Statistics: Mean, Median and Mode

1Summary 重點摘要

Mean = total ÷ count; median = the middle value after sorting; mode = the most frequent value. One data set can have three different answers — each describes the 'centre' differently.

2Vocabulary cards Vocabulary

3Step by step Steps

1. The meanThe mean is the sum of all values divided by how many there are. For 1, 2, 3, 4, 5: the sum is 15 over 5 values, so the mean is 15 ÷ 5 = 3. Using every value, it is the most common measure of centre — the 'average score' and 'average temperature' of daily life are means. Its weakness is sensitivity to outliers: if five monthly incomes are 10k, 11k, 12k, 13k and 1000k, the mean is dragged far upward by the 1000k and no longer represents most people.
2. The medianThe median is the middle value after sorting the data in order. For 1, 2, 3, 4, 5 the third value, 3, is the median. With an even count, as in 1, 2, 3, 4, the two middle values 2 and 3 are averaged: (2 + 3) ÷ 2 = 2.5. Sorting first is the step most often forgotten — taking the 'middle' of unsorted data gives a wrong answer. Unaffected by outliers, the median suits widely spread data such as incomes and flat prices.
3. The modeThe mode is the value that occurs most often. In 1, 2, 2, 3, 3, 3, 4 the counts are one, two and three, so the mode is 3. The mode also works for categories: a shoe shop says 'size 38 sells best' without any average foot length. A data set may have no mode (every value occurs once) or more than one mode (two values tie for the lead). Only counting is needed — no adding or subtracting — making the mode the most direct of the three measures.
4. Comparing the three: exam scoresFive students score 60, 70, 70, 80, 90. Mean = (60 + 70 + 70 + 80 + 90) ÷ 5 = 370 ÷ 5 = 74; the median is the 3rd sorted value, 70; the mode is the most frequent, also 70. The three results — 74, 70, 70 — sit close together because the data is fairly even with no outliers. The lesson: the same data can be measured by three different 'rulers' of centre, and different answers are definitions at work, not mistakes; always check which measure the question asks for.
5. Choosing the right measureThe mean is standard but one outlier can wreck it: for 1, 2, 3, 4, 100 the mean is 22, though most values lie between 1 and 4 — 22 represents nobody; the median of this set, 3, sits far closer to the majority. The mode answers 'most popular' questions — sizes, colours, classes. When reporting skewed data such as incomes or flat prices, quote the median alongside the mean; for symmetric data without oddities, the mean alone suffices. News reports pairing average and median flat prices follow exactly this logic.
6. Recurring errors and answer formatThree recurring errors: forgetting to sort before finding the median; with an even count, taking the n ÷ 2-th value instead of averaging the two middle ones; and reporting the frequency instead of the value as the mode — in 1, 2, 2, 3 the mode is 2, not 'twice'. The standard answer format is: state the definition or formula, substitute the data, then declare 'mean 74, median 70, mode 70' as separate items, each matched to its measure, so no credit is lost to mixing them up.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Mean = sum ÷ count
• Median = middle after sorting (average two if even)
• Mode = most frequent
• Always sort before finding median
• Outliers distort mean, median is better for skewed data