How to Calculate Median from Frequency Table

Median from Frequency Table Calculator

function calculateMedianFromFrequencyTable() { var lowerBoundaryMedianClass = parseFloat(document.getElementById('lowerBoundaryMedianClass').value); var frequencyMedianClass = parseFloat(document.getElementById('frequencyMedianClass').value); var cumulativeFreqBeforeMedian = parseFloat(document.getElementById('cumulativeFreqBeforeMedian').value); var totalObservations = parseFloat(document.getElementById('totalObservations').value); var classWidth = parseFloat(document.getElementById('classWidth').value); var resultDiv = document.getElementById('medianResult'); if (isNaN(lowerBoundaryMedianClass) || isNaN(frequencyMedianClass) || isNaN(cumulativeFreqBeforeMedian) || isNaN(totalObservations) || isNaN(classWidth)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; resultDiv.style.color = '#dc3545'; return; } if (frequencyMedianClass <= 0 || totalObservations <= 0 || classWidth = totalObservations / 2) { resultDiv.innerHTML = "Cumulative frequency before median class (cf) must be less than n/2."; resultDiv.style.color = '#dc3545'; return; } var n_over_2 = totalObservations / 2; var median = lowerBoundaryMedianClass + ((n_over_2 – cumulativeFreqBeforeMedian) / frequencyMedianClass) * classWidth; resultDiv.innerHTML = "The Median is: " + median.toFixed(4) + ""; resultDiv.style.color = '#28a745'; }

Understanding and Calculating the Median from a Frequency Table

The median is a measure of central tendency that represents the middle value in a dataset when it's ordered from least to greatest. Unlike the mean, the median is not affected by extreme outliers, making it a robust measure for skewed distributions. When data is presented in a frequency table, especially with grouped class intervals, calculating the exact median requires a specific formula.

What is a Frequency Table?

A frequency table is a tabular representation of data that shows the number of times each value or range of values (class interval) appears in a dataset. For grouped data, it lists class intervals and their corresponding frequencies (how many data points fall into that interval).

Why Calculate Median from a Frequency Table?

When you have a large dataset grouped into class intervals, you don't have access to the individual data points. In such cases, you cannot simply order all values and pick the middle one. The formula for the median from a frequency table allows us to estimate the median value within the median class interval, providing a good approximation of the true median.

Steps to Calculate the Median from a Frequency Table

To use the calculator above, you first need to perform a few preliminary steps to identify the necessary values from your frequency table:

1. Calculate Cumulative Frequencies: Add a column to your frequency table for cumulative frequency. This is the running total of frequencies. For each class, its cumulative frequency is the sum of its frequency and the frequencies of all preceding classes.
2. Find n/2: Calculate half of the total number of observations (n). The total number of observations (n) is the sum of all frequencies. This value tells you where the median position lies.
3. Identify the Median Class: Locate the class interval in the cumulative frequency column where n/2 falls for the first time. This class interval is your "median class."
4. Extract the Required Values: From the median class and the class immediately preceding it, identify the following:
   • L (Lower Boundary of Median Class): The lower real limit of the median class interval. If your classes are 10-19, 20-29, the lower boundary for 20-29 would be 19.5. If they are 10-20, 20-30, the lower boundary for 20-30 is 20.
   • f (Frequency of Median Class): The frequency of the median class itself.
   • cf (Cumulative Frequency of Class Before Median Class): The cumulative frequency of the class interval immediately preceding the median class.
   • n (Total Number of Observations): The sum of all frequencies.
   • w (Class Width): The width of the median class interval (upper boundary – lower boundary).
5. Apply the Formula: Use the formula for the median of grouped data:

   Median = L + [ (n/2 – cf) / f ] * w

Example Calculation

Let's consider a frequency table showing the scores of 50 students in an exam:

Scores (Class Interval)	Frequency (f)	Cumulative Frequency (cf)
0-10	5	5
10-20	12	17
20-30 (Median Class)	18	35
30-40	10	45
40-50	5	50

From the table:

• Total number of observations (n) = 50
• n/2 = 50 / 2 = 25
• The cumulative frequency just greater than or equal to 25 is 35, which corresponds to the class interval 20-30. So, the median class is 20-30.
• Lower boundary of the median class (L) = 20
• Frequency of the median class (f) = 18
• Cumulative frequency of the class before the median class (cf) = 17 (from the 10-20 class)
• Class width (w) = 30 – 20 = 10

Now, applying the formula:

Median = 20 + [ (25 – 17) / 18 ] * 10

Median = 20 + [ 8 / 18 ] * 10

Median = 20 + 0.4444… * 10

Median = 20 + 4.4444…

Median ≈ 24.44

This calculator simplifies the final step by allowing you to input the pre-identified values (L, f, cf, n, w) directly, providing a quick way to compute the median once you've analyzed your frequency table.