Problem Solving & Data Analysis · SAT Math

Statistics on the Digital SAT

Statistics questions on the Digital SAT focus on the measures of center (mean, median, mode) and spread (range, standard deviation), plus reading values off charts and tables. The math is mostly arithmetic — the SAT tests whether you know which measure to use and how the measures change when the data changes. 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~10 min read

Measures of center: mean, median, mode

A measure of center summarizes a dataset with a single number. The SAT uses all three:

Mean (average): sum of values divided by count. Sensitive to outliers.

Median: the middle value when the data is sorted. (For an even count, the average of the two middle values.) Robust to outliers.

Mode: the value that appears most often. A dataset can have no mode, one mode, or multiple modes.

Outliers shift the mean but barely move the median. If one student in a class of 20 scores 1500 while everyone else scores around 1000, the mean rises noticeably but the median barely changes. The SAT tests this intuition.

Measures of spread: range and standard deviation

Spread tells you how clustered or scattered the data is around the center.

Range: the maximum value minus the minimum value. Cares only about the extremes.

Standard deviation: roughly the average distance from the mean. The SAT doesn't ask you to calculate it — it asks you to compare standard deviations between datasets.

Rule of thumb for standard deviation: data that clusters tightly around the mean has a small standard deviation. Data that's spread out has a large one. If two datasets have the same mean but one is more "packed" together, its standard deviation is smaller.

Weighted averages

A regular mean treats every value equally. A weighted average assigns each value a weight (importance) and computes:

weighted mean = Σ(value × weight) / Σ(weights)

Example. A class has 20 students who scored an average of 80, and 30 students who scored an average of 90. What is the overall class average?

Weighted: (80 × 20 + 90 × 30) / (20 + 30) = (1600 + 2700) / 50 = 86.

Not 85 (the simple average of 80 and 90). Weights matter.

What happens when you add or remove a value?

The SAT loves to ask "if you add value X, what happens to the mean / median / standard deviation?" Predict before you compute:

  • Adding a value equal to the mean: mean stays the same. Median might shift slightly toward it. Standard deviation decreases (data is more clustered).
  • Adding an outlier above the mean: mean increases. Median barely moves. Standard deviation increases.
  • Removing the smallest value: mean increases. Median shifts up. Standard deviation usually decreases.

Practice prediction: before doing any calculation, ask "is this new value above or below the current mean?" That tells you which direction the mean shifts.

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Solving with Desmos

Desmos' statistics features make these problems much faster. Three techniques:

1. Use stats(L) for instant calculations

Create a list: L = [12, 15, 18, 20, 22]. Then type mean(L), median(L), stdev(L). Desmos returns the values immediately.

2. Build a table to organize data

For data spread across categories or weighted averages, create a Desmos table with two columns: values and weights. Then compute sum(values · weights) / sum(weights).

3. Verify standard deviation comparisons by graphing

For two datasets, type each as a list and graph the points. Visual comparison of clustering tells you which has the smaller standard deviation — no calculation needed.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Computing mean when the question asks for median

Median is the middle value, not the average. Sort the data first, then pick the middle (or average the two middles if even count). The mean and median can be very different.

Forgetting to sort before finding the median

The data {7, 2, 9, 4, 5} has median 5 (sorted: 2, 4, 5, 7, 9). Picking 9 because it's in the middle of the unsorted list is the #1 median mistake.

Treating a weighted average as a simple average

If 20 students scored 80 and 30 scored 90, the class average isn't (80+90)/2 = 85. It's (20·80 + 30·90)/(20+30) = 86. The bigger group pulls the mean toward its value.

Computing standard deviation by hand

The SAT never asks for the actual SD value — only comparisons. If you're computing it, you're wasting time. Compare visually (clustering) instead.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

What is the median of the dataset 5, 12, 7, 8, 15, 4, 10?

  1. 7
  2. 8
  3. 10
  4. 12

Click to reveal solution →

Answer: (B) 8

Sort: 4, 5, 7, 8, 10, 12, 15. There are 7 values; the middle one (4th) is 8.

2

The mean of five numbers is 14. If a sixth number is added and the mean rises to 16, what is the sixth number?

  1. 18
  2. 22
  3. 26
  4. 30

Click to reveal solution →

Answer: (C) 26

Sum of original 5: 5 × 14 = 70.

New sum after adding sixth: 6 × 16 = 96.

Sixth number: 96 - 70 = 26.

3

In a class of 30 students, the average exam score was 78. After a new student scored 92 on the same exam joined the class, what is the new average (rounded to the nearest tenth)?

  1. 78.0
  2. 78.3
  3. 78.5
  4. 79.0

Click to reveal solution →

Answer: (C) 78.5

Original sum: 30 × 78 = 2340.

New sum: 2340 + 92 = 2432. New count: 31.

New mean: 2432 / 31 ≈ 78.45 → rounds to 78.5.

4

Two classes take the same test. Class A (with 20 students) has a mean of 75. Class B (with 30 students) has a mean of 85. What is the mean of the combined group?

  1. 79
  2. 80
  3. 81
  4. 82

Click to reveal solution →

Answer: (C) 81

Total sum: 20 · 75 + 30 · 85 = 1500 + 2550 = 4050.

Total students: 50. Combined mean: 4050 / 50 = 81.

5

The mode of the dataset 3, 5, 5, 7, 9, 9, 9, 12 is:

  1. 5
  2. 7
  3. 9
  4. 9.5

Click to reveal solution →

Answer: (C) 9

The mode is the most frequent value. 9 appears three times; 5 appears twice; everything else appears once.

6

Dataset X: 10, 10, 10, 10, 10. Dataset Y: 2, 6, 10, 14, 18. Both have mean 10. Which has the smaller standard deviation?

  1. Dataset X
  2. Dataset Y
  3. They are equal
  4. Cannot be determined

Click to reveal solution →

Answer: (A) Dataset X

Dataset X has every value exactly at the mean — zero spread, so standard deviation = 0. Dataset Y is spread out, so its SD is positive. X has the smaller SD.

7

A teacher records test scores: 72, 85, 90, 78, 65, 88, 95, 80. What is the range?

  1. 20
  2. 25
  3. 30
  4. 35

Click to reveal solution →

Answer: (C) 30

Max = 95, Min = 65. Range = 95 - 65 = 30.

8

The mean of four numbers is 10. If a fifth number, equal to 20, is added to the dataset, what happens to the median?

  1. It increases
  2. It decreases
  3. It stays the same
  4. Cannot be determined without the original values

Click to reveal solution →

Answer: (D) Cannot be determined without the original values

Without knowing the actual numbers (only their mean), we can't determine the median or how it changes. Mean alone doesn't pin down the data distribution.

Frequently asked questions

What's the difference between mean, median, and mode?

+
Mean is the average (sum ÷ count). Median is the middle value when sorted. Mode is the most-frequent value. All three are 'measures of center' but they capture different things — and they can be very different from each other in skewed data.

How do I calculate standard deviation on the SAT?

+
You don't. The SAT never asks for the actual value of standard deviation — it asks you to compare standard deviations between datasets, or predict how SD changes when data is added/removed. Use the rule: clustered data has low SD, spread-out data has high SD.

What's a weighted average?

+
An average where some values count more than others. The formula is Σ(value × weight) / Σ(weights). For example, combining two classes with different student counts requires weights — you can't just average the two class averages.

How does adding an outlier affect the mean vs median?

+
An outlier shifts the mean noticeably (since the mean uses every value). The median barely moves because it only depends on the middle position. This is why the median is preferred when data has outliers — it's 'robust.'

What's the range in statistics?

+
Range is the difference between the maximum and minimum values in a dataset. It's the simplest measure of spread but ignores everything between the extremes.

Keep going

Percent problems

Another core PSDA topic

Ratios and proportions

Foundational to data analysis

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