Averages from a Frequency Table
Finding averages from a frequency table lets you calculate the mean, median, and mode far more efficiently than working from a raw list. This page covers discrete data (whole-number values like goals scored or number of siblings), where the answers are exact. You will learn how to use the \(f \times x\) column to find the mean, use cumulative frequency to locate the median, and read off the mode directly from the table. Use the Practice Questions below to build your skills from Starter to Master level.
A frequency table is a compact way to record how often each value occurs in a data set. Once your data is in a table you can calculate the mean, find the mode, and locate the median far more efficiently than working from a raw list, especially when \(n\) is large.
What is a frequency table?
Instead of listing every single value, a frequency table groups identical values together and records how many times each one appears. The column labelled \(f\) (frequency) tells you how many data points have that value. The sum of the \(f\) column, written \(\sum f\), gives the total number of data points \(n\).
When calculating the mean, an extra \(f \times x\) column is added; multiplying each value by its frequency gives the total contribution of that value to the overall sum.
Core Ideas: The Three Averages from a Frequency Table
Each average answers a different question about the data. Here is what each one means and how to find it from a frequency table.
📊 Mode
The most common value; the one that appears most often.
No calculation needed; find the row with the biggest frequency and read off \(x\).
One mode: one value has a strictly higher frequency than all others. Example: if \(f = 8\) for \(x = 3\) and all other frequencies are lower, the mode is 3.
Two modes (bimodal): two values share the joint-highest frequency. Both are modes. Example: if \(x = 2\) and \(x = 5\) both have \(f = 7\), the modes are 2 and 5.
No mode: if every value appears the same number of times, there is no mode; do not write 0, just state "no mode".
📐 Mean
The fair share value; the total spread equally across all data points.
Add an \(f \times x\) column. Sum it to get \(\sum fx\). Divide by \(\sum f\).
Never just average the \(x\) column; that ignores how many times each value occurs.
🎯 Median
The middle value; half the data is below it, half above.
Find the position number, then build a running cumulative frequency total until you reach that position; the \(x\) value in that row is the median.
For even \(n\), the median is the mean of the two middle positions.
⚙️ Key Formulae: Discrete Frequency Table
Let \(x\) = each data value and \(f\) = its frequency. Add an \(f \times x\) column to your table:
\(\sum\) means "the sum of". \(\sum fx\) = total of every data value. \(\sum f = n\) = total number of values.
Worked Examples
💡 Worked Example 1: All Three Averages from a Frequency Table
A teacher records the number of merits awarded to 20 students in one term.
Raw data: 3 5 2 4 1 0 4 5 5 3 4 4 5 1 3 3 4 5 5 2
Find the mean, median, and mode.
| Merits \((x)\) | Frequency \((f)\) | \(f \times x\) | Cumulative \(f\) |
|---|---|---|---|
| 0 | 1 | 0 | 1 |
| 1 | 2 | 2 | 3 |
| 2 | 2 | 4 | 5 |
| 3 | 4 | 12 | 9 |
| 4 | 5 | 20 | 14 |
| 5 | 6 | 30 | 20 |
| Total | \(\sum f = 20\) | \(\sum fx = 68\) |
Take the mean of the 10th and 11th values.
💡 Worked Example 2: Locating the Median Carefully
The frequency table shows the number of siblings of 30 students. Find the mean, median, and mode.
| Siblings \((x)\) | Frequency \((f)\) | \(f \times x\) | Cumulative \(f\) |
|---|---|---|---|
| 0 | 6 | 0 | 6 |
| 1 | 11 | 11 | 17 |
| 2 | 8 | 16 | 25 |
| 3 | 4 | 12 | 29 |
| 4 | 1 | 4 | 30 |
| Total | \(\sum f = 30\) | \(\sum fx = 43\) |
Take the mean of the 15th and 16th values.
🔑 Key Points
- Always add an \(f \times x\) column before calculating the mean; never just average the \(x\) values.
- The mode is the \(x\) value with the highest \(f\); no calculation needed.
- For the median, find position \(\frac{n+1}{2}\), then read down the cumulative frequency column until you reach that position.
- For even \(n\), the median is the mean of the two middle positions.
- Always check \(\sum f = n\) before you start; a wrong total ruins every answer.
⚠️ Common Mistakes
- Averaging the \(x\) values directly: ignores frequency, gives the wrong mean.
- Wrong median position: always use cumulative frequency, not the row with the largest \(f\).
- Confusing mode and median: mode is most common; median is the middle by position.
- Off-by-one: median position uses \(\frac{n+1}{2}\), not \(\frac{n}{2}\).