Mean, Median, Mode

1Summary 重點摘要

Learn three averages: mean, median, mode, when to use each.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: MeanThe mean is the most commonly used average: add all the data to find the total, then divide by the number of values, that is, mean = total ÷ count. For example, for 2, 4 and 6 the total is 12, and 12 ÷ 3 = 4, so the mean is 4. It reflects the general level of the data, but it has a weakness: a single outlier pulls the mean up or down. For example, the mean of 2, 4, 6 is 4; after adding the outlier 100 the total becomes 112, and 112 ÷ 4 = 28 — the mean jumps from 4 to 28 at once, showing how large the outlier's influence is.
2. Step 2: MedianThe median is the middle value when the data are arranged from smallest to largest, and its position is not affected by outliers. Sorting first is essential: for example, 7, 2, 5 becomes 2, 5, 7, and the middle value 5 is the median. With 7 values the median is the 4th; in general, the median of n values lies in position (n + 1) ÷ 2. For example, with 3, 5, 7, 9, 100 arranged from small to large, the median is the 3rd value, 7, barely affected by the 100; this is its greatest difference from the mean.
3. Step 3: Even countIf there is an even number of values, there are two middle numbers, and the median is the average of these two. For example, after sorting 2, 3, 5, 8, the two middle numbers are 3 and 5, so the median = (3 + 5) ÷ 2 = 4. It does not matter that 4 was not in the original data; the median need not be one of the observed values. For example, the median of 1, 2, 4, 9 = (2 + 4) ÷ 2 = 3; when the two middle values are equal, say 5 and 5, the median is 5 — the method is exactly the same.
4. Step 4: ModeThe mode is the value that appears most often. Count how many times each value occurs, and the one with the highest frequency is the mode. For example, in 2, 3, 5, 5, 7 the value 5 appears twice while the others appear once, so the mode is 5. There can be more than one mode; if all values occur equally often, the data set has no mode. For example, in 4, 4, 4, 9 the mode is 4; for 1, 2, 3, 4 each value appears once, so there is no mode. The mode suits categorical data especially well, such as the most popular colour or size.
5. Step 5: Which bestWhich average to use depends on the data: when outliers are present the mean is distorted, so the median is preferred; when you want the most popular choice, the mode is best, such as the best-selling shoe size; in ordinary situations the mean is used most, because it uses information from every value. In an exam, state your reason before giving your conclusion. Example: four sales in an estate at 3.0, 3.2, 3.4 and 20 million pull the mean up to 7.4 million, while the median of 3.3 million better represents a typical transaction.
6. Step 6: InterpretAfter finding the mean, the median and the mode, compare them to understand the distribution. If the three are close together, the data are roughly symmetric; if the mean is clearly larger than the median, a very large outlier has pulled the mean up. When interpreting, quote the actual numbers and state what the conclusion applies to; only then is the analysis complete. For example, in 2, 2, 3, 5, 8: the mean is 4, the median 3 and the mode 2; the mean exceeding the median signals a longer right tail, with larger values pulling the mean up.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Mean: all sum divide by count
• Median: middle one after sorted
• Mode: most frequent one
• Even count median average two middle
• Have outlier use median most accurate