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How to find median from frequency density graph?

Published in Data Analysis 3 mins read

Finding the median from a frequency density graph involves a few steps that use the graph's data to determine the middle value of a dataset. Here's how you do it:

Steps to Find the Median

  1. Calculate Frequencies:

    • Start with the leftmost bin of the graph.
    • Multiply each bin's width by its frequency density. This gives you the frequency (or count) for that bin.
    • For example, if a bin is 5 units wide and has a frequency density of 10, then the frequency for that bin is 5 * 10 = 50.
    • Repeat this for each bin in the graph.

  2. Calculate Cumulative Frequencies:

    • Once you have the frequency for each bin, sum these up progressively.
    • For the first bin, the cumulative frequency is just its frequency.
    • For the second bin, the cumulative frequency is its frequency plus the cumulative frequency of the first bin.
    • Continue this process until you reach the last bin, giving you the total frequency.
    • This is the cumulative frequency up to that bin.

  3. Locate the Median Bin:

    • Determine the total frequency of the dataset (the final cumulative frequency).
    • Divide this total frequency by 2 to find the position of the median value. For instance, if the total frequency is 130, the median position is 130 / 2 = 65.
    • Examine the cumulative frequencies to find the bin where the cumulative count first equals or exceeds half the total frequency. This bin contains the median.

  4. Interpolate Within the Median Bin:

  • Once the median bin is found, use linear interpolation to estimate the precise median value. This step is crucial for accurate results, as the median rarely falls exactly at the bin's edge.

  • Formula for Linear Interpolation:

    Median = L + [(N/2 - CF_lower) / freq_median ] * W

    Where:

    • L is the lower bound of the median bin.
    • N is the total frequency.
    • CF_lower is the cumulative frequency of the bin below the median bin.
    • freq_median is the frequency of the median bin.
    • W is the width of the median bin.

Example:

Let's illustrate this with an example: Suppose a graph represents data of house sizes.

Bin Width Frequency Density Frequency Cumulative Frequency
0-10 10 2 20 20
10-20 10 4 40 60
20-30 10 5 50 110
30-40 10 1 10 120
40-50 10 1 10 130

  • Step 1: You multiply the width by the density to find the frequencies (already calculated in the table above).

  • Step 2: The cumulative frequencies are calculated and shown in the table above.

  • Step 3: The total frequency is 130, so half is 65. The cumulative frequency crosses this within the third bin (20-30), where it becomes 110. This is the median bin.

    • We will use the interpolation formula using information from the third bin which will be our median bin.
    • L = 20
    • N= 130
    • CF_lower = 60
    • freq_median = 50
    • W = 10
    • Median = 20 + [(130/2 - 60) / 50] * 10
    • Median = 20 + [5/50] * 10
    • Median = 20 + 1 = 21

Therefore, the median house size is estimated as 21.

By following these steps, you can effectively find the median from a frequency density graph.

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