Calculating the Median for Grouped Frequency Data

The question asks us to find the median of the given grouped frequency distribution. The median is the middle value in a dataset that is ordered from least to greatest. For grouped data, we first need to find the median class and then use a specific formula to calculate the exact median value within that class.

Understanding the Given Grouped Frequency Distribution

Here is the provided distribution:

Steps to Calculate the Median of Grouped Data

To calculate the median for this grouped frequency distribution, we follow these steps:

1. Find the total number of observations (N), which is the sum of all frequencies.
2. Calculate \( \frac{N}{2} \). The median corresponds to the value at the \( \frac{N}{2} \)-th position.
3. Create a cumulative frequency (CF) column. Cumulative frequency for a class is the sum of the frequencies of that class and all preceding classes.
4. Identify 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.
5. Use the formula for the median of grouped data:

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

   Where:
   • \( L \) is the lower limit of the median class.
   • \( N \) is the total frequency.
   • \( C \) is the cumulative frequency of the class just preceding the median class.
   • \( f \) is the frequency of the median class.
   • \( h \) is the class size (upper limit - lower limit).
6. Substitute the values into the formula and compute the median.

Performing the Median Calculation

Let's apply the steps to our frequency distribution.

1. Total frequency \( N \):
\( N = 1 + 2 + 4 + 6 + 4 + 3 = 20 \)

2. Calculate \( \frac{N}{2} \):
\( \frac{N}{2} = \frac{20}{2} = 10 \)

The median is the value corresponding to the 10th observation.

3. Create the cumulative frequency column:

4. Identify the median class: The first cumulative frequency greater than or equal to 10 is 13, which falls in the class interval 30-40. Therefore, the median class is 30-40.

5. Identify the values for the formula from the median class (30-40):

• Lower limit of the median class, \( L = 30 \)
• Total frequency, \( N = 20 \)
• \( \frac{N}{2} = 10 \)
• Cumulative frequency of the class preceding the median class (20-30), \( C = 7 \)
• Frequency of the median class (30-40), \( f = 6 \)
• Class size, \( h = 10 - 0 = 10 \) (or 20-10, 30-20, etc.)

6. Apply the formula:

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

\( \text{Median} = 30 + \left( \frac{10 - 7}{6} \right) \times 10 \)

\( \text{Median} = 30 + \left( \frac{3}{6} \right) \times 10 \)

\( \text{Median} = 30 + \left( 0.5 \right) \times 10 \)

\( \text{Median} = 30 + 5 \)

\( \text{Median} = 35 \)

Thus, the median of the given grouped frequency distribution is 35.

Revision Table: Grouped Data Median

Additional Information on Measures of Central Tendency

The median is one of the main measures of central tendency, which are used to describe the center of a dataset. Other common measures include the mean and the mode.

• Mean: The arithmetic average of the data. Calculated by summing all values and dividing by the total number of values. For grouped data, it involves using class marks.
• Mode: The value that appears most frequently in the dataset. For grouped data, it is estimated using a formula after identifying the modal class (the class with the highest frequency).
• Median: The middle value when the data is ordered. It is less affected by extreme values (outliers) than the mean.

Choosing the appropriate measure of central tendency depends on the type of data and the distribution's shape. For skewed distributions, the median is often preferred over the mean as a representative center.