Mean, Median, Mode

1Summary 重點摘要

Learn three types of averages, know how to calculate and when to use which.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: MeanThe mean is the most common measure of central tendency: add all the values, then divide by the count. For 2, 4, 6, 8, 10: the sum is 2 + 4 + 6 + 8 + 10 = 30, the count is 5, and the mean is 30 ÷ 5 = 6. The mean's strength is that it uses every value, giving a precise result, which suits evenly spread data such as scores or heights. Its weakness is sensitivity to outliers: one unusually large or small value can drag the mean well away from the typical case, as the salary example in step five shows. Formula: mean = sum of values ÷ number of values.
2. Step 2: MedianThe median is the middle value after sorting the data from smallest to largest. With an odd count, the single middle value is the median: for 1, 3, 8, 20, 35 the median is 8. With an even count, average the middle two: for 2, 5, 7, 9, 12, 30 the median is (7 + 9) ÷ 2 = 8. The essential step is sorting first: taking the middle of 9, 1, 7 as 1 is wrong — after sorting, 1, 7, 9 has median 7. The median depends only on position and is completely untouched by the size of extreme values, which makes it especially reliable when outliers are present.
3. Step 3: ModeThe mode is the value that occurs most often. In 1, 2, 2, 3, 3, 3, 4, the 3 appears three times, more than any other value, so the mode is 3. If all values occur equally often, as in 1, 2, 3, 4, there is no mode; if two values tie for the lead, as in 2, 2, 5, 5, 9, both 2 and 5 are modes. The mode's unique worth is with categorical data: a shoe shop counting its best-selling size or a drink shop its most popular flavour handles data that are not numerical at all, so neither mean nor median applies, but the mode does. Tally the values in a table as you count; the eye alone easily misses one.
4. Step 4: Which average is bestEach of the three averages fits a different situation. When the data are evenly spread with no outliers, the mean is best because it uses all the information; when outliers exist, the median is more reliable because it depends only on middle positions; categorical data admit only the mode. An intuitive case: house numbers 8, 10, 12, 14, 16 have a meaningful mean of 12; but asked for 'the most popular door colour', only the mode can answer. The decision path: first ask whether the data are numerical or categorical; if numerical, check for unusually large or small values — with them choose the median, without them the mean. If the question names the measure, simply follow it.
5. Step 5: A salary example (the effect of outliers)A realistic example shows the power of outliers: five people in an office earn $10000, $12000, $11000, $13000 and $1000000 a month (the boss). The mean is (10000 + 12000 + 11000 + 13000 + 1000000) ÷ 5 = 1046000 ÷ 5 = $209200, far above any ordinary employee's salary and thoroughly misleading; the median of $12000 reflects the typical income. This is why government statistics and property reports quote median incomes and prices rather than means. When reading statistical news, first note which average is used, then consider whether outliers exist — only then can you judge the figure fairly.
6. Step 6: Common exam question typesThree question types recur in exams. Type one: given a data set, find all three averages at once — for 4, 7, 7, 9, 13 the mean is 40 ÷ 5 = 8, the median is 7 and the mode is 7; all three must be written, and one missing costs marks. Type two: work backwards for an unknown — 'the mean of four numbers is 9; three of them are 7, 8 and 15; find the fourth': the sum is 9 × 4 = 36, so the fourth is 36 − (7 + 8 + 15) = 6. Type three: compare the mean and the median to discuss outlier effects. For type two, the restoring formula 'mean × count = sum' is the key that unlocks the question.
7. Step 7: Common mistakes summaryEach average has its own typical errors. Mean: missing a value, dividing by the wrong count, or forgetting that 'mean × count = sum' serves as a check. Median: taking the middle without sorting (the most common slip), or with an even count, forgetting to average the middle two and simply picking one of them. Mode: quoting the frequency instead of the value — in 2, 2, 5 the value 2 appears twice, and the mode is 2, not '2 times'; also remember to answer 'no mode' when nothing repeats rather than forcing one. Quick checks after answering: the mode really must occur most often; the median must leave equal counts on both sides; the mean times the count must rebuild the sum.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Mean = Sum ÷ Count
• Median = Middle one when sorted
• Mode = Appears most times
• Use median when there's outlier
• Must sort before finding median