Understanding the Median of Grouped Frequency Data

The median is the middle value in a dataset that has been ordered from least to greatest. For a grouped frequency distribution, we cannot find the exact median, but we can calculate an approximate value using a specific formula. The median is the value that divides the distribution into two equal halves.

First, let's look at the given grouped frequency distribution table showing weekly wages and the number of workers:

The table contains an unknown frequency represented by 'k'. To find the median, we first need the total number of observations (N) and the cumulative frequency. The cumulative frequency is the running total of frequencies. Let's add a cumulative frequency column.

The total number of workers, N, is the sum of all frequencies, which is \(35 + k\). The median corresponds to the value that lies at the \((\frac{N}{2})^{\text{th}}\) position. The median class is the class interval where the cumulative frequency is greater than or equal to \(\frac{N}{2}\) for the first time.

The formula for calculating the median of a grouped frequency distribution is:

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

Where:

• L is the lower boundary of the median class.
• N is the total frequency.
• CF is the cumulative frequency of the class preceding the median class.
• f is the frequency of the median class.
• h is the class width of the median class.

Finding the Value of k and the Median Class

Since the options for the median are around Rs. 3,300 to Rs. 3,500, the median value likely falls within the class interval 3050 - 3550. If this is the median class:

• L = 3050
• f = k
• CF (of the preceding class) = 15
• h = \(3550 - 3050 = 500\)
• \(N = 35 + k\)

The position \(\frac{N}{2}\) must lie within the cumulative frequency of this class (15 + k) and be greater than the cumulative frequency of the previous class (15).

Using the median formula and the provided answer options (specifically Rs. 3,383 which is the correct answer), we can infer the value of k. Let's assume the median is approximately 3383:

\(3383 \approx 3050 + \left(\frac{\frac{35+k}{2} - 15}{k}\right) \times 500\)

\(3383 - 3050 \approx \left(\frac{\frac{35+k}{2} - 15}{k}\right) \times 500\)

\(333 \approx \left(\frac{\frac{35+k-30}{2}}{k}\right) \times 500\)

\(333 \approx \left(\frac{\frac{5+k}{2}}{k}\right) \times 500\)

\(333 \approx \left(\frac{5+k}{2k}\right) \times 500\)

\(\frac{333}{500} \approx \frac{5+k}{2k}\)

\(0.666 \approx \frac{5+k}{2k}\)

\(0.666 \times 2k \approx 5+k\)

\(1.332k \approx 5+k\)

\(1.332k - k \approx 5\)

\(0.332k \approx 5\)

\(k \approx \frac{5}{0.332} \approx 15.06\)

Since frequency must be a whole number, k is most likely 15.

Calculating the Median with k = 15

Let's recalculate using k = 15.

Total frequency \(N = 35 + 15 = 50\).

\(\frac{N}{2} = \frac{50}{2} = 25\).

Now let's find the cumulative frequencies again with k=15:

• 2050 - 2550: 5
• 2550 - 3050: 15
• 3050 - 3550: \(15 + 15 = 30\)
• 3550 - 4050: \(30 + 8 = 38\)
• ... and so on

The value 25 falls in the class 3050 - 3550, as its cumulative frequency (30) is the first one greater than 25. So, the median class is indeed 3050 - 3550.

Using the median formula:

• L = 3050
• \(N/2 = 25\)
• CF = 15 (cumulative frequency before the median class)
• f = 15 (frequency of the median class)
• h = 500

\text{Median} = 3050 + \left(\frac{25 - 15}{15}\right) \times 500

\text{Median} = 3050 + \left(\frac{10}{15}\right) \times 500

\text{Median} = 3050 + \left(\frac{2}{3}\right) \times 500

\text{Median} = 3050 + \frac{1000}{3}

\text{Median} \approx 3050 + 333.33

\text{Median} \approx 3383.33

This value, approximately Rs. 3,383, closely matches one of the given options.

Revision Table: Key Terms for Median Calculation

Additional Information: Measures of Central Tendency

The median is one of the main measures of central tendency used to describe the center of a dataset. Other important measures include the Mean and the Mode.

• Mean: The average of all values in the dataset. Calculated by summing all values and dividing by the total number of values. For grouped data, an approximate mean can be calculated using class midpoints.
• Mode: The value that appears most frequently in the dataset. For grouped data, the mode lies in the class with the highest frequency (modal class), and its approximate value can be calculated using a formula.
• Comparison: The mean is affected by extreme values, while the median is not, making the median a better measure of central tendency for skewed distributions. The mode represents the most common value or class. Choosing the appropriate measure depends on the nature of the data and the goal of the analysis.