When data is grouped into class intervals, each individual value is replaced by the class it fell into. You can still estimate the mean and identify the modal class and the class containing the median, but you cannot state exact values such as the highest score. This skill sits within statistics and distributions and builds on weighted means.
What information is lost in a grouped table?
A table that says 10 < t ≤ 20 has frequency 9 tells you nine people took between 10 and 20 minutes. It does not tell you whether they took 11 minutes or 19. That lost detail is why every answer from grouped data comes with a word such as “estimate”.
Two habits keep you safe. Use the midpoint of each class for any calculation. Describe results as an estimate, a class or a range of possible values, never as an exact raw value.
How to work with a grouped table, step by step
- Add the frequencies to get the total number of values.
- Find the midpoint of each class: add the two limits and halve.
- Multiply each midpoint by its frequency and add the results.
- Divide by the total frequency for the estimated mean.
- For the modal class, pick the class with the largest frequency (equal class widths).
- For the median class, find the middle position and use the running total of frequencies.
Worked example
Twenty students recorded how long they took to walk to school.
| Time t (minutes) | Frequency |
|---|---|
| 0 < t ≤ 10 | 4 |
| 10 < t ≤ 20 | 9 |
| 20 < t ≤ 30 | 5 |
| 30 < t ≤ 40 | 2 |
Step 1, total: 4 + 9 + 5 + 2 = 20.
Step 2, midpoints: 5, 15, 25 and 35.
Step 3, products: 4 × 5 = 20, 9 × 15 = 135, 5 × 25 = 125, 2 × 35 = 70. The sum is 350.
Step 4, estimated mean: 350 ÷ 20 = 17.5 minutes.
Step 5, modal class: the highest frequency is 9, so the modal class is 10 < t ≤ 20.
Step 6, median class: the middle of 20 values lies between the 10th and 11th. The running totals are 4, 13, 18, 20. The total first reaches 10 or more in the second class, so the median is in 10 < t ≤ 20.
Bounds on the mean: if every student were at the lower end of their class, the mean would be (0 + 90 + 100 + 60) ÷ 20 = 12.5. If every student were at the upper end, it would be (40 + 180 + 150 + 80) ÷ 20 = 22.5. The true mean lies between 12.5 and 22.5 minutes, and 17.5 sits in the middle of that.
The mistake to watch for
A common slip is to read an exact value out of a class.
Mistaken answer: “The longest walk took 40 minutes, so the range is 40 minutes.”
The table only says the longest walk was in the class 30 < t ≤ 40. It could be 31 minutes. Also, the shortest is only known to be above 0.
The correction is to rewrite the claim so the table supports it: “the longest walk was more than 30 minutes and at most 40”. The same care applies to the mean: say “an estimate of 17.5 minutes”, not “the mean is 17.5”.
Check yourself
Try these, then open each answer.
1. Twenty students’ heights: 140 < h ≤ 150 has frequency 3, 150 < h ≤ 160 has frequency 10, and 160 < h ≤ 170 has frequency 7. Estimate the mean height.
Show answer
Midpoints: 145, 155 and 165. Products: 3 × 145 = 435, 10 × 155 = 1550, 7 × 165 = 1155. Sum = 3140.
3140 ÷ 20 = 157 cm (an estimate).
2. In the same table, which class contains the median?
Show answer
There are 20 values, so the median lies between the 10th and 11th. Running totals are 3, 13, 20. The 10th and 11th values are both in the second class.
The median is in 150 < h ≤ 160.
3. A student says, “At least one person is exactly 169 cm tall.” Is the table enough to support that?
Show answer
No. The table only shows that 7 people are taller than 160 cm and at most 170 cm. Their exact heights are not recorded, so no particular value such as 169 cm can be claimed.
Where this leads next
Once you can handle what a grouped table does and does not say, move on to comparing spread and centre together. The statistics and distribution explorer lets you change the class limits and watch the estimate move, and the non-calculator working trainer is useful for the midpoint multiplications. Try the mixed practice set afterwards.
Students often calculate the estimated mean correctly but then describe it as exact in the final sentence. Checking that wording is an everyday part of online one-to-one Mathematics tuition.