Average (mean)

1Summary 重點摘要

Mean = sum ÷ count.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The formulaMean = total ÷ count. For the three numbers 3, 4 and 5, find the total first and then divide by the count: the total is 12, the count is 3, and the mean is 4. Why learn the mean? Because it stands for the whole set with a single number and is the most used measure in statistics — the class mean summarises overall results, and a month's mean temperature describes the weather. Once the formula is remembered, every mean question follows the same fixed plan: add first, divide second, always in that order. A common mistake is to remember it as 'total ÷ 2', or to forget to count the values first; before computing, ask yourself two questions: what is the total, and how many values are there, and begin only when both are answered.
2. Step 2: Add firstAdd all the numbers: 3 + 4 + 5 = 12. Why add first? Because the mean shares out a grand total, and if the total is wrong the mean goes wrong with it. Add in order one by one, or use pairing when the data are many: 3 + 5 = 8 first, then 4 more to reach 12 — whichever is easier is fine, and the result is the same. After adding, check once that every number in the question has been included, leaving none out. For example, the total of 45, 52, 48 and 55: 45 + 55 = 100 and 52 + 48 = 100, altogether 200 — pairing makes even mental addition quick. A common mistake is to leave out one value or copy a number wrongly; line the data up neatly and strike each one off as it is added, and nothing will be missed.
3. Step 3: Divide by the countDivide the total by the count: 12 ÷ 3 = 4, so the mean is 4. Why divide by the count? Because the mean says 'how much per share', and sharing the total 12 equally into 3 shares gives exactly 4 each. The count is the number of values; count them one by one and never miscount or skip, and with many values be extra careful — 5 values divide by 5, 8 values divide by 8. The quotient is the mean of the set and goes straight into the answer. For example, a total of 200 and a count of 4 give mean = 200 ÷ 4 = 50. A common mistake is a correct total divided by the wrong count, such as 4 values divided by 3; when the data are messy, point and mark each value, confirm the count, and only then divide.
4. Step 4: What it meansLevelling 3, 4 and 5 sweets gives everyone 4: this is the meaning of the mean — moving from those with more to those with less until all are equal, while the total never changes. Why does the total stay the same? Because what is given comes from the surplus, so it is only a redistribution, with nothing added and nothing taken away. The mean always sits between the largest and the smallest value: here the largest is 5 and the smallest is 3, and the mean 4 stands in between, bigger than some numbers and smaller than others. If a computed mean exceeds the largest or falls below the smallest, a mistake has surely been made. A common mistake is to think the mean must be one of the original numbers, although the mean of 3, 4, 6 is not; another is to confuse the mean with the middle value, which are different ideas.
5. Step 5: CheckVerify by multiplying: mean × count = total. Here 4 × 3 = 12, exactly the original total, so the mean is right. Why can multiplication check it? Because division and multiplication are inverse operations: dividing the total by the count gives the mean, and multiplying the mean by the count should restore the total, the two steps confirming each other. This check is simple, reliable and takes under a minute, and it should follow every mean you find. For example, total 200, count 4, mean 50: checking, 50 × 4 = 200, all correct. A common mistake is to re-add the total and repeat the very same slip; the multiplication check works from the count, re-examining the divisor, and combined with re-adding it gives double insurance.
6. Step 6: Finding a hidden valueA reverse question: three tests average 80 marks, the first two are 75 and 82, and the third is wanted. First turn the mean back into a total: total = 80 × 3 = 240; then subtract the known values: the third = 240 − 75 − 82 = 83 marks. Why find the total first? Because the mean hides the total: the question hands you the average while one score is missing, and only restoring the total and peeling off the known scores reveals the hidden one. Check: 75 + 82 + 83 = 240, and 240 ÷ 3 = 80, matching the question. A common mistake is to compute 80 − 75 − 82 directly and get a negative number; a mean can never be subtracted piece by piece — multiply back by the count to rebuild the total first, and that is the key opening move of every reverse question.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Mean = total ÷ count
• Reverse it: mean × count = total
• The mean need not be one of the values
• Bigger than the mean = above average; smaller = below
• Hidden value: subtract the known values from the total