Mean, Median and Range on the SAT
The three definitions
The mean is the total divided by the number of values. The median is the middle value once the data is in order, or the average of the two middle values when there is an even number of them. The range is the largest value minus the smallest.
If you remember one thing, make it this: mean $\times$ count $=$ total. Most of the harder mean questions turn out to be questions about totals.
The SAT asks for these numbers directly, but it also asks how they change. What happens to the mean and the median when an outlier is removed, when a value is added, or when every value goes up by the same amount?
What you will be asked
- A missing value and a known mean. Total $=$ mean $\times$ count. Subtract the values you know.
- Two groups with different means. Add the two totals, then divide by the combined count. Don't average the two means.
- An outlier. It pulls the mean toward it. The median barely moves.
- A frequency table. Count down to the middle position for the median. For the mean, multiply each value by its frequency and add.
- Every value increased by $5$. Mean and median go up by $5$. Range and standard deviation stay the same.
- Every value doubled. Mean, median and range all double.
Three examples
Example 1: a missing value
Five numbers have a mean of $20$. Four of them are $18$, $22$, $15$ and $25$. Find the fifth.
- Total $= 20 \times 5 = 100$.
- The four known values add to $80$.
- The fifth number is $100 - 80 = 20$.
Example 2: combining two groups
A class of $20$ has a mean score of $75$. A class of $30$ has a mean score of $80$. What is the mean of all $50$ students?
- Totals: $20 \times 75 = 1{,}500$ and $30 \times 80 = 2{,}400$.
- Combined: $\dfrac{3{,}900}{50} = 78$.
- Averaging the two means gives $77.5$, which ignores that the second class is bigger.
Example 3: the median from a frequency table
In a quiz, $2$ students scored $1$, $5$ scored $2$, $8$ scored $3$ and $5$ scored $4$. Find the median.
- There are $20$ scores, so the median is the average of the $10$th and $11$th in order.
- Positions $1$ to $2$ are $1$'s, $3$ to $7$ are $2$'s, $8$ to $15$ are $3$'s.
- The $10$th and $11$th are both $3$, so the median is $3$.
Common mistakes
- Finding the median without sorting. Put the values in order first.
- Averaging two means. When the groups are different sizes, combine the totals instead.
- Using frequencies as values. In a frequency table, the frequency says how many times each value appears. It is not a data value.
- Forgetting the even-count rule. With $6$ values, the median is the average of the $3$rd and $4$th.
- Thinking an outlier moves the median a lot. It changes the mean much more than the median.
Practice questions
For the mean questions, think in totals. For the median questions, put the data in order first.
Easy
What is the mean of the data set $4, 8, 9, 11, 13$?
- A$9$
- B$8$
- C$11$
- D$45$
Show solution
Answer: A. The total is $45$ and there are $5$ values: $\dfrac{45}{5} = 9$. Choice D is the total, not the mean.
EasyStudent-produced response
What is the median of the data set $12, 3, 7, 9, 15, 6$?
Enter your answer.
Show solution
Answer: $8$. In order: $3, 6, 7, 9, 12, 15$. With six values, the median is the average of the $3$rd and $4$th: $\dfrac{7 + 9}{2} = 8$.
MediumStudent-produced response
The mean of five numbers is $20$. Four of the numbers are $18$, $22$, $15$ and $25$. What is the fifth number?
Enter your answer.
Show solution
Answer: $20$. The total of all five is $5 \times 20 = 100$. The four known numbers add to $80$, so the fifth is $20$.
Medium
For the data set $2, 4, 4, 5, 7, 30$, which statement is true?
- AThe mean and the median are equal.
- BThe median is greater than the mean.
- CThe mean is greater than the median.
- DThe range is less than the mean.
Show solution
Answer: C. The mean is $\dfrac{52}{6} \approx 8.7$ and the median is $\dfrac{4 + 5}{2} = 4.5$. The outlier $30$ pulls the mean up. The range is $28$, so choice D is false.
Medium
A class of $20$ students has a mean test score of $75$. Another class of $30$ students has a mean score of $80$. What is the mean score of all $50$ students?
- A$77.5$
- B$79$
- C$78$
- D$76$
Show solution
Answer: C. Totals: $1{,}500$ and $2{,}400$. Combined mean: $\dfrac{3{,}900}{50} = 78$. Choice A averages the two means without weighting.
MediumStudent-produced response
What is the range of the data set $14, 9, 22, 17, 5, 11$?
Enter your answer.
Show solution
Answer: $17$. The largest value is $22$ and the smallest is $5$: $22 - 5 = 17$.
Hard
A data set of $10$ values has a mean of $12$. When one value, $30$, is removed, what is the mean of the remaining values?
- A$10$
- B$9$
- C$11$
- D$12$
Show solution
Answer: A. The total is $120$. Without $30$ it is $90$, spread over $9$ values: $\dfrac{90}{9} = 10$.
Hard
The data set $6, 7, 7, 8, 9, 40$ is changed by replacing $40$ with $10$. Which statement describes the effect?
- AThe mean and the median both decrease.
- BThe median decreases and the mean stays the same.
- CThe mean decreases and the median stays the same.
- DNeither the mean nor the median changes.
Show solution
Answer: C. The median is the average of the middle two values, $7$ and $8$, both before and after: $7.5$. The mean drops from $\dfrac{77}{6}$ to $\dfrac{47}{6}$.
HardStudent-produced response
In a quiz, $2$ students scored $1$ point, $5$ scored $2$ points, $8$ scored $3$ points, and $5$ scored $4$ points. What is the median score?
Enter your answer.
Show solution
Answer: $3$. There are $20$ scores. The $10$th and $11$th in order both fall in the group of $3$'s (positions $8$ to $15$), so the median is $3$.
Hard
Every value in a data set is increased by $5$. Which statement is true about the new data set?
- AThe mean and the range both increase by $5$.
- BThe mean and the median increase by $5$, and the range does not change.
- CThe mean increases by $5$, and the median does not change.
- DNothing changes, because every value moved by the same amount.
Show solution
Answer: B. Adding $5$ to every value moves the whole data set up by $5$, so the mean and median each increase by $5$. The gap between the largest and smallest values stays the same, so the range does not change.
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