The average (mean)

1Summary 重點摘要

Average = sum ÷ count: what each would get if things were shared evenly. And backwards: sum = average × count.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is the mean?The mean is the most widely used average: add up all the values, then divide by how many there are. For the data 5, 10, 15: the sum is 5 + 10 + 15 = 30, the count is 3, and the mean is 30 ÷ 3 = 10. Intuitively it means 'sharing the total equally': if three classmates hold 5, 10 and 15 dollars, pooling the money and splitting it gives each 10 dollars. The mean shows the central level of a data set; average test scores and average team heights work the same way. As a formula: mean = sum ÷ count, and both parts are essential.
2. 2. How to calculate the mean?Calculating the mean takes only two steps. For the data 5, 10, 15: step one, find the sum, 5 + 10 + 15 = 30; step two, divide by the count, 30 ÷ 3 = 10, so the mean is 10. A decimal example: for 4.5, 6.5, 7, 10, the sum is 4.5 + 6.5 + 7 + 10 = 28, the count is 4, and the mean is 28 ÷ 4 = 7. Two errors are common: missing one value while adding, and dividing by the wrong count — four values divided by 3. Count the values before you start and re-check the sum term by term afterwards to cut mistakes sharply.
3. 3. Class scores examplePractise with a real setting: five students score 80, 85, 90, 95 and 100 in a quiz. The sum is 80 + 85 + 90 + 95 + 100 = 450, the count is 5, and the mean score is 450 ÷ 5 = 90 marks. This mean of 90 represents the group's general level and lets you compare classes or sittings as a whole. Note that the mean need not equal anyone's actual score: here someone does score exactly 90, a coincidence; the mean can also be absent from the data yet remain a fair centre. If the next quiz's mean rises to 92, the whole group has improved — rather than judging by one high scorer.
4. 4. What the mean meansThe mean is a central value, not everybody's value. A mean score of 90 describes the class's general level, yet the reality may be that some score above 90 (say 95 or 100) and some below (say 80 or 85). A mean compresses all the differences into one number, which makes comparison easy but hides individual spread: two classes with the same mean may be one where everyone sits between 88 and 92, and another where half score full marks and half fail. So when reading statistics, the mean gives only the overall impression; to see the full picture, pair it with the maximum and the minimum.
5. 5. The medianThe median is the middle value once the data are sorted from smallest to largest. For 5, 10, 15: sorted they are still 5, 10, 15, the middle one is 10, so the median is 10. With an even count, say 4, 8, 10, 12, two values sit in the middle (8 and 10), and the median is their average: (8 + 10) ÷ 2 = 9. The most common mistake is skipping the sorting: taking the 'middle' of 3, 15, 5 as 15 is wrong — after sorting, 3, 5, 15 has median 5. The median's strength is that extreme values cannot move it, and the next step compares all three averages in detail.
6. 6. The modeThe mode is the value that appears most often. For 5, 10, 10, 15: the 5 appears once, the 10 appears twice, the 15 once, so the most frequent value is 10 and the mode is 10. The mode suits categorical data especially well — the class's favourite colour, the shoe size a shop sells most — data for which a mean makes no sense but a mode does. Watch two special cases: if every value appears equally often, as in 2, 4, 6, 8, there is no mode; and if two values tie for the lead, as in 2, 2, 5, 5, 8, then both 2 and 5 are modes (a bimodal set) and both must be written.
7. 7. Which average to use?Each of the three averages has its use, and the key is whether extreme values exist. Take 1, 2, 3, 100: the mean is (1 + 2 + 3 + 100) ÷ 4 = 106 ÷ 4 = 26.5, while the median is (2 + 3) ÷ 2 = 2.5. The single 100 drags the mean up to 26.5, yet three of the four values are below 4, so 26.5 cannot represent a typical value; the median 2.5 is far closer to reality. The rule: use the mean for tidy data (it uses all the information), the median when outliers exist (it resists dragging), and the mode for categories. Hong Kong reports quote median property prices and salaries for exactly this reason.
8. 8. Common mistakes and checkingAvoid three frequent errors when finding averages. First, missing or double-counting a value in the sum, turning 5, 10, 15 into 25 or 35; the fix is re-checking term by term after adding. Second, dividing by the wrong count, such as dividing five values by 4; the fix is circling all the values and counting them first. Third, seeing 'average' and assuming the mean is wanted, when the question may ask for the median or the mode — always read which one is required. A checking trick: mean × count should rebuild the sum; 10 × 3 = 30 matches the original total, so the calculation is right. This reverse check saves many careless marks in exams.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Average: sum ÷ count
• Average represents typical level
• Median: middle number sorted
• Mode: most frequent
• With outliers, median more accurate