Statistics questions appear in the Problem Solving and Data Analysis section. They're usually straightforward if you know the definitions and a few key tricks.
Mean (Average)
The mean is the sum of all values divided by the number of values.
Example 1: Find the mean of .
The Missing Value Trick
The SAT loves asking: "If the average of 5 numbers is 12, and four of them are 8, 10, 14, and 16, what is the fifth number?"
Total sum =
Sum of known values =
Missing value =
Median
The median is the middle value when data is sorted in order.
- Odd count: the middle number
- Even count: the average of the two middle numbers
Example 2: Find the median of .
Sorted: . Median = .
Example 3: Find the median of .
Median =
Effect of Outliers
The mean is sensitive to outliers; the median is not.
Data:
- Mean = (pulled up by 100)
- Median = (unaffected)
Mode
The mode is the most frequent value. A data set can have no mode, one mode, or multiple modes.
Example 4: In , the mode is (appears 3 times).
Range
Example 5: For , range = .
Standard Deviation (Conceptual)
The SAT won't ask you to calculate standard deviation โ just understand it.
Standard deviation measures how spread out the data is from the mean.
- Data close together โ small standard deviation
- Data spread out โ large standard deviation
Example 6: Which has a larger standard deviation?
- Set A:
- Set B:
Set B โ the values are much more spread out.
Key Rule
Adding the same number to every value does NOT change the standard deviation (it shifts the data but doesn't change the spread).
Multiplying every value by the same number DOES change the standard deviation.
Reading Data from Tables
Example 7: A survey recorded the number of books read:
| Books | Frequency |
|---|---|
| 0 | 3 |
| 1 | 5 |
| 2 | 7 |
| 3 | 4 |
| 4 | 1 |
Total students:
Mean:
Median: The 10th and 11th values. Counting through the frequencies: positions 1-3 are 0, positions 4-8 are 1, positions 9-15 are 2. Both the 10th and 11th values are . Median = .
Practice Problems
Problem 1: The average of 6 test scores is 85. If one score of 70 is removed, what is the new average?
Solution
Total = . Remove 70: . New average = .
Problem 2: Find the median of: .
Solution
Sorted: . Median =
Problem 3: If every student in a class scores 5 points higher on a retest, what happens to the mean and standard deviation?
Solution
Mean increases by 5. Standard deviation stays the same.
Key Takeaways
- Mean = sum รท count. Use the "total sum" trick for missing values.
- Median = middle value when sorted. Not affected by outliers.
- Standard deviation measures spread โ know it conceptually, not computationally.
- Adding a constant shifts the mean but doesn't change the spread.
- For frequency tables, multiply each value by its frequency before summing.
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