Averages, Range and Data from Tables
Three averages, one measure of spread, and a set of table methods built on top of them. The methods are short — the marks are lost in which number you divide by and in misreading the table.
1. The four measures
| Measure | How to find it | Watch out for |
|---|---|---|
| Mean | total of values ÷ number of values | dividing by the wrong count |
| Median | middle value, once in order | forgetting to sort first |
| Mode | the most frequent value | it’s the value, not the frequency |
| Range | largest − smallest | it’s a single number, not an interval |
Example: 4, 7, 2, 9, 4, 6
- Mean = (4+7+2+9+4+6) ÷ 6 = 32 ÷ 6 = 5.33 (3 s.f.)
- Order first: 2, 4, 4, 6, 7, 9 → Median = (4+6)/2 = 5
- Mode = 4
- Range = 9 − 2 = 7
Put the data in order before finding the median. Taking the middle of an unsorted list is one of the most common errors in the topic — and the answer will usually look plausible.
With an even number of values, the median is the mean of the middle two. With an odd number, it is the single middle value.
The mode is the value that occurs most, not how many times it occurs. If 6 appears four times and nothing else appears more, the mode is 6, not 4. Similarly, the “modal day” question wants the day, but a “modal height” wants the height.
The range is a single number. “From 2 to 9” is not a range; 7 is. And range measures spread, not a typical value.
2. Median position
For n values in order:
The median is at position (n + 1) / 2
- n = 9 → position (9+1)/2 = 5th value
- n = 40 → position 20.5 → the mean of the 20th and 21st values
Add one, then divide by two. Multiplying, or dividing n by 2 alone, gives the wrong position — a recorded error. For n = 40 the median is between the 20th and 21st, not the 20th.
3. Averages from a frequency table
| Score (x) | Frequency (f) | f × x |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 5 | 10 |
| 3 | 8 | 24 |
| 4 | 4 | 16 |
| Total | 20 | 53 |
Mean = Σfx ÷ Σf — the total of the (f × x) column divided by the total frequency.
Mean = 53 ÷ 20 = 2.65
Divide by the total FREQUENCY, not by the number of rows. Here you divide by 20, not by 4. This is the single biggest error in the whole topic — students divide by how many lines are in the table.
Reading the other averages from the table:
- Mode = the value with the highest frequency → score 3 (frequency 8)
- Median = the (20+1)/2 = 10.5th value. Counting: 3 values of 1, then up to 8 covers score 2, up to 16 covers score 3. The 10th and 11th are both 3, so the median is 3
- Range = 4 − 1 = 3
Count through the frequencies to locate the median — a running (cumulative) total is the safest way. Don’t take the middle row of the table.
4. Grouped data — estimating the mean
When data is grouped, you no longer have the individual values, so you use the midpoint of each class.
| Height (cm) | Frequency (f) | Midpoint (x) | f × x |
|---|---|---|---|
| 0 < h ≤ 10 | 4 | 5 | 20 |
| 10 < h ≤ 20 | 7 | 15 | 105 |
| 20 < h ≤ 30 | 6 | 25 | 150 |
| 30 < h ≤ 40 | 3 | 35 | 105 |
| Total | 20 | 380 |
Estimated mean = 380 ÷ 20 = 19 cm
The midpoint is the average of the two class boundaries — (0 + 10)/2 = 5. Forgetting to divide by 2 and using the upper bound instead was a repeated error.
It is an ESTIMATE, and you should say so. The question will say “estimate the mean” precisely because the original values are lost inside the groups. If asked why it is an estimate, the answer is that the exact data values are not known — only the class they fall in.
The modal class is the class with the highest frequency — here 10 < h ≤ 20.
Give the class, not the frequency. The answer is the interval “10 < h ≤ 20”, not “7”.
The class containing the median: find the (n+1)/2 th value by running through the cumulative frequencies, then name the class it falls in.
5. Working backwards from a mean
A common higher-mark question: you are given the mean and asked for a missing value.
total = mean × number of values
Example: after 4 tests a student has a mean of 72. What must they score on the 5th test for a mean of 75?
- Total needed for 5 tests = 75 × 5 = 375
- Total so far = 72 × 4 = 288
- Required score = 375 − 288 = 87
Example with an unknown in a table: if the mean of the frequency table is given, form an equation with the unknown as a letter, using Σfx ÷ Σf = mean, and solve.
Multiply the mean by the count to recover the total. That single step unlocks nearly every “missing value” question.
6. Reading charts and tables
Check the key or scale first. In a pictogram, one symbol often represents several items — reading it as 1 was a specific documented error. On a bar chart, check what one gridline is worth before reading any bar.
Give the units. Heights in cm, times in minutes — a bare number can lose the final mark.
Choosing an average to comment on:
- The mean uses every value, but is distorted by extreme values
- The median is unaffected by extremes, so it is better for skewed data
- The mode is the only average available for non-numerical data (favourite colour, brand)
A comparison question wants two things: an average and the spread. “Team A has a higher median score (7 vs 5), so they generally performed better, and a smaller range (4 vs 11), so they were more consistent.” Both halves earn marks.
7. Mistakes that cost marks
Dividing by the number of rows instead of the total frequency.
Not ordering the data before finding the median.
Using n/2 instead of (n+1)/2 for the median position.
Giving the frequency instead of the value for the mode.
Giving a frequency instead of the class for the modal class.
Using the upper bound instead of the midpoint in grouped data.
Forgetting to halve when finding a midpoint.
Writing the range as an interval rather than a single number.
Confusing mean, median and mode with one another.
Misreading a pictogram key or a chart scale.
Forgetting that a grouped mean is an estimate.
Omitting units.
Frequently asked questions
How do I find the mean from a frequency table? Multiply each value by its frequency, add those up, and divide by the total frequency: Σfx ÷ Σf.
Why do I divide by the total frequency? Because that is how many data values there are — the rows are categories, not individual pieces of data.
How do I estimate the mean of grouped data? Use the midpoint of each class as x, then Σfx ÷ Σf.
Why is it only an estimate? Because the exact values are unknown — you only know which class each falls into.
What is the modal class? The class with the highest frequency. Give the interval.
How do I find the median position? (n + 1) ÷ 2. If it lands on a half, take the mean of the two values either side.
What is the range? Largest − smallest: a single number describing spread.
When is the median better than the mean? When there are extreme values, which pull the mean away from the typical value.
How do I find a missing value given the mean? Total = mean × number of values, then subtract the total you already have.
Quick revision checklist
- I can find mean, median, mode and range from a list
- I order the data before taking the median
- I use (n+1)/2 for the median position
- I average the middle two when n is even
- I give the value for the mode, not its frequency
- I can find the mean from a frequency table using Σfx ÷ Σf
- I divide by the total frequency
- I can locate the median in a frequency table by cumulative counting
- I can find midpoints correctly and estimate a grouped mean
- I can identify the modal class and give it as an interval
- I know why a grouped mean is only an estimate
- I can work backwards from a given mean
- I check keys and scales before reading a chart
- I can compare two data sets using an average and the spread
- I include units
These notes cover averages, range and data from tables in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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