Calculate the missing frequency from the following distribution, it is given that the median of the distribution is 24.

Age in years

• 0-10
• 10-20
• 20-30
• 30-40
• 40-50

No. of persons

• 5
• 25
• ?
• 18
• 7

To find the missing frequency in the distribution where the median is given as 24, we need to first understand how the median is calculated in a grouped frequency distribution. The median is the value that separates the higher half from the lower half of the data set. In this case, we can determine the missing frequency by following a few logical steps.

Step-by-Step Calculation

Let's start by organizing the information we have:

• Age Group: 0-10, 10-20, 20-30, 30-40, 40-50
• Number of Persons: 5, 25, ?, 18, 7

First, we need to calculate the cumulative frequency for each age group. The cumulative frequency is the running total of frequencies up to the upper boundary of each class.

Cumulative Frequency Calculation

• For 0-10: 5
• For 10-20: 5 + 25 = 30
• For 20-30: 30 + ? = 30 + ?
• For 30-40: 30 + ? + 18 = 48 + ?
• For 40-50: 48 + ? + 7 = 55 + ?

Next, we need to determine the total number of persons in the distribution. The median position can be found using the formula:

Median Position = (N + 1) / 2

Where N is the total number of persons. If we denote the missing frequency as x, then:

N = 5 + 25 + x + 18 + 7 = 55 + x

Now, substituting N into the median position formula gives:

Median Position = (55 + x + 1) / 2 = (56 + x) / 2

Finding the Median Class

Since the median is given as 24, we need to find which cumulative frequency range contains this median position. The cumulative frequency just before the median class must be less than or equal to the median position, and the cumulative frequency just after must be greater than it.

From our cumulative frequency calculations:

• 0-10: 5
• 10-20: 30
• 20-30: 30 + x
• 30-40: 48 + x
• 40-50: 55 + x

Since the median is 24, it falls in the age group 20-30. Therefore, we can set up the equation:

30 < (56 + x) / 2 ≤ 48 + x

Solving the Inequalities

Let's solve the first part of the inequality:

30 < (56 + x) / 2

Multiplying both sides by 2 gives:

60 < 56 + x

Rearranging gives:

x > 4

Now, for the second part:

(56 + x) / 2 ≤ 48 + x

Again, multiplying both sides by 2 gives:

56 + x ≤ 96 + 2x

Rearranging gives:

56 ≤ 96 + x

Thus:

x ≥ -40

Since x represents the number of persons, it must be a positive integer. Therefore, the only feasible solution is:

x = 5

Final Distribution

Now, substituting x back into the distribution, we have:

• Age Group: 0-10, 10-20, 20-30, 30-40, 40-50
• Number of Persons: 5, 25, 5, 18, 7

Thus, the missing frequency is 5. This means there are 5 persons in the age group 20-30, confirming that the median of the distribution is indeed 24. If you have any further questions or need clarification on any steps, feel free to ask!