A data set can have hundreds of values. Statistics boils them down to a few numbers: one for the center, and one for how spread out the values are. Choosing the right summary matters as much as calculating it.
Measures of center
| Measure | How to find it | Best for |
|---|---|---|
| Mean | Add the values and divide by how many there are | Data without extreme outliers |
| Median | The middle value once the data is sorted | Data with outliers or a skewed shape |
| Mode | The value that appears most often | Categories, like favorite color |
Example Find the mean, median, mode and range of 4, 8, 6, 5, 3, 8, 9
- Sort: 3, 4, 5, 6, 8, 8, 9. There are 7 values.
- Mean: the sum is 43, so the mean is 43/7 ≈ 6.14.
- Median: the 4th value, 6.
- Mode: 8, the only value that repeats.
- Range: 9 − 3 = 6.
With an even number of values, the median is the mean of the two middle values. For 2, 5, 7, 10, the median is (5 + 7)/2 = 6.
Sort before finding the median
The median of 7, 3, 9 is 7, the middle of the sorted list 3, 7, 9. Taking the middle of the unsorted list gives 3, which is wrong.
Why outliers matter
One extreme value pulls the mean toward it. The median barely moves.
Example Salaries at a small company, in thousands of dollars: 40, 42, 45, 47, 50 and 300
- Mean: 524/6 ≈ 87.3.
- Median: (45 + 47)/2 = 46.
Five of the six people earn under 50, so a “typical salary” of 87.3 misleads. The median, 46, describes them better. This is why home prices and incomes are usually reported as medians.
Measures of spread
The range is the largest value minus the smallest. It’s quick but depends entirely on two values.
The interquartile range (IQR) is the spread of the middle half of the data: Q3 − Q1, where Q1 is the median of the lower half and Q3 is the median of the upper half.
Example Find the IQR of 1, 3, 5, 7, 9, 11, 13, 15
- Lower half: 1, 3, 5, 7, so Q1 = 4.
- Upper half: 9, 11, 13, 15, so Q3 = 12.
- IQR = 12 − 4 = 8.
A common rule flags a value as an outlier if it is more than 1.5 × IQR below Q1 or above Q3. Here that means below −8 or above 24.
Standard deviation
Standard deviation measures the typical distance of the values from the mean. A small standard deviation means the values cluster tightly. A large one means they’re spread out.
To find the population standard deviation:
- Find the mean.
- Subtract the mean from each value and square the result.
- Find the mean of those squares. That is the variance.
- Take the square root.
Example Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9
- The mean is 40/8 = 5.
- Squared distances from 5: 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.
- Variance: 32/8 = 4.
- Standard deviation: √4 = 2.
For a sample instead of a whole population, divide by n − 1 instead of n in step 3. Here that gives 32/7 ≈ 4.57 and a standard deviation of about 2.14. Your calculator shows both, usually labeled σ for population and s for sample.
On the SAT you’ll rarely calculate a standard deviation. You will be asked to compare two data sets and say which has the larger one, so learn to see spread at a glance.
How changes affect the summary
| Change to every value | Mean and median | Range, IQR and standard deviation |
|---|---|---|
| Add a constant k | Increase by k | Don’t change |
| Multiply by k | Multiply by k | Multiply by k (use the absolute value of k) |
Adding 10 points to everyone’s test score raises the average by 10, but the scores are just as spread out as before.
Weighted mean
When some values count more than others, multiply each by its weight and add.
Example Tests are 50% of a grade, quizzes 30% and homework 20%. A student averages 82 on tests, 90 on quizzes and 95 on homework.
0.5(82) + 0.3(90) + 0.2(95) = 41 + 27 + 19 = 87.
This is exactly how a GPA works. See Weighted vs. Unweighted GPA or try the GPA calculator.
Check yourself
Find the mean of 12, 15, 9, 20, 14.
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14. The sum is 70, and 70/5 = 14.
Find the median of 7, 3, 9, 1, 12, 5.
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6. Sorted: 1, 3, 5, 7, 9, 12. The middle two are 5 and 7.
The mean of five numbers is 18. Four of them are 15, 20, 17 and 22. What is the fifth?
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16. The total must be 5 × 18 = 90, and the four known values add to 74.
Every value in a data set is increased by 10. What happens to the standard deviation?
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It stays the same. Shifting every value doesn’t change how far apart they are.
Find the population standard deviation of 1, 3, 5.
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About 1.63. The mean is 3, the squared distances are 4, 0 and 4, the variance is 8/3.
A student scored 85, 90 and 78 on three tests. What do they need on the fourth to average 85?
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87. Four tests averaging 85 total 340, and 85 + 90 + 78 = 253.
Practice
Questions students ask
Can a data set have more than one mode?
Yes. If two values tie for most frequent, the data is bimodal. If every value appears once, there is no mode.
When should I use the median instead of the mean?
When the data has outliers or is skewed, like incomes or house prices. For data that’s roughly symmetric, like heights, the mean and median are close and the mean is the usual choice.
What does a standard deviation of zero mean?
Every value is the same. There’s no spread at all.