Probability and Statistics on the Digital SAT: Complete Guide

Probability and statistics are part of Problem-Solving and Data Analysis.

High-value skills include:

• probability;
• conditional probability;
• two-way tables;
• mean/median;
• spread and standard deviation;
• samples and populations;
• margin of error;
• interpreting studies.

See the rest of our SAT Math guides for other Data Analysis and Algebra topics.

Basic probability

[ P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}} ]

If 18 of 60 students choose option A:

[ P(A)=\frac{18}{60}=0.30 ]

= 30%.

Two-way tables

Probability a randomly selected student plays sport:

[ \frac{54}{100} ]

Probability a Grade 11 student plays sport:

[ \frac{24}{40} ]

The denominator changes because the condition restricts the group.

Conditional probability

[ P(A|B)=\frac{P(A\cap B)}{P(B)} ]

But on the SAT, a table often makes counting easier than memorizing the formula.

“What proportion of Grade 11 students play sport?”

Condition = Grade 11.

Denominator = all Grade 11 students.

The same fraction-setup habits also carry over to linear equations word problems.

Mean

[ \text{mean}=\frac{\text{sum}}{n} ]

If one data point increases by 10 in a set of 5 values, the mean rises by:

[ 10/5=2 ]

This shortcut is often faster than recomputing everything. For messier datasets, checking your arithmetic with Desmos can also save time.

Median

Middle value after ordering.

An extreme outlier can change the mean substantially while leaving median almost unchanged.

Standard deviation

You generally do not need to compute a long standard deviation formula manually.

Understand:

• larger SD = more spread;
• smaller SD = tighter clustering around the mean.

If two sets have the same mean but one is more dispersed, it has larger standard deviation.

Sampling

A sample should represent the population you want to study.

Bad sample:

Survey only varsity athletes to estimate how all students feel about school sports.

Selection bias.

Random sampling is often used to reduce systematic bias.

Observational study vs experiment

Observational study

Researchers observe without assigning treatment.

Can reveal associations.

Randomized experiment

Researchers randomly assign treatments.

Provides stronger evidence for causal conclusions when well designed.

Do not infer causation from observational data automatically.

Margin of error

Larger random samples generally lead to smaller margins of error, all else equal.

If a poll says:

52% ± 3%

the plausible interval is roughly:

49% to 55%.

Do not interpret margin of error as “3% of respondents answered incorrectly.”

Mini-practice

1

A bag contains 5 red, 3 blue, and 2 green tokens. Probability of blue?

2

In a class, 12 of 20 girls and 9 of 15 boys completed an assignment. What proportion of the whole class completed it?

3

Two datasets have equal means. Set A is tightly clustered; Set B is widely spread. Which has larger standard deviation?

4

A survey of only morning commuters is used to estimate the travel habits of all city residents. What is the main concern?

A. arithmetic error
B. selection bias
C. sample is too random
D. causation

5

A randomized experiment assigns participants to two treatments. Why is random assignment useful?

Answers

1. 3/10
2. 21/35 = 60%
3. Set B
4. B
5. It helps balance other characteristics across groups, strengthening causal comparison.

Common traps

• wrong denominator in conditional probability;
• mean vs median confusion;
• standard deviation treated as average;
• causal claims from observational studies;
• assuming larger sample removes every type of bias.

Careless algebra here is similar to what trips students up in quadratic equations — slow down and double-check denominators and signs.

Official references

• College Board Problem-Solving and Data Analysis: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving

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