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.
Standard deviation measures spread. 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. 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?
In Unit 7 (Inference for Quantitative Data: Means), 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 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.
Keep studying AP Statistics Unit 7
Variance (Units 1 & 7)
Variance 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.
Confidence Interval (Unit 7)
Standard deviation drives the width of every confidence interval 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)
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.
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.
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.
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.
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.
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.
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.
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.
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.