The mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.
The mean is the fair share: move blocks from the tall towers into the gaps until every tower is the same height. The total stays 24, spread over 4 towers, so each gets 6.
Formal definition and notation
For n numerical data values x1, x2, …, xn, the mean is x = x1 + x2 + ⋯ + xnn. It is a measure of center: if every value were replaced by the mean, the total would stay the same. It is also the balance point of the data on a number line, because the distances of the values above the mean add up to the distances of the values below it. Every value affects the mean, including outliers.
The mean does not have to be one of the data values or a whole number: the mean of 1, 2, 2 and 4 is 2.25.
The mean can be affected by outliers. For 2, 3, 4 and 31 the mean is 10, larger than three of the four values; the median is 3.5.
Adding a value above the mean raises it; adding a value below the mean lowers it; adding a value equal to the mean leaves it unchanged.
If you add the same number to every value, the mean goes up by that number. If you multiply every value by 2, the mean doubles.
For symmetric data, the mean and median are close. For skewed data, the mean is pulled toward the long tail.
Mean formula
Mean formula
mean = sum of the valuesnumber of values
sum of the values
add every data value, including zeros and repeats
number of values
how many data values there are, n
Turned around, the formula finds a total: sum = mean × number of values. That is how you solve missing-value problems.
Try it: find the balance point
Interactive
These are 8 quiz scores. Edit the numbers, drag a dot, or press Add an outlier. The readout shows the sum divided by the count, and the dashed blue line marks the mean. Watch how far one outlier moves the mean compared with the median.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Mean examples, step by step
Example 1: Find the mean
Warm-up
Find the mean of 12, 15, 9, 20 and 14.
Add the values: 12 + 15 + 9 + 20 + 14 = 70.
Count them: there are 5 values.
Divide: 70 ÷ 5 = 14.
Answer: The mean is 14.
Example 2: Find a missing score
Core
Ana’s mean score on five tests is 84. Four of her scores are 80, 90, 78 and 85. What is her fifth score?
Turn the mean into a total: 84 × 5 = 420 points on all five tests.
Add the four known scores: 80 + 90 + 78 + 85 = 333.
Subtract: 420 − 333 = 87.
Check: (333 + 87) ÷ 5 = 84.
Answer: Her fifth score is 87.
Example 3: A new value changes the mean
Test-style
The mean of 6 numbers is 15. When a seventh number is added, the mean becomes 17. What is the seventh number?
Total of the first 6 numbers: 6 × 15 = 90.
Total of all 7 numbers: 7 × 17 = 119.
The seventh number is the difference: 119 − 90 = 29.
Trap: 17 − 15 = 2 is the change in the mean, not the new number.
Answer: The seventh number is 29.
Common mistakes
✗ MistakeThe mean of 4, 8 and 12 is 4 + 8 + 12 = 24.
✓ InsteadDivide the sum by the count: 24 ÷ 3 = 8.
Do not forget to divide by how many values there are. The sum is a total, not a typical value.
✗ MistakeFor 0, 8, 10 and 6, the mean is 24 ÷ 3 = 8, skipping the 0.
✓ Instead0 is a data value: 24 ÷ 4 = 6.
Zeros add nothing to the sum, but they still count as values, so they change the number you divide by.
✗ MistakeClass A (20 students) averaged 80 and class B (30 students) averaged 90, so all 50 averaged (80 + 90) ÷ 2 = 85.
The range measures spread (max minus min), not center. 5, 5, 5 and 1, 5, 9 both have mean 5, but ranges 0 and 8.
Quick check: 3 questions
Question 1 of 3
What is the mean of 3, 8, 4, 10 and 5?
Show answer and explanation
Answer: A. The sum is 30 and there are 5 values, so the mean is 30 ÷ 5 = 6. 5 is the median, 30 forgets to divide, and 7 is the range.
Question 2 of 3
The mean of four numbers is 12. Three of them are 10, 15 and 9. What is the fourth?
Show answer and explanation
Answer: B. The four numbers must add to 4 × 12 = 48. The three known ones add to 34, so the fourth is 48 − 34 = 14. 12 assumes the missing value equals the mean, and 34 is the partial sum.
Question 3 of 3
The data are 5, 7, 9 (mean 7). Which change increases the mean?
Show answer and explanation
Answer: C. A new value above the mean pulls it up: adding 10 gives 31 ÷ 4 = 7.75. Adding 7 keeps the mean at 7, adding 5 lowers it to 6.5, and removing 9 lowers it to 6.
Mean: frequently asked questions
Is the mean the same as the average?
Usually, yes. When people say the average, they almost always mean the arithmetic mean: add the values and divide by how many there are. Strictly, average can refer to any measure of center, including the median and mode, which is why statistics books prefer the word mean. On tests, average without another word means the mean.
Can the mean be a decimal or a number that is not in the data?
Yes. The mean of 2, 3 and 6 is 11 ÷ 3 ≈ 3.67, which is not a data value and not a whole number. That is fine even when the data count whole things: a mean of 2.5 children per family describes the group, not any one family. Round only at the end, if the problem asks you to.
When is the mean not a good summary?
When the data are skewed or include outliers. Incomes, home prices and wait times often have a few very large values that pull the mean up, so it no longer describes a typical value. In those cases the median is the better measure of center. The mean works well for roughly symmetric data without outliers.
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