SAT Math · Problem-Solving and Data Analysis

Mean and median

Q: What is the difference between the mean and the median?

The mean is the arithmetic average: add all the values and divide by how many there are. The median is the middle value when the data is in order. The mean is sensitive to outliers, and the median is not.

Q: How do you find the median with an even number of values?

Put the values in order and take the average of the two middle values. For 3, 5, 6, 9, 12, 13, the median is (6 + 9) ÷ 2 = 7.5.

Q: How does an outlier affect the mean and median?

An outlier pulls the mean toward itself, because every value counts toward the sum. The median barely changes, because it depends only on which values are in the middle, not how extreme the ends are.

Q: When is the median better than the mean?

Use the median when the data is skewed or has outliers, such as incomes, home prices, or wait times. The median then describes a typical value better than the mean does.

Q: How do you find a missing value from the mean?

Multiply the mean by the number of values to get the total, then subtract the values you know. If 6 scores must average 88, they must sum to 528.

Q: How can Aristotle help with SAT data analysis?

Aristotle is a voice AI SAT tutor. Your child talks through a statistics question out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including measures of center and spread.

How to find each one, and why an outlier pulls the mean but barely moves the median.

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Updated October 30, 2026

$\text{mean} = \dfrac{\text{sum}}{\text{count}}$

An outlier at 20 drags the mean to 6. The median stays at 4.5, inside the cluster.

What are the mean and the median?

\text{mean} = \dfrac{\text{sum}}{\text{count}}

\text{median} = \text{middle of the sorted list}

Both are one number standing in for the middle of a data set. The mean shares the total out equally across every value. The median is the value that splits the sorted list into two equal halves.

Seven values: median 4 (the fourth value), mean 5 (35 shared seven ways).

The mean is where the data balances

\text{below: } (5-1)+(5-4) = 5

\text{above: } (6-5)+(9-5) = 5

Put the dots on a seesaw. It balances at the mean, because the distances below the mean add up to exactly the distances above it. Every value pushes on the mean, and the farther away it sits, the harder it pushes.

1, 4, 6 and 9 balance on a fulcrum at 5.

An outlier drags the mean, not the median

\text{mean } 4 \to 6

\text{median } 4 \to 4.5

Start with 2, 3, 3, 4, 5, 5, 6: mean 4, median 4. Add a 20 and the seesaw must shift far right to balance, so the mean jumps to 6. The median only counts positions, so it barely moves.

Adding 20 drags the mean to 6; the median only reaches 4.5.

Why outliers move the mean

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
Data: 2, 3, 3, 4, 5, 5, 6, 6. Change the last 6 to 20. What happens to the sum?
Answer
It grows by 14, so the mean grows by $14 \div 8 = 1.75$ .
Question 2
Does the median change?
Answer
No. The two middle values are still 4 and 5, so it stays 4.5.
Question 3
So which is better for incomes or home prices?
Answer
The median, because a few huge values pull the mean up.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

Find the mean and median of 4, 7, 8, 10, 11.

Question 2

What is the median of 9, 3, 12, 5, 13, 6?

Question 3

The mean of 5 test scores is 87. What must a sixth score be for the mean of all 6 to be 88?

Question 4

The data set 12, 14, 15, 15, 16, 18, 45 has an outlier at 45. If 45 is removed, which decreases more, the mean or the median?

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The definitions

Mean

$\text{mean} = \dfrac{\text{sum of values}}{n}$

$n$ is the number of values. Every value affects the mean, including outliers.

Median

$\text{middle value, in order}$

With an even count, average the two middle values.

Sum from the mean

$\text{sum} = \text{mean} \times n$

The key step for questions that add or remove a value and ask for the new mean.

Range

$\text{range} = \max - \min$

A simple measure of spread. Standard deviation measures how far values typically sit from the mean.

Which way the mean moves
Shape of the data	Mean vs. median	Better center
symmetric	about equal	either
skewed right (high outlier)	$\text{mean} > \text{median}$	median
skewed left (low outlier)	$\text{mean} < \text{median}$	median

Try it

Drag the outlier

Drag the last value and watch which measure follows it.

Last value6

Data2, 3, 3, 4, 5, 5, 6, 6

Mean34 ÷ 8 = 4.25

Median(4 + 5) ÷ 2 = 4.5

Drag the last value out to 30. The mean chases the outlier; the median barely moves.

Common mistakes

Not sorting before finding the median

Put the values in order before you pick the middle.

Picking one middle value from an even count

With six values there is no single middle. Average the third and fourth values.

Averaging the means of two groups

For groups of different sizes, add the sums and divide by the total count.

Reading a frequency table as a list

“3 appears 4 times” adds 3 four times and counts as four values.

How it shows up on the SAT

Problem-Solving and Data Analysis is about 15% of SAT Math. Expect dot plots and tables, a missing value from a mean, and which measure an outlier changes.

What Aristotle tracks in one-variable data: distributions and measures of center and spread

5 SAT Math skills, including:

Interpreting data distributions from tables, histograms, dot plots, and box plots
Calculating and comparing mean, median, and range
Comparing distributions with different means and the same standard deviation
Comparing distributions with different standard deviations

What is the difference between the mean and the median?

The mean is the arithmetic average: add all the values and divide by how many there are. The median is the middle value when the data is in order. The mean is sensitive to outliers, and the median is not.

How do you find the median with an even number of values?

How does an outlier affect the mean and median?

When is the median better than the mean?

How do you find a missing value from the mean?

How can Aristotle help with SAT data analysis?

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