-60 class (CF = 7).
The 8th item is in the 60-70 class (CF = 8).
The 9th to 10th items are in the 70-80 class (CF = 10).

Since the 5th item falls within the cumulative frequency range of the 50-60 class, the median class is 50-60.

Applying the Median Formula for Grouped Data

The formula for the median (M) of grouped data is:

$M = L + \frac{\frac{N}{2} - C}{f} \times h$

Where:

$L$ is the lower boundary of the median class. For the class 50-60, $L = 50$.
$\frac{N}{2}$ is the median position, which is 5.
$C$ is the cumulative frequency of the class *preceding* the median class. The class before 50-60 is 40-50, and its cumulative frequency is 4. So, $C = 4$.
$f$ is the frequency of the median class. The frequency of the 50-60 class is 3. So, $f = 3$.
$h$ is the class width. For the class 50-60, $h = 60 - 50 = 10$.

Now, substitute these values into the median formula:

$M = 50 + \frac{5 - 4}{3} \times 10$

$M = 50 + \frac{1}{3} \times 10$

$M = 50 + \frac{10}{3}$

$M = 50 + 3.333...$

$M = 53.333...$

This can also be written as $M = 53 \frac{1}{3}$.

Calculating 3M

The question asks for the value of 3M. We have calculated $M = 53 \frac{1}{3}$.

First, convert the mixed number to an improper fraction: $53 \frac{1}{3} = \frac{(53 \times 3) + 1}{3} = \frac{159 + 1}{3} = \frac{160}{3}$.

Now, multiply M by 3:

$3M = 3 \times \frac{160}{3}$

$3M = 160$

So, the value of 3M is 160.

Summary of Calculation

We found the median (M) of the grouped data to be $53 \frac{1}{3}$. Then, we calculated 3 times this median value, which resulted in 160.

Revision Table: Key Statistics Terms

Term	Definition	Used In Calculation
Frequency (f)	The number of times a value or range of values appears.	Summing frequencies to find N.
Total Frequency (N)	The sum of all frequencies in the distribution.	Used to find the median position ($N/2$).
Cumulative Frequency (C)	The running total of frequencies up to a certain class.	Used to identify the median class and in the median formula.
Median Class	The class interval containing the median value.	Identified using cumulative frequency. Its properties (L, f) are used in the formula.
Lower Boundary (L)	The lowest value in a class interval.	Used in the median formula.
Class Width (h)	The difference between the upper and lower boundaries of a class.	Used in the median formula.

Additional Information: Measures of Central Tendency

The median is one of the key measures of central tendency, which are values that represent the center or typical value of a dataset. Other important measures include the mean and the mode.

Mean: The average of all values in a dataset. Calculated by summing all values and dividing by the number of values. For grouped data, it's calculated using midpoints and frequencies.
Median: The middle value in a dataset when arranged in order. For grouped data, it's found using the formula demonstrated above. The median is less affected by extreme values (outliers) than the mean.
Mode: The value that appears most frequently in a dataset. For grouped data, the modal class is the class with the highest frequency, and the mode is estimated using a specific formula.

Understanding how to calculate these measures for both ungrouped and grouped data is fundamental in statistics and data analysis.

Prepp Team · Answer

$53 \frac{1}{3}$

Prepp Team · Answer

60

Prepp Team · Answer

180

Finding the Median of Grouped Data

Calculating Total Frequency and Median Position

Identifying the Median Class

Applying the Median Formula for Grouped Data

Calculating 3M

Summary of Calculation

Revision Table: Key Statistics Terms

Additional Information: Measures of Central Tendency

Similar Questions

Important Questions from Elementary Statistics

UPSC CDS 2026 Mock Test Series

CDS Mock Test Series

Start Your Preparation with Prepp Mobile App