One number can describe a whole data set, but only if you choose the right one. Learn the mean, median, mode and range, with worked examples, frequency tables and real-life cases.
Seven quiz scores side by side do not tell a story on their own. You would have to hold all seven numbers in your head at once. Statistics solves this by compressing a whole data set into one representative number. The catch is that there are three different “representative numbers” to choose from, and choosing the wrong one can quietly give a false picture.
Data means information that has been collected, such as test marks, daily temperatures, the number of goals in a match, or the colours that students like best. There are two main kinds:
An average is a single value that represents the centre of the data. This article covers the three averages in your syllabus (mean, median and mode) and one measure of spread (the range).
The mean is what most people simply call “the average”. Add up every value, then divide by how many values there are.
Mean = sum of all values ÷ number of values
Five students score 70, 75, 80, 85 and 90 on a test. Find the mean.
Step 1: Add: 70 + 75 + 80 + 85 + 90 = 400.
Step 2: Divide by 5: 400 ÷ 5 = 80.
Answer: 80
The mean of five numbers is 12. Four of the numbers are 10, 11, 14 and 9. Find the fifth number.
Step 1: The total of all five numbers = mean × number of values = 12 × 5 = 60.
Step 2: The four known numbers add up to 10 + 11 + 14 + 9 = 44.
Step 3: The missing number = 60 − 44 = 16.
Answer: 16. Check: (10 + 11 + 14 + 9 + 16) ÷ 5 = 60 ÷ 5 = 12. ✓
The mean uses every value, which is its strength. It is also its weakness: one unusually high or low value, called an outlier, can pull the mean away from where most of the data sits.
The median is the middle value once the data is arranged in order from smallest to largest. If there are two middle values, the median is halfway between them. With n values, the median is at position (n + 1) ÷ 2.
Odd: for 3, 7, 9, 15, 21 (already in order), there are 5 values, so the median is at position (5 + 1) ÷ 2 = 3. The median is 9.
Even: for 4, 8, 10, 14 there are 4 values, so the median is at position 2.5, halfway between the 2nd and 3rd values: (8 + 10) ÷ 2 = 9.
Warning: if the numbers are not in order, put them in order first. For 7, 3, 12, 9, 4, 15 the ordered list is 3, 4, 7, 9, 12, 15, so the median is (7 + 9) ÷ 2 = 8.
Because the median depends only on position, not on the exact size of each value, it stays close to the middle even when one value is extreme.
The mode is the value that appears most often. Unlike the mean and median, the mode also works for non-numeric data, such as favourite colours or types of pet.
In the data set 4, 6, 6, 6, 9, 12, the value 6 appears three times, more than any other value, so the mode is 6. A data set can have two modes (called bimodal), or no mode at all if every value appears the same number of times.
When the same value appears many times, data is often shown in a frequency table. The frequency f tells you how many times each value x occurs.
| Goals (x) | Frequency (f) | x × f |
|---|---|---|
| 0 | 3 | 0 |
| 1 | 5 | 5 |
| 2 | 6 | 12 |
| 3 | 4 | 12 |
| 4 | 2 | 8 |
| Total | 20 | 37 |
Mean = total of x × f ÷ total frequency = 37 ÷ 20 = 1.85 goals.
Mode = the value with the highest frequency (6), so the mode is 2 goals.
Median: there are 20 values, so the median lies between the 10th and 11th values. Running totals of frequency: 3 (for 0), 8 (for 1), 14 (for 2). The 10th and 11th values are both in the “2 goals” group, so the median is 2 goals.
Two sets of data can have the same mean and still be very different. The range measures spread: range = largest value − smallest value.
Class A scores: 60, 65, 70, 75, 80. Class B scores: 40, 55, 70, 85, 100.
Mean of A = 350 ÷ 5 = 70. Mean of B = 350 ÷ 5 = 70. The means are equal.
Range of A = 80 − 60 = 20. Range of B = 100 − 40 = 60.
Meaning: Class A is consistent. Class B has very strong and very weak students. A teacher would plan the two classes differently, even though the averages match.
The chart below shows seven quiz scores. Six students scored in a similar range, but one student, perhaps rushed or unwell that day, scored far lower. Watch what this does to the mean compared with the median.
Check the numbers: the seven scores add up to 82 + 85 + 79 + 88 + 84 + 91 + 12 = 521, and 521 ÷ 7 ≈ 74.4. In order, the scores are 12, 79, 82, 84, 85, 88, 91, so the median is the 4th value, 84. Without the outlier, the mean of the other six scores is 509 ÷ 6 ≈ 84.8. So the one low score pulled the mean down by about 10 points, while the median barely moved.
| Average | Best Used When | Weakness |
|---|---|---|
| Mean | Data is fairly evenly spread, with no extreme values | Easily distorted by outliers |
| Median | Data contains outliers or is unevenly spread | Ignores the exact size of every value |
| Mode | Data is categorical, or you need the most common result | May not exist, or may not be near the “typical” value |
A company has 8 workers who each earn Rs 40,000 a month, and an owner who earns Rs 400,000. What is the mean, median and mode? Which is fairest to describe a “typical” salary?
Mean: (8 × 40,000 + 400,000) ÷ 9 = 720,000 ÷ 9 = Rs 80,000.
Median: the 5th of 9 values, which is Rs 40,000.
Mode: Rs 40,000.
Answer: the median (and mode) of Rs 40,000 describes a typical worker. The mean of Rs 80,000 is double what eight of the nine people earn, so it is misleading here.
National income statistics show why this choice matters. If a small number of people earn extremely high incomes, the mean income is pulled well above what a typical person earns. The median income, the exact middle of the distribution, is much less affected by those few high earners. This is why economists and journalists usually report median income when they describe what a “typical” household earns.
Sometimes data is given in groups (classes), such as heights from 150 to 160 cm. We do not know the exact values, so we use the midpoint of each class to estimate the mean.
| Height h (cm) | Frequency f | Midpoint x | f × x |
|---|---|---|---|
| 150 ≤ h < 160 | 5 | 155 | 775 |
| 160 ≤ h < 170 | 12 | 165 | 1,980 |
| 170 ≤ h < 180 | 8 | 175 | 1,400 |
| Total | 25 | 4,155 |
Estimated mean = 4,155 ÷ 25 = 166.2 cm. It is only an estimate, because we used midpoints instead of the real heights.
Modal class: the class with the highest frequency is 160 ≤ h < 170. Median class: the 13th value lies in the second class (running totals 5, then 17), so it is also 160 ≤ h < 170.
A batter scores 45, 12, 78 (not out), 0 and 30 in five innings. In cricket, the batting average is the total runs divided by the number of times the batter was dismissed, not the number of innings.
Total runs = 45 + 12 + 78 + 0 + 30 = 165. Dismissals = 5 − 1 = 4. Batting average = 165 ÷ 4 = 41.25.
If you divided by 5 innings, you would get 33, which is a different (and lower) number. The “not out” innings is why the rule is different.
Averages also appear when:
1. Mean = 40 ÷ 5 = 8. Median = middle value = 8. Mode = 8 (it appears twice).
2. Total = 7 × 6 = 42. Known total = 4 + 6 + 8 + 9 + 5 = 32. Sixth number = 42 − 32 = 10.
3. The median, because the mansion is an outlier that would pull the mean far above a typical house price.
1. 12 + 15 + 9 + 20 + 14 = 70, and 70 ÷ 5 = 14.
2. Ordered: 3, 4, 7, 9, 12, 15. Median = (7 + 9) ÷ 2 = 8. Range = 15 − 3 = 12.
3. The total of the six numbers = 8.5 × 6 = 51. New total = 51 + 12 = 63. New mean = 63 ÷ 7 = 9.
4. Total of x × f = 1×2 + 2×5 + 3×8 + 4×3 + 5×2 = 2 + 10 + 24 + 12 + 10 = 58. Mean = 58 ÷ 20 = 2.9. Mode = 3 (highest frequency, 8). Median: the 10th and 11th values; running totals are 2, 7, 15, so both are 3, and the median is 3.
Students often learn the three averages as three separate recipes. It helps to start with a question: “If I had to describe this class in one number, which number would I choose, and why?” Let students collect their own data (for example, the number of siblings in the class, or time spent on homework), then compute all three averages and discuss which one tells the truth best. Watch for the typical errors listed above, especially forgetting to sort before finding the median and averaging averages. When reading news at home, ask your child “is this a mean or a median?” This builds the critical habit that statistics is meant to teach.
None of the three averages is universally “correct”. Each answers a slightly different question. Part of thinking statistically is recognising which question is being asked, and choosing the average that answers it honestly rather than the one that looks the most impressive.
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