Frequency Tables
Ask 30 people their shoe size and you get a messy list of 30 numbers. A frequency table tidies that list into a few rows that say "this many people had this value". On this page you'll build frequency tables with tally marks, learn to read them properly, and find the mode, median, range and mean straight from the table, including tables where the data is put into groups.
National curriculum KS3: describe, interpret and compare observed distributions of a single variable through appropriate tables and graphs involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range)1 · Collecting data: the tally chart
Data is just information, usually numbers, that you collect to answer a question. Suppose you roll a dice 30 times and write down each score. That list, written in the order it happened, is called raw data: nothing has been sorted or counted yet.
Raw data is hard to read. Is 6 more common than 1? You can't tell at a glance. So we count how many times each value turns up. The number of times a value appears is its frequency (how frequently it happens).
Counting a long list in your head is easy to get wrong, so we use a tally chart. Go through the list once, in order. For each item, draw a small stroke | in its row. The fifth stroke goes across the other four, making a "gate" of five. Gates make it quick to count at the end: two gates and three strokes is 5 + 5 + 3 = 13. Finally write the count as a number in the frequency column.
Raw data (the highlighted item is next). Tap the row of the table that it belongs to.
Two habits that stop mistakes. Cross each item off as you tally it (the widget greys them out), so you never count something twice or miss it. And when you finish, add up the frequency column: it must equal the number of items you started with. If you rolled the dice 30 times, the frequencies must add up to 30.
Once you have a frequency table you can draw a bar chart from it: one bar per row, as tall as the frequency. That's covered on the statistical charts page.
2 · Discrete and continuous data
Before organising data, it helps to know what kind it is. There are two kinds of number data.
- Discrete data is counted (or can only take certain separate values). Number of goals, number of pets, the score on a dice, shoe size. You can have 2 pets or 3 pets, but not 2.6 pets. There are gaps between the possible values.
- Continuous data is measured. Height, time, mass, temperature. A height could be 152 cm, or 152.4 cm, or 152.43 cm: any value in a range is possible; the only limit is how accurately you measure.
Why does it matter? Discrete data with only a few possible values fits in an ordinary frequency table with one row per value (sections 3 to 5). Continuous data has so many possible values that every measurement might be different, so we put it into groups (section 6). Test yourself:
3 · Reading a frequency table
Here is a finished frequency table. It shows the number of goals scored in each of 20 football matches. Read each row as a sentence: the row 2 | 5 means "5 matches had 2 goals".
The left column holds the values (often called x). The right column holds the frequencies (often called f). The table is just a short way of writing the full list of data. Press Unpack to turn it back into that list.
Total frequency = how many data values there are. Add up the frequency column. In the goals table, 4 + 6 + 5 + 3 + 2 = 20, so there were 20 matches. Adding the value column (0 + 1 + 2 + 3 + 4 = 10) means nothing at all: those are just the possible scores, each written once.
4 · Mode, median and range from a table
Quick reminder from the averages and range page: the mode is the most common value, the median is the middle value when the data is in order, and the range is largest value − smallest value. A frequency table is already in order, so all three come straight out of it.
- Mode: find the biggest frequency, then give the value in that row. The mode is a value, never the frequency.
- Median: with n values the middle one is in position (n + 1) ÷ 2. Count down the table, adding up frequencies as you go (a running total, also called cumulative frequency), until you reach that position.
- Range: the largest value that actually appears minus the smallest one. Skip any value with frequency 0, because nobody had it.
Common mistakes.
- Giving the frequency as the mode. In the shoe table the biggest frequency is 8, but the mode is size 5.
- Giving the position as the median. (25 + 1) ÷ 2 = 13 tells you to look for the 13th pupil; the median is that pupil's shoe size, 5.
- Finding the middle of the value column. The shoe sizes 3 to 8 have middle 5.5, but that ignores how many pupils have each size.
5 · The mean from a frequency table
The mean shares the total out equally: add up all the data values, then divide by how many values there are. With a frequency table you need to be careful about both parts.
Think about the row 2 | 8 in the "people in 30 cars" table: 8 cars each had 2 people. Unpacked, that's 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2, which is a quick 2 × 8 = 16 people. So we add a third column, f × x (frequency times value), to get each row's contribution to the total. Then
mean = total of f × x ÷ total of f
Step through it one line at a time.
The three classic mean mistakes (for the cars table, where the correct mean is 65 ÷ 30 ≈ 2.17):
- Dividing by the number of rows: 65 ÷ 5 = 13. There were 30 cars, not 5.
- Dividing by the total of the value column: 65 ÷ 15 ≈ 4.33. That total means nothing.
- Adding the value column and dividing: (1 + 2 + 3 + 4 + 5) ÷ 5 = 3. This treats every row as if it happened once and ignores the frequencies.
Sense check: the mean must lie between the smallest and largest values (here between 1 and 5) and is usually near the rows with big frequencies. 13 people in a car is clearly impossible.
6 · Grouped frequency tables
Here are the times, in minutes, that 20 pupils spent on homework last night. Almost every time is different, so a row for each value would be no tidier than the raw list. Instead we sort the data into groups called class intervals, such as "10 to 20 minutes".
But which group does exactly 20 minutes go in, "10 to 20" or "20 to 30"? Inequality symbols settle it. The class 10 ≤ t < 20 means "t is at least 10 and less than 20":
- ≤ means "less than or equal to", so 10 is included (drawn as a filled dot ●).
- < means "less than", so 20 is not included (an open dot ○). 20 belongs to the next class, 20 ≤ t < 30.
Drag each time into its class (or tap a time, then tap a row). Watch the number line: filled dots show where each class starts, open dots where it stops.
Once the data is grouped, we lose the exact values: the table only says that 7 pupils took somewhere from 20 up to 30 minutes. So:
- Instead of a mode we give the modal class: the class with the biggest frequency.
- Instead of the exact median we say which class contains the median, using position (n + 1) ÷ 2 and running totals just like before.
- To estimate the mean we pretend every value in a class sits at its midpoint (the middle of the class: for 10 ≤ t < 20 it is (10 + 20) ÷ 2 = 15). Then we use f × midpoint exactly like f × x.
Why "estimate"? Some values in a class are below the midpoint and some are above. If they roughly balance out, the estimate is close, but you can't be sure because the real values are hidden. That's why the answer must be called an estimate, and why you should never say it is exact. Also: the range can only be estimated from a grouped table, for the same reason. For more on describing the shape of data, see distributions and correlation.
7 · Practice: spot the mistake
Every question is new. Type a number and press Check. If you're wrong, the feedback tries to work out which of the common mistakes you made: adding the value column, dividing by the number of rows, dividing by the total of the values, giving the frequency as the mode, or giving the position as the median.
Worked examples
Make a frequency table from raw data
20 pupils were asked how many brothers and sisters they have:
1, 2, 0, 1, 3, 1, 2, 2, 0, 1, 4, 1, 2, 0, 1, 3, 2, 1, 1, 2
| Siblings | Tally | Frequency |
|---|---|---|
| 0 | 3 | |
| 1 | 8 | |
| 2 | 6 | |
| 3 | 2 | |
| 4 | 1 |
Mode, median and range: shoe sizes of 25 pupils
| Size x | Freq f | Running total |
|---|---|---|
| 3 | 2 | 2 |
| 4 | 5 | 7 |
| 5 | 8 | 15 |
| 6 | 6 | 21 |
| 7 | 3 | 24 |
| 8 | 1 | 25 |
The mean from a frequency table: people in 30 cars
| People x | Cars f | f × x |
|---|---|---|
| 1 | 12 | 12 × 1 = 12 |
| 2 | 8 | 8 × 2 = 16 |
| 3 | 5 | 5 × 3 = 15 |
| 4 | 3 | 3 × 4 = 12 |
| 5 | 2 | 2 × 5 = 10 |
| Total | 30 | 65 |
An even total: median and mean of goals in 20 matches
| Goals x | Freq f | Running total | f × x |
|---|---|---|---|
| 0 | 4 | 4 | 0 |
| 1 | 6 | 10 | 6 |
| 2 | 5 | 15 | 10 |
| 3 | 3 | 18 | 9 |
| 4 | 2 | 20 | 8 |
| Total | 20 | 33 |
Grouped data: journey times of 30 pupils
| Time t (min) | f | Mid | f × mid |
|---|---|---|---|
| 0 ≤ t < 10 | 7 | 5 | 35 |
| 10 ≤ t < 20 | 11 | 15 | 165 |
| 20 ≤ t < 30 | 8 | 25 | 200 |
| 30 ≤ t < 40 | 4 | 35 | 140 |
| Total | 30 | 540 |
Exam-style: a missing frequency
A table shows the scores of 15 people: score 1 has frequency 6, score 2 has frequency k, score 3 has frequency 4. Find k and the mean score.
Quick check
What you've learned
- Raw data is the list as collected. A tally chart counts it in one pass: one stroke per item, every fifth stroke closing a gate of five.
- The frequency of a value is how many times it occurs. A frequency table lists each value with its frequency.
- Discrete data is counted (separate values, like goals); continuous data is measured (any value in a range, like time). Continuous data usually goes in a grouped table.
- Total frequency = the total of the frequency column = how many data values there are. Never add up the value column.
- Mode = the value with the biggest frequency (not the frequency itself). Range = largest value − smallest value that actually occur.
- Median: find position (n + 1) ÷ 2, then count down the table with running totals. If the position ends in .5, average the two values either side.
- Mean = total of f × x ÷ total of f. Never divide by the number of rows or by the total of the values. The answer must lie between the smallest and largest values.
- In a grouped table such as 10 ≤ t < 20, 10 is included and 20 is not. Give the modal class and the class containing the median, and estimate the mean using midpoints: total of f × midpoint ÷ total of f. It is only an estimate because the exact values are lost.