Calculating the Mean from a Cumulative Frequency Distribution

The problem asks us to find the mean of a distribution where the median value is given as 24. The distribution is presented in a "less than" cumulative frequency format, and we need to determine the value of an unknown constant '\(k\)' before calculating the mean.

Step 1: Understanding the Cumulative Frequency Table

The provided data represents cumulative frequencies. To work with it for median and mean calculations, we first need to determine the actual frequencies for each class interval and find the value of '\(k\)'.

The median value of 24 falls between 20 and 30. This means the median class is 20-30.

Let's represent the data in a more usable format:

Step 2: Calculating the value of 'k' using the Median

The median is given as 24. For a grouped frequency distribution, the median formula is:

Median \(= L + \frac{\frac{N}{2} - CF}{f} \times h\)

Where:

• \(L\) = Lower limit of the median class = 20
• \(N\) = Total number of students = Cumulative frequency of the last class = \(55 + k\)
• \(CF\) = Cumulative frequency of the class preceding the median class = 30
• \(f\) = Frequency of the median class = \(k\)
• \(h\) = Class width = 10

Substituting the values into the median formula:

\(24 = 20 + \frac{\frac{55+k}{2} - 30}{k} \times 10\)

Subtract 20 from both sides:

\(4 = \frac{\frac{55+k - 60}{2}}{k} \times 10\)

\(4 = \frac{k - 5}{2k} \times 10\)

\(4 = \frac{5(k - 5)}{k}\)

Cross-multiply:

\(4k = 5(k - 5)\)

\(4k = 5k - 25\)

Solve for \(k\):

\(k = 25\)

Step 3: Determining the Final Frequency Distribution

Now that we know \(k=25\), we can update the frequencies and find the total number of students (\(N\)):

• Total students \(N = 55 + k = 55 + 25 = 80\).

Let's create a table with class intervals, frequencies (\(f\)), mid-values (\(x\)), and the product \(f \times x\):

Step 4: Calculating the Mean

The formula for the mean of a grouped frequency distribution is:

Mean \(= \frac{\sum (f \times x)}{\sum f}\)

From the table above:

• \(\sum (f \times x) = 1970\)
• \(\sum f = N = 80\)

Therefore, the mean is:

Mean \(= \frac{1970}{80}\)

Mean \(= \frac{197}{8}\)

Mean \(= 24.625\)

The mean of the distribution is 24.625.