Lesson 26: Mean, Median, Mode

§Text Passage

1Summary 重點摘要

Three averages: mean (sum ÷ count), median (middle value), mode (most frequent). Learn to calculate all three.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: MeanMean = sum ÷ count. For example, five pupils in a class score 60, 70, 80, 90 and 100: the sum is 400, and the mean is 400 ÷ 5 = 80 marks. Why is the mean the most used? Because it employs all the data and captures the level of the whole set with one number — the class mean sums up overall results. Add first, then divide, and count the values carefully. Another example: the four numbers 10, 20, 30, 36 have total 96, and the mean is 96 ÷ 4 = 24. A common mistake is a correct total divided by the wrong count, or forgetting that a 0 in the data still counts as one value; zero is a value too, so it must be included when counting.
2. Block 2: MedianThe median is the number sitting exactly in the middle after the data are sorted from small to large. For example, 3, 5, 7 sorted keep 5 in the middle, so the median is 5; with an even number of values, such as 2, 4, 6, 8, there are two middle numbers 4 and 6, and the median = (4 + 6) ÷ 2 = 5, the average of the two middle values. Why sort first? Because 'the middle position' only makes sense in ordered data; a number picked from unsorted data is not the median. For example, the data 9, 1, 5 sort to 1, 5, 9, and the median is 5, not 9. With an odd count the median is always an original value; with an even count it may be newly computed. A common mistake is to take the middle without sorting, or, for an even count, to forget to add the two middle values and divide by 2.
3. Block 3: ModeThe mode is the value that appears most often in the data. For example, among 1, 2, 2, 3, 2, 5 the number 2 appears 3 times while the others appear once each, so the mode is 2. Why is a mode needed? Because some questions care most about 'what most people choose': a shoe shop wants its best-selling size, an election its most-supported option, and the mode answers directly. A set may have several modes — among 1, 1, 2, 2, 3 both 1 and 2 are modes — and if every value appears equally often, there is no mode. A common mistake is to give the number of appearances instead of the value, saying the mode is '3 times'; the mode is the value itself, not the frequency, so read carefully what the question asks for.
4. Block 4: What's the Difference?The three representatives have different characters. The mean uses all the data but is strongly affected by extreme values — one huge number drags it upward. The median looks only at the middle position and ignores extremes. The mode counts appearances and suits popular preferences. Why learn all three? Because the same data can give the three measures different values, and the question decides which fits. For example, among 2, 3, 3, 4, 100 the mean 22.4 is pulled up by the extreme 100, while the median 3 and the mode 3 represent the majority better. A common mistake is to reach for the mean in every question; when the data hold extreme values, look at the distribution first and decide whether the median or the mode is the fairer choice.
5. Block 5: Example: SalariesA company pays five monthly salaries: $10000, $12000, $15000, $18000 and $100000, the last one being the boss's. The mean = (10000 + 12000 + 15000 + 18000 + 100000) ÷ 5 = 155000 ÷ 5 = $31000, but the median is $15000. Why do the two differ so much? Because the boss's $100000 is an extreme value that pulls the mean up: $31000 exceeds the pay of the other four and fails to reflect ordinary staff, whereas the median $15000, untouched by the extreme, describes the true situation better. A common mistake is to read the mean $31000 as a typical monthly wage; when reading statistics, always check for extreme values and judge whether the mean is distorted.
6. Block 6: Example: Exam ScoresA set of exam scores 50, 60, 60, 70, 80: the mean = (50 + 60 + 60 + 70 + 80) ÷ 5 = 320 ÷ 5 = 64; sorted, the exact middle is 60, so the median = 60; and 60 appears twice, more than any other, so the mode = 60. Why are the three so close here? Because the data are even and hold no extreme values, so the mean, the median and the mode naturally land near one another, and any of them may serve as the representative. Contrast the salary example: with an extreme value the three spread far apart, and choosing the wrong representative misleads. A common mistake is to assume the three always agree; they are three different meanings of 'average' — close for symmetric data, far apart for skewed data — and any report should state which one is used.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Mean = sum ÷ count
• Median = middle value
• Mode = most frequent
• Mean affected by extremes
• Median not affected by extremes