On the test
Problem-Solving and Data Analysis is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I look at a set of numbers, I first ask two simple questions: where is the middle, and how spread out are they? I think about sharing cooking oil among friends. If everyone puts their oil into one big pot and we share it equally, each person gets the mean. If we line the friends up from the one with least oil to the one with most, the person in the middle shows the median. But two groups can both have the same middle and still be very different: in one group everyone has about the same, in another one person has almost all of the oil. That difference is the spread. The SAT asks you to read the middle and the spread of one list of numbers, and to see what happens when a number changes.
The words mean, median and standard deviation sound heavy, but you already use these ideas every day. Once you know what each word means, these are some of the fastest points on the test.
What is inside this part
Finding the average, the middle value, or the most common value of a data set.
mean from a list · median from a table · missing value given the mean
Describing or comparing how far apart the values are, using range, IQR or standard deviation.
range · interquartile range · comparing standard deviations
Describing whether a histogram or dot plot is symmetric or skewed, and what that means for mean and median.
skewed left or right · mean vs median · outliers
Finding how the mean, median or range change when a value is added, removed or changed.
adding an outlier · combining two groups · adding the same amount to every value
How to know a question is from this part
The question gives a list, table, dot plot, histogram or box plot of one kind of value and asks about its center or spread.
- The words mean, average, median, mode, range or standard deviation.
- A frequency table with a value column and a count column.
- A histogram or dot plot with one measurement along the bottom.
- 'Which data set has the greater standard deviation?' or 'which statement must be true?'.
- A value is added, removed, or two groups are combined.
The lesson
1. What 'one-variable data' means
One-variable data is a list of values about only one thing: for example the heights of 10 students, or the marks of a class on one test. Each value is one data point. Together they are a data set.
The SAT will show the data as a list, a frequency table, a dot plot, a histogram or a box plot. Your job is to describe the set: where is its center, how spread out is it, and what is its shape.
2. Measures of center: mean, median and mode
The mean is the average. Add all the values and divide by how many values there are. mean = sum ÷ count.
The median is the middle value after you put the numbers in order from smallest to largest. If there are two middle numbers (an even count), the median is halfway between them.
The mode is the value that appears most often. A set can have no mode, one mode, or several.
Find the mean and median of 3, 5, 6, 9, 12.
Show the steps
- Sum = 3 + 5 + 6 + 9 + 12 = 35. Count = 5.
- Mean = 35 ÷ 5 = 7.
- The numbers are already in order. The middle (third) value is 6, so the median is 6.
mean 7, median 6
3. Measures of spread: range, IQR and standard deviation
The range is the biggest value minus the smallest value. It is easy, but one extreme number can change it a lot.
The interquartile range (IQR) is the spread of the middle half of the data. It is the third quartile minus the first quartile (Q3 − Q1). On the SAT it is often given in a box plot.
The standard deviation tells you how far the values usually are from the mean. A small standard deviation means the values are close together. A large one means they are spread out. The SAT will NOT ask you to calculate a standard deviation by hand. It asks you to compare: which set has the larger one?
Set A is 10, 20, 30, 40, 50 and set B is 28, 29, 30, 31, 32. Both have mean 30. Which has the greater standard deviation?
Show the steps
- Look at how far values sit from 30.
- In A they are 20, 10, 0, 10, 20 away. In B they are 2, 1, 0, 1, 2 away.
- A is more spread out.
Set A has the greater standard deviation
4. Reading frequency tables
A frequency table says how many times each value appears. To find the mean, multiply each value by its frequency, add these products, and divide by the total count.
To find the median, add up the frequencies from the smallest value until you reach the middle position. If there are 20 values, the median is halfway between the 10th and 11th values.
Scores: 1 (4 students), 2 (6 students), 3 (8 students), 4 (2 students). Find the median.
Show the steps
- Total = 4 + 6 + 8 + 2 = 20 students.
- The middle is between the 10th and 11th values.
- Running total: score 1 covers students 1 to 4, score 2 covers 5 to 10, score 3 covers 11 to 18.
- The 10th is 2 and the 11th is 3, so the median is 2.5.
2.5
5. Shape: skewed and symmetric
A distribution is symmetric when the left and right sides look like mirror images. Then the mean and median are about the same.
A distribution is skewed right when there is a long tail of large values. The mean is pulled up to the right, so mean > median. It is skewed left when the tail goes to small values, and then mean < median.
Think of house prices in a town: a few very expensive houses pull the mean above the median.
6. What happens when the data changes
Adding or removing a value changes the mean. A very large or very small value (an outlier) pulls the mean a lot, but moves the median only a little. The range can change a lot too.
If you add the same number to every value, the mean and median go up by that number, but the range and standard deviation do not change. If you multiply every value by a number, the mean, median, range and standard deviation all get multiplied (the standard deviation by the positive size of that number).
The sum is the key to many mean questions: sum = mean × count. When a value is added or removed, update the sum first, then divide again.
The mean of 6 numbers is 20. A seventh number is added and the mean becomes 22. What is the seventh number?
Show the steps
- Old sum = 6 × 20 = 120.
- New sum = 7 × 22 = 154.
- Seventh number = 154 − 120 = 34.
34
7. Combining groups: weighted mean
If two groups have different sizes, you cannot just average their two means. The bigger group counts more. Add the two sums, then divide by the total count.
combined mean = (sum of group 1 + sum of group 2) ÷ (count 1 + count 2). Each sum is mean × count.
Class A has 20 students with mean 80. Class B has 30 students with mean 70. What is the combined mean?
Show the steps
- Sum A = 20 × 80 = 1,600. Sum B = 30 × 70 = 2,100.
- Total = 3,700 for 50 students.
- Mean = 3,700 ÷ 50 = 74.
74, not 75
8. Common traps
Trap 1: not putting the numbers in order before finding the median.
Trap 2: using the middle of the table instead of the middle of the data when frequencies are given.
Trap 3: thinking the median and mean are always equal.
Trap 4: calculating the standard deviation. You almost never need to; compare how far values are from the center.
Trap 5: averaging two means without thinking about group sizes.
See it
| Measure | What it tells you | Changed a lot by an outlier? |
|---|---|---|
| Mean | The average (sum ÷ count) | Yes |
| Median | The middle value in order | No, only a little |
| Mode | The most common value | No |
| Range | Largest minus smallest | Yes |
| IQR | Spread of the middle half (Q3 − Q1) | No |
| Standard deviation | Typical distance from the mean | Yes |
- Mean = 30
- Values very close together
- Small standard deviation
- Range = 4
- Mean = 30
- Values far apart
- Large standard deviation
- Range = 40
The data 2, 4, 5, 7, 9, 12, 15. The big value 15 pulls the mean slightly above the median.
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