Step By Step Guide To Calculating The Median From A Frequency Table With Easy Examples

Understanding how to find the median from a frequency table is a fundamental skill in statistics, data analysis, and everyday decision-making. Whether you're analyzing test scores, customer age groups, or product ratings, the median provides a central value that isn't skewed by extreme outliers—unlike the mean. This guide walks you through the process clearly and systematically, using real-world examples and actionable steps.

Why the Median Matters

The median represents the middle value in an ordered dataset. It’s especially useful when dealing with skewed distributions, such as income levels or house prices, where a few high values can distort the average. In grouped or tabulated data, like frequency tables, the raw values aren’t listed individually, so we must use cumulative frequencies and interpolation techniques to estimate the median accurately.

Finding the median from a frequency table involves more than just picking the middle number. You need to interpret class intervals, calculate running totals, and sometimes apply a formula for grouped data. Mastering this process improves your analytical precision and confidence in handling real datasets.

Understanding Frequency Tables

A frequency table organizes data into categories or intervals, showing how often each value or range occurs. Here's a simple example:

Score Range	Frequency	Cumulative Frequency
0–10	3	3
11–20	7	10
21–30	12	22
31–40	6	28
41–50	2	30

In this table, “Score Range” defines the class intervals, “Frequency” shows how many observations fall within each interval, and “Cumulative Frequency” tracks the running total up to that point. This structure is essential for locating the median group.

Tip: Always sort your frequency table in ascending order of class intervals before beginning calculations. An unsorted table leads to incorrect cumulative frequencies and flawed results.

Step-by-Step Guide to Finding the Median

1. Determine the total number of observations (n)
   Add all frequencies to get the total sample size. In our example: 3 + 7 + 12 + 6 + 2 = 30.
2. Find the median position
   Use the formula: Median Position = (n + 1) / 2 For n = 30: (30 + 1) / 2 = 15.5 This means the median lies between the 15th and 16th values.
3. Locate the median class
   Look at the cumulative frequency column to find the first class where the cumulative total exceeds the median position.
   - Cumulative frequency up to 11–20: 10 (less than 15.5)
   - Cumulative frequency up to 21–30: 22 (greater than 15.5)
   So, the median lies in the 21–30 class.
4. Apply the median formula for grouped data
   When data is grouped, use the interpolation formula:

   Median = L + [(n/2 - CF) / f] × w

   Where:
   - L = Lower boundary of the median class
   - n = Total number of observations
   - CF = Cumulative frequency before the median class
   - f = Frequency of the median class
   - w = Width of the class interval
5. Plug in the values and calculate
   From our example:
   - L = 20.5 (lower boundary of 21–30; assumes continuous data)
   - n = 30 → n/2 = 15
   - CF = 10 (cumulative frequency before 21–30)
   - f = 12 (frequency of 21–30)
   - w = 10 (class width: 30.5 – 20.5)
   
   Median = 20.5 + [(15 - 10) / 12] × 10
   = 20.5 + (5 / 12) × 10
   = 20.5 + 4.17 ≈ 24.67

This result indicates that the median score is approximately 24.67, meaning half of the students scored below this value and half above.

“Interpreting grouped data correctly is crucial in research. The median from a frequency table gives a realistic center point when raw data isn't available.” — Dr. Alan Reyes, Statistics Educator

Mini Case Study: Analyzing Customer Age Data

A retail manager wants to understand the typical age of customers visiting a new store. She collects data over a week and groups it into a frequency table:

Age Group	Frequency	Cumulative Frequency
10–19	8	8
20–29	15	23
30–39	10	33
40–49	5	38
50–59	2	40

Total observations (n) = 40
Median position = (40 + 1)/2 = 20.5 → between 20th and 21st values
Cumulative frequency reaches 23 in the 20–29 group → median class is 20–29

Using the formula:
L = 19.5, n/2 = 20, CF = 8, f = 15, w = 10
Median = 19.5 + [(20 - 8)/15] × 10 = 19.5 + (12/15)×10 = 19.5 + 8 = 27.5

The median customer age is 27.5 years. This insight helps tailor marketing strategies toward young adults rather than assuming an older demographic.

Common Mistakes and How to Avoid Them

• Forgetting to sort the table: Always ensure class intervals are in ascending order.
• Misidentifying the median class: Double-check cumulative frequencies against the median position.
• Using the wrong lower boundary: For discrete ranges like 21–30, use 20.5 as the lower limit if assuming continuity.
• Confusing n/2 with (n+1)/2: Use (n+1)/2 to find the median position, but n/2 in the grouped formula.

Tip: If your data is discrete (e.g., whole numbers), adjust class boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit to maintain accuracy.

Checklist: Calculating the Median from a Frequency Table

• ✅ Ensure the frequency table is sorted in ascending order
• ✅ Calculate total number of observations (n)
• ✅ Compute cumulative frequencies
• ✅ Determine median position: (n + 1)/2
• ✅ Identify the median class using cumulative frequency
• ✅ Note the lower boundary (L), class width (w), frequency (f), and prior cumulative (CF)
• ✅ Apply the grouped median formula: L + [(n/2 - CF)/f] × w
• ✅ Interpret the result in context

FAQ

Can I find the exact median from a frequency table?

No—since individual values aren’t listed, the median is estimated using interpolation. The result is a close approximation based on assumptions about uniform distribution within the class.

What if the data is ungrouped but in a frequency table?

If each row represents a single value (e.g., score of 5 appears 3 times), list all values in order and pick the middle one directly. No interpolation is needed.

Do I always use (n+1)/2 for grouped data?

You use (n+1)/2 to find the median position in the ordered list, but in the grouped formula, n/2 is standard. Both approaches converge on the same median class in large samples.

Final Thoughts and Next Steps

Calculating the median from a frequency table transforms raw, organized data into meaningful insights. Whether you’re a student tackling exam questions or a professional analyzing survey responses, mastering this method builds statistical literacy and decision-making power. The key is practice: work through multiple examples, verify your steps, and always interpret your results in context.

💬 Put this knowledge into action. Find a frequency table in a report or textbook and calculate its median. Share your example and result in the comments to help others learn!