Understanding the Median of a Frequency Distribution

The question asks us to find the median of a given frequency distribution. A frequency distribution shows how often each distinct value occurs in a dataset. The median is the middle value of a dataset when it is arranged in ascending order. For a frequency distribution, finding the median involves calculating the total number of observations and then identifying the value that corresponds to the middle position.

The given frequency distribution is:

• Values (x): 1, 2, 3, 5
• Frequencies (f): 4, 6, 9, 7

This means the value '1' appears 4 times, the value '2' appears 6 times, the value '3' appears 9 times, and the value '5' appears 7 times.

Calculating the Total Frequency

The total number of observations, denoted by $\Sigma f$ or $N$, is the sum of all frequencies:

\( N = \Sigma f = 4 + 6 + 9 + 7 = 26 \)

So, there are a total of 26 observations in this dataset.

Determining the Median Position

For a discrete frequency distribution, the median is located at the position calculated using the formula:

Median Position \( = \frac{N+1}{2} \)

Substituting the total frequency $N = 26$:

Median Position \( = \frac{26+1}{2} = \frac{27}{2} = 13.5 \)

This means the median is the value located at the 13.5th position when the data is arranged in ascending order.

Finding the Median Value using Cumulative Frequency

To find the value at the 13.5th position, we need to calculate the cumulative frequencies. Cumulative frequency (CF) for a value is the sum of its frequency and the frequencies of all preceding values. This helps us determine which value corresponds to a given position.

We are looking for the value at the 13.5th position. Let's look at the cumulative frequencies:

• The value '1' covers positions 1 through 4.
• The value '2' covers positions 5 through 10.
• The value '3' covers positions 11 through 19.
• The value '5' covers positions 20 through 26.

The 13.5th position falls within the range of positions covered by the value '3' (which is positions 11 to 19). Therefore, the median value is 3.

Conclusion: Median of the Distribution

Based on the calculation of the median position and the cumulative frequency table, the value at the 13.5th position is 3.

The median of the frequency distribution is 3.

Revision Table: Key Statistics Concepts

Additional Information: Handling Different Median Positions

When calculating the median position using \( \frac{N+1}{2} \):

• If $N$ is odd, the median position will be a whole number (e.g., 5th position). The median is the value exactly at that position.
• If $N$ is even, the median position will be a number ending in .5 (e.g., 13.5th position). For discrete data like this, the median is typically the value corresponding to the cumulative frequency that is greater than or equal to this position. If the position falls exactly on a cumulative frequency value, the median is usually taken as the average of the value corresponding to that CF and the value corresponding to the next CF. However, for grouped data or continuous data, interpolation might be used. In this specific case with discrete data and the 13.5th position falling within a range covered by a single 'x' value, that 'x' value is the median.

Understanding the difference between calculating the median for raw data, discrete frequency distributions, and continuous frequency distributions is important.