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Math foundations
SAT mean vs. median: what one outlier does to your data
Drag a single data point and watch the mean chase it while the median holds still. Then use that intuition to answer SAT data questions without heavy arithmetic.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- The mean uses every value, so one extreme value can pull it far; the median depends only on the middle of the ordered list.
- When a data set has a long tail to the right, the mean is usually greater than the median.
- Many SAT data questions ask what must change when one value changes. Reason about the definitions before you calculate.
In this article
Two ways to describe the center
The mean is the balance point of a data set: add the values and divide by how many there are. The median is the middle value after you put the data in order. With an even number of values, the median is the mean of the two middle values.
Both describe a “typical” value, but they react differently to change. Try it below: nine students reported how many minutes they read yesterday. One student, shown in red, reported 60 minutes. Drag that value and watch the two center lines.
Try it yourself · interactive
Move one value. Which center moves with it?
- Mean
- 23
- Median
- 20
- Range
- 48
Why the outlier pulls the mean but not the median
With the red value at 60, the sum is 207 and the mean is \(\displaystyle 207 \div 9 = 23\). Change 60 to 25 and the sum drops by 35, so the mean drops to about 19.1. The median is 20 in both cases: the fifth of nine ordered values never changed.
That is the whole idea in one sentence: the mean feels every value’s size, while the median only cares about order. An unusually large or small value is an outlier, and it can make the mean a misleading summary of a “typical” student.
- Increase the largest value: the mean rises, the median stays the same, and the range grows.
- Remove a high outlier: the mean falls, and the median shifts at most slightly.
- Add the same number to every value: both the mean and the median rise by that number, and the range stays the same.
Read the shape: skew tells you which center is bigger
Picture a histogram of household incomes or commute times: most values bunch on the left, and a few large values stretch a long tail to the right. Those tail values drag the mean toward them, so the mean usually sits to the right of the median.
Flip the picture for a tail to the left, such as scores on an easy quiz where most students did well and a few did poorly: the mean is usually less than the median. When the distribution is roughly symmetric, the mean and the median are close.
- Long right tail (right-skewed): mean usually greater than median.
- Long left tail (left-skewed): mean usually less than median.
- Roughly symmetric: mean ≈ median.
Spread: same center, different story
Two data sets can share a mean and still look very different. The lists {8, 9, 10, 11, 12} and {2, 6, 10, 14, 18} both have mean 10 and median 10. The second list’s values sit much farther from the center, so its range (16 versus 4) and its standard deviation are larger.
The SAT typically asks you to compare standard deviations, not compute them. Look at how tightly the values cluster around the center: tighter clustering means a smaller standard deviation.
The study lab
Equal centers, different spreads
Both lists have mean 10; the second list’s values are spread farther from it.
- List A: 8, 9, 10, 11, 12 (row y = 2)
- List B: 2, 6, 10, 14, 18 (row y = 1)
How SAT questions test this
Expect questions that change one value, add or remove a value, or compare two distributions shown in a table, dot plot, or histogram. Before calculating, ask which measure depends on size (mean, range, standard deviation) and which depends only on position (median).
When the data come in a frequency table, remember that each row stands for repeated values. A value of 4 with frequency 6 contributes 24 to the sum and six positions to the ordered list.
Your turn
Quick practice
A data set has 11 different values. The greatest value is increased by 30. Which of the following must be true?
Hint: Which measure uses the size of every value, and which uses only the middle position?
Practice: try these before checking the answers
Use definitions first, arithmetic second.
- 1. Find the mean and median of 3, 7, 7, 9, 14.
- 2. In the list above, 14 becomes 44. What are the new mean and median?
- 3. Which list has the larger standard deviation: 20, 21, 22, 23, 24 or 2, 12, 22, 32, 42?
- 4. A histogram has a long tail to the right. Which is probably larger, the mean or the median?
Sources & further reading
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