---
title: "Standard Deviation (σ) — AP Stats Definition & Exam Guide"
description: "Standard deviation (σ) measures how spread out data is from the mean. On the AP Stats exam, knowing σ vs. not knowing it decides whether you run a z-test or t-test."
canonical: "https://fiveable.me/ap-stats/key-terms/standard-deviation-s"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 4"
---

# Standard Deviation (σ) — AP Stats Definition & Exam Guide

## Definition

Standard deviation (σ) measures the typical distance of values from the mean; on the AP Stats exam, σ refers to the POPULATION standard deviation, and whether σ is known (z-test) or unknown (t-test) determines which inference procedure you use for a population mean.

## What It Is

[Standard deviation](/ap-stats/unit-1/investigative-question-revisited-data-collection/study-guide/f842Kr6YNnYX4G0dtAC8 "fv-autolink") measures [spread](/ap-stats/key-terms/spread "fv-autolink"). It tells you, roughly, how far a typical value sits from the mean. A small standard deviation means the data huddles close to the mean; a large one means the data is scattered.

The symbol matters in [AP Stats](/ap-stats "fv-autolink"). **σ (sigma) is the population standard deviation**, a fixed parameter describing the entire population. **s is the sample standard deviation**, a statistic you calculate from your data to estimate σ. This distinction drives Topic 7.4. In the real world you almost never know σ, so when you test a claim about a population mean, you plug in s instead. Substituting s for σ adds extra uncertainty, and that is exactly why you use a **t-distribution** instead of the normal (z) distribution. The whole z-versus-t decision boils down to one question: do you actually know σ?

## Why It Matters

In Unit 7 ([Inference for Quantitative Data: Means](/ap-stats/unit-4 "fv-autolink")), σ is the gatekeeper for choosing your test. Learning objective AP Stats 7.4.A says it directly: the appropriate test for a [population mean](/ap-stats/key-terms/population-mean "fv-autolink") with **unknown σ** is a one-sample t-test. That same logic carries into hypothesis setup (7.4.B, where H₀: μ = μ₀) and condition checking (7.4.C, where you verify independence and normality before trusting the test). Standard deviation also feeds the standard error formula, s/√n, which sits in the denominator of every t-statistic and confidence interval for means. If you misread whether σ is given, you pick the wrong procedure and lose points before you even compute anything.

## Connections

### [Variance (Units 1 & 7)](/ap-stats/key-terms/variance)

[Variance](/ap-stats/key-terms/variance "fv-autolink") is just standard deviation squared (σ²). Standard deviation is more useful for interpretation because it lives in the same units as your data, so "σ = 3 inches" actually means something, while "σ² = 9 square inches" does not.

### Population Mean (μ) (Unit 7)

σ and μ are the two parameters that define a population's center and spread. Topic 7.4 tests claims about μ, and σ (or its stand-in, s) controls how much your sample mean x̄ is allowed to wobble around μ by [chance](/ap-stats/unit-3 "fv-autolink").

### [Confidence Interval (Unit 7)](/ap-stats/key-terms/confidence-interval)

Standard deviation drives the width of every [confidence interval](/ap-stats/key-terms/confidence-interval "fv-autolink") for a mean through the formula s/√n. Bigger spread means a wider interval, and a bigger sample size shrinks it. Same machinery, different inference goal.

### [Independence Condition (Unit 7)](/ap-stats/key-terms/independence-condition)

Before you use s in a t-test, the conditions in 7.4.C have to hold: random data collection, n ≤ 10% of N when sampling without replacement, and an approximately normal sampling distribution (n > 30, or no strong skew if smaller). Spread calculations mean nothing if the sample itself is biased.

## On the AP Exam

Multiple-choice questions love the z-versus-t decision. A typical stem describes a study and asks which test is appropriate, and the answer hinges on whether σ is known. One practice question asks when a one-sample t-test is NOT appropriate; another asks which conditions would make a z-test better than a t-test (the answer involves a known population σ). On FRQs, the inference question (usually one of the six free-response questions) expects you to name the test, and "one-sample t-test for a population mean" is correct precisely because σ is unknown. Writing "z-test" when σ isn't given is one of the most common ways to lose the procedure point. You'll also use s when checking conditions and computing the standard error s/√n in your test statistic.

## Standard Deviation (σ) vs Sample standard deviation (s)

σ is the population standard deviation, a fixed (usually unknown) parameter. s is the sample standard deviation, a statistic computed from your data to estimate σ. The AP exam exploits this constantly. If a problem gives you σ, you can use a z-test. If you only have s (which is almost always), you must use a t-test, because estimating σ with s adds uncertainty that the t-distribution's heavier tails account for.

## Key Takeaways

- Standard deviation measures the typical distance of data values from the mean, in the same units as the data.
- σ is the population standard deviation (a parameter); s is the sample standard deviation (a statistic that estimates σ).
- When σ is unknown, the correct procedure for testing a population mean is a one-sample t-test, per learning objective 7.4.A.
- Standard deviation appears in the standard error formula s/√n, which controls the size of your test statistic and the width of your confidence interval.
- For matched pairs, you find the differences first, then run a one-sample t-test on those differences using their standard deviation.
- Before using s in inference, verify the conditions: random sampling, n ≤ 10% of the population, and approximate normality of the sampling distribution.

## FAQs

### What is standard deviation in AP Stats?

Standard deviation measures how spread out data is around the mean. In AP Stats, σ specifically means the population standard deviation, while s means the sample standard deviation calculated from your data.

### Do I use a z-test or t-test when σ is unknown?

Use a one-sample t-test. The CED is explicit (7.4.A): when the population standard deviation σ is unknown, you estimate it with s, and that extra uncertainty requires the t-distribution. A z-test is only appropriate when σ is actually known, which is rare.

### Is standard deviation the same as standard error?

No. Standard deviation describes the spread of individual data values, while standard error (s/√n) describes the spread of the sampling distribution of x̄. Standard error shrinks as sample size grows; standard deviation does not.

### What's the difference between standard deviation and variance?

Variance is the standard deviation squared (σ² vs. σ). Standard deviation is preferred for interpretation because it's in the same units as the data, which is why exam answers describe spread using standard deviation, not variance.

### How does standard deviation work in a matched pairs test?

First subtract to get one list of differences (and define the order of subtraction). Then treat those differences as a single sample: compute their mean and standard deviation, and run a one-sample t-test on μd just like any population mean test.

## Related Study Guides

- [4.4 Setting Up a Test for a Population Mean or Population Mean Difference](/ap-stats/unit-4/setting-up-test-for-population-mean/study-guide/1gAGgo2P3abc5G0sLvMr)

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