SAT Math: Mean, Median, Mode and Range
Learn How to Find the Average, Middle Value and Spread of Data
A list of numbers can describe many different things: test scores, temperatures, distances, prices, study times, or the number of items sold each day.
Instead of examining every number separately, we can summarize the data using four important measurements:
• Mean
• Median
• Mode
• Range
Each one describes the data in a different way.
The mean tells you the average.
The median tells you the middle value.
The mode tells you which value occurs most often.
The range tells you how far apart the smallest and largest values are.
Understanding the difference between these four measurements is essential for solving SAT Math questions involving data.
1. What Is a Data Set?
A data set is a collection of values.
For example:
20, 15, 30, 25, 10
These five numbers form a data set.
For many questions involving the median, it is helpful to arrange the values from smallest to largest:
10, 15, 20, 25, 30
This is called ascending order.
You can also arrange numbers from largest to smallest:
30, 25, 20, 15, 10
This is descending order.
For finding the median, the order of the values is especially important.
2. Mean: The Average
The mean is found by adding all the values and dividing the result by the number of values.
Formula
Mean = Sum of all values ÷ Number of values
Using symbols:
M = S ÷ n
where:
M = mean
S = sum of all values
n = number of values
The basic idea is simple:
Add everything, then divide by how many values there are.
3. Worked Example: Finding the Mean
Find the mean of:
6, 8, 10, 12, 14
Step 1: Add the values
6 + 8 + 10 + 12 + 14 = 50
Step 2: Count the values
There are 5 values.
Step 3: Divide
Mean = 50 ÷ 5
Mean = 10
Answer
10
4. A Useful Mean Shortcut
Sometimes a data set follows a regular pattern.
Consider:
8, 10, 12, 14, 16
The values are evenly spaced.
The mean is:
(8 + 16) ÷ 2 = 12
So the mean is 12.
Another example:
15, 20, 25, 30, 35
Mean = (15 + 35) ÷ 2
Mean = 50 ÷ 2
Mean = 25
This shortcut works for an evenly spaced data set.
It should not be used automatically for every list of numbers.
For example:
2, 3, 4, 10, 20
The first and last values have an average of:
(2 + 20) ÷ 2 = 11
But the actual mean is:
39 ÷ 5 = 7.8
So always check the data before using the shortcut.
5. The Most Useful Mean Formula
Many SAT questions give you the mean and the number of values but do not give you the total.
You can reverse the mean formula:
Total = Mean × Number of values
Using symbols:
S = M × n
This relationship is extremely useful when solving missing-number questions.
6. Worked Example: Finding the Total
A data set contains 8 numbers.
The mean is 15.
What is the sum of the numbers?
Total = 15 × 8
Total = 120
Answer
120
You do not need to know the individual values.
7. Finding a Missing Number
Suppose the mean is known and one value is missing.
The fastest method is:
Find the required total.
Add the known values.
Subtract the known total from the required total.
Formula
Missing value = Required total − Known total
8. Worked Example: Missing Number
The mean of five numbers is 18.
Four of the numbers are:
12, 15, 20, 21
What is the fifth number?
Step 1: Find the required total
Total = 18 × 5
Total = 90
Step 2: Find the total of the known values
12 + 15 + 20 + 21 = 68
Step 3: Find the missing value
90 − 68 = 22
Answer
22
Check
12 + 15 + 20 + 21 + 22 = 90
90 ÷ 5 = 18
The answer is correct.
9. How Adding a Number Changes the Mean
Suppose the mean of a data set is 20.
Now add another number.
What happens to the mean?
It depends on the new number.
If the new number is greater than the mean
The mean increases.
If the new number is less than the mean
The mean decreases.
If the new number equals the mean
The mean stays the same.
This is one of the most useful shortcuts for SAT Math questions.
10. Worked Example: Adding a New Value
The mean of four numbers is 12.
A fifth number, 20, is added.
What is the new mean?
Step 1: Find the original total
12 × 4 = 48
Step 2: Add the new value
48 + 20 = 68
Step 3: Divide by the new number of values
68 ÷ 5 = 13.6
Answer
13.6
The mean increased because 20 is greater than the original mean of 12.
11. A Faster Way to Think About the Change
Suppose the original mean is M.
If you add a value x:
• x > M → the mean increases
• x < M → the mean decreases
• x = M → the mean stays the same
You can often answer a question using this comparison alone.
You do not always need to calculate the new mean.
12. Removing a Number
The same idea works when a value is removed.
If the removed value is greater than the original mean, the mean decreases.
If the removed value is less than the original mean, the mean increases.
If the removed value equals the original mean, the mean stays the same.
13. Worked Example: Removing a Number
The mean of 6 numbers is 18.
One of the numbers, 30, is removed.
What is the new mean?
Original total
18 × 6 = 108
Remove 30
108 − 30 = 78
Five values remain
New mean = 78 ÷ 5
New mean = 15.6
Answer
15.6
The mean decreased because 30 was greater than the original mean.
14. Median: The Middle Value
The median is the middle value after the numbers have been arranged in order.
For example:
3, 7, 9, 12, 15
The middle value is 9.
Therefore:
Median = 9
Unlike the mean, the median is not found by adding every number and dividing.
The first step is always to put the values in order.
15. Median With an Odd Number of Values
When there is an odd number of values, there is one exact middle value.
Example
Find the median of:
14, 5, 9, 20, 7
First arrange the values:
5, 7, 9, 14, 20
There are 5 values.
The middle value is the third value.
Therefore:
Median = 9
Answer
9
16. Finding the Median Position
If there are n values and n is odd, the position of the median is:
(n + 1) ÷ 2
For 5 values:
(5 + 1) ÷ 2 = 3
So the median is the third value.
For 7 values:
(7 + 1) ÷ 2 = 4
So the median is the fourth value.
This can be useful when a data set contains many values.
17. Median With an Even Number of Values
When there are an even number of values, there are two middle values.
The median is the mean of those two values.
Example
Find the median of:
4, 8, 12, 16, 20, 24
The two middle values are 12 and 16.
Median = (12 + 16) ÷ 2
Median = 28 ÷ 2
Median = 14
Answer
14
18. An Unordered Data Set
Never identify the median before arranging the numbers.
Consider:
18, 5, 12, 9, 20
Arrange them:
5, 9, 12, 18, 20
The middle value is 12.
Answer
12
The original position of a number does not matter.
Only its position after sorting matters.
19. Mode: The Most Frequent Value
The mode is the value that appears most often.
Example
Find the mode of:
4, 7, 7, 9, 10, 7, 12
The number 7 appears three times.
The other numbers appear only once.
Therefore:
Mode = 7
Answer
7
20. More Than One Mode
A data set can have more than one mode.
Consider:
2, 4, 4, 6, 6, 8
The number 4 appears twice.
The number 6 also appears twice.
Therefore, both 4 and 6 are modes.
Answer
4 and 6
A data set with two modes is called bimodal.
21. No Mode
A data set can also have no mode.
Example:
3, 5, 8, 11, 14
Every value occurs exactly once.
Therefore, there is no mode.
The mode is determined by frequency, not by which number is largest or smallest.
22. Range: Measuring the Spread
The range measures the difference between the largest and smallest values.
Formula
Range = Largest value − Smallest value
Example
Find the range of:
8, 13, 5, 20, 11
Largest value = 20
Smallest value = 5
Range = 20 − 5
Range = 15
Answer
15
23. Range Is Not the Largest Value
Suppose the data set is:
4, 7, 10, 18
The largest value is 18.
The smallest value is 4.
Therefore:
Range = 18 − 4
Range = 14
The range is 14, not 18.
24. Finding All Four Measures
Consider:
2, 4, 6, 8, 10
Mean
Mean = (2 + 4 + 6 + 8 + 10) ÷ 5
Mean = 30 ÷ 5
Mean = 6
Median
The middle value is 6.
Median = 6
Mode
Every value appears once.
There is no mode.
Range
Range = 10 − 2
Range = 8
Answers
Mean = 6
Median = 6
Mode = No mode
Range = 8
25. What Happens When an Outlier Is Added?
Consider:
5, 6, 7, 8, 9
Now add 100.
The new data set becomes:
5, 6, 7, 8, 9, 100
The value 100 is much larger than the other values.
New mean
Mean = (5 + 6 + 7 + 8 + 9 + 100) ÷ 6
Mean = 135 ÷ 6
Mean = 22.5
New median
The two middle values are 7 and 8.
Median = (7 + 8) ÷ 2
Median = 7.5
New range
Range = 100 − 5
Range = 95
The unusually large value has a strong effect on the mean and range.
The median is much less affected.
This is an important concept when interpreting data.
26. Adding the Same Number to Every Value
Suppose the original data set is:
4, 6, 8, 10, 12
Now add 5 to every value:
9, 11, 13, 15, 17
The original mean is:
40 ÷ 5 = 8
The new mean is:
65 ÷ 5 = 13
The mean increased by 5.
The original median is 8.
The new median is 13.
The median also increased by 5.
But the range remains:
12 − 4 = 8
and:
17 − 9 = 8
Important rule
When the same number is added to every value:
• Mean increases by that number.
• Median increases by that number.
• Mode increases by that number, if a mode exists.
• Range stays unchanged.
27. Multiplying Every Value by the Same Positive Number
Consider:
2, 4, 6, 8, 10
Multiply every value by 3:
6, 12, 18, 24, 30
The original mean is 6.
The new mean is:
18
The original median is 6.
The new median is:
18
The original range is:
10 − 2 = 8
The new range is:
30 − 6 = 24
Important rule
When every value is multiplied by the same positive number:
• Mean is multiplied by that number.
• Median is multiplied by that number.
• Mode is multiplied by that number, if a mode exists.
• Range is multiplied by that number.
28. SAT Question: Adding a Value
A data set has a mean of 25.
A new value of 40 is added.
What happens to the mean?
A. It decreases.
B. It increases.
C. It stays the same.
D. It becomes 40.
Solution
The new value is 40.
The original mean is 25.
Since:
40 > 25
the mean increases.
Answer
B. It increases.
There is no need to calculate the new mean.
29. SAT Question: Finding a Missing Value
The mean of 6 numbers is 24.
Five of the numbers are:
18, 21, 25, 27, 30
What is the sixth number?
Step 1: Find the required total
24 × 6 = 144
Step 2: Find the known total
18 + 21 + 25 + 27 + 30 = 121
Step 3: Subtract
144 − 121 = 23
Answer
23
30. SAT Question: Finding the Median
A data set contains:
17, 5, 12, 9, 21
What is the median?
A. 9
B. 12
C. 17
D. 21
Solution
Arrange the values:
5, 9, 12, 17, 21
The middle value is 12.
Answer
B. 12
31. SAT Question: Finding the Range
A data set contains:
14, 22, 9, 30, 18
What is the range?
Solution
Largest value = 30
Smallest value = 9
Range = 30 − 9
Range = 21
Answer
21
32. Comparing Two Data Sets
Consider:
Data Set A:
10, 15, 20, 25, 30
Data Set B:
18, 19, 20, 21, 22
Both sets have a mean of 20.
Both sets also have a median of 20.
But their ranges are different.
Data Set A
Range = 30 − 10
Range = 20
Data Set B
Range = 22 − 18
Range = 4
Therefore, the two sets have the same mean and median but different amounts of spread.
This illustrates why one statistic cannot always describe an entire data set.
33. Common SAT Mistakes
Mistake 1: Forgetting to divide by the number of values
For:
4, 6, 8, 10
The sum is 28.
The mean is:
28 ÷ 4 = 7
Mistake 2: Finding the median before sorting
For:
9, 2, 15, 6, 4
First arrange:
2, 4, 6, 9, 15
The median is 6.
Mistake 3: Confusing mode with the largest value
The mode is the most frequently occurring value.
It has nothing to do with which value is largest.
Mistake 4: Confusing range with the largest value
Range = Largest − Smallest
Mistake 5: Assuming every statistic changes when one value changes
A change in one value can affect different statistics in different ways.
Always focus on the exact quantity the question asks about.
34. Practice Questions
Try these before looking at the solutions.
Question 1
Find the mean of:
8, 12, 16, 20, 24
Question 2
Find the median of:
17, 5, 12, 9, 21
Question 3
Find the mode of:
3, 5, 5, 7, 8, 5, 9
Question 4
Find the range of:
14, 22, 9, 30, 18
Question 5
The mean of 7 numbers is 16.
What is their total?
Question 6
The mean of 4 numbers is 18.
Three of the numbers are:
12, 20, 25
Find the fourth number.
Question 7
The data set is:
4, 6, 8, 10, 12
A number of 20 is added.
Does the mean increase, decrease, or stay the same?
Question 8
Find the median of:
6, 10, 14, 18, 22, 26
Question 9
The mean of 5 numbers is 30.
One of the numbers, 50, is removed.
What happens to the mean?
A. It increases.
B. It decreases.
C. It stays the same.
D. There is not enough information.
Question 10
A data set is:
2, 4, 6, 8, 100
Which measure is strongly affected by the unusually large value?
A. Mean
B. Median
C. Mode
D. None of these
35. Complete Solutions
Solution 1
Mean = (8 + 12 + 16 + 20 + 24) ÷ 5
Mean = 80 ÷ 5
Mean = 16
Answer: 16
Solution 2
Arrange the values:
5, 9, 12, 17, 21
The middle value is 12.
Answer: 12
Solution 3
The number 5 appears three times.
Answer: 5
Solution 4
Largest value = 30
Smallest value = 9
Range = 30 − 9
Range = 21
Answer: 21
Solution 5
Total = Mean × Number of values
Total = 16 × 7
Total = 112
Answer: 112
Solution 6
Required total:
18 × 4 = 72
Known total:
12 + 20 + 25 = 57
Missing value:
72 − 57 = 15
Answer: 15
Solution 7
First find the original mean:
(4 + 6 + 8 + 10 + 12) ÷ 5
= 40 ÷ 5
= 8
The added value is 20.
Since 20 > 8, the mean increases.
Answer: The mean increases.
Solution 8
The two middle values are 14 and 18.
Median = (14 + 18) ÷ 2
Median = 32 ÷ 2
Median = 16
Answer: 16
Solution 9
Original total:
30 × 5 = 150
Remove 50:
150 − 50 = 100
Four values remain.
New mean:
100 ÷ 4 = 25
The mean decreases from 30 to 25.
Answer: B. It decreases.
Solution 10
The value 100 is much larger than the other values.
It pulls the mean upward considerably.
The median is still 6.
Answer: A. Mean
36. Final Revision Sheet
Mean
Mean = Total ÷ Number of values
Median
Arrange the values and find the middle.
If there are two middle values, find their mean.
Mode
The value that occurs most frequently.
Range
Range = Largest − Smallest
Important shortcut
Total = Mean × Number of values
Missing value
Missing value = Required total − Known total
Adding a value
Value > Mean → Mean increases
Value < Mean → Mean decreases
Value = Mean → Mean stays the same
Changing every value
Adding the same number to every value:
Mean changes by that number.
Median changes by that number.
Range does not change.
Multiplying every value by the same positive number:
Mean is multiplied by that number.
Median is multiplied by that number.
Range is multiplied by that number.
37. The Main Idea to Remember
Mean, median, mode, and range all describe the same data from different perspectives.
The mean uses every value.
The median depends on the ordered position of the values.
The mode depends on frequency.
The range depends only on the smallest and largest values.
When an SAT Math question changes a data set, do not automatically assume that all four measurements change in the same way.
Identify the measurement being tested, apply the appropriate rule, and calculate only what is necessary.
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