Sample standard deviation (symbol: s) is a statistic that measures how spread out the values in a sample are around the sample mean. Because it's calculated from sample data, s varies from sample to sample, and it stands in for the unknown population standard deviation (σ) in t-based inference procedures.
Sample standard deviation is the typical distance between the data points in your sample and the sample mean. The symbol is a lowercase s, and that letter choice matters. Lowercase Roman letters (like s and x̄) are statistics computed from samples, while Greek letters (like σ and μ) are parameters describing the whole population.
The formula is s = √[Σ(xᵢ − x̄)² / (n − 1)]. Notice you divide by n − 1, not n. That tweak makes s a better estimate of σ, since samples tend to slightly underestimate population spread. Here's the bigger idea, though. Because s comes from a sample, it changes every time you draw a new sample. Four random samples of household incomes from the same city might give you standard deviations of 15,800, $13,200, and $14,500. None of those samples is broken. That sample-to-sample variation is exactly what Unit 5 wants you to expect and reason about.
Sample standard deviation lives in Topic 5.1 (Introducing Statistics: Why Is My Sample Not Like Yours?) and supports learning objective AP Stats 5.1.A, which asks you to identify questions suggested by variation in statistics from samples taken from the same population. The essential knowledge here is that variation in sample statistics may be random or not. Seeing four different values of s from four samples should prompt the question "is this just random sampling variability, or is something else going on?"
It also shows up in Topic 9.6 (Skills Focus: Selecting an Appropriate Inference Procedure). When you don't know σ, which is almost always on the AP exam, you use s instead, and that swap is the whole reason t-procedures exist. So s isn't just a Unit 1 computation. It's the bridge between describing data and doing inference.
Keep studying AP Statistics Unit 5
Population Standard Deviation (Unit 5)
σ describes the spread of the entire population and is a fixed (usually unknown) number, while s estimates it from a sample and changes with every new sample. On the exam, knowing σ means z-procedures and knowing only s means t-procedures.
Variance (Unit 1)
Sample variance (s²) is just the sample standard deviation squared. Variance is handy for the math (it adds nicely for independent random variables), but standard deviation is what you interpret, because it's in the same units as the data.
Mean (Unit 1)
You can't compute s without x̄ first, since every deviation is measured from the sample mean. The pair travels together. Mean tells you the center, s tells you how far values typically stray from it.
Hypothesis Test (Units 6-9)
Whenever you test a claim about a mean or a slope without knowing σ, s appears in the standard error of your test statistic. That substitution is why those tests use the t-distribution instead of the normal distribution.
Multiple choice loves the symbol question. You should instantly match s to sample standard deviation, σ to population standard deviation, x̄ to sample mean, and μ to population mean. Other MCQ stems give you several samples from the same population with different standard deviations and ask what statistical question that variation suggests, which is straight from LO 5.1.A.
On FRQs, s does its real work inside inference. The 2024 Q6 investigative task had a statistician estimating the mean price of a whistle from sample data, and the 2025 Q6 task involved analyzing reading comprehension data from a study of 100 children. In tasks like these, you use s to build standard errors, run t-procedures, and interpret variability. You almost never get σ on the exam, so recognizing "σ unknown, use s, use t" is a skill the exam rewards over and over.
Population standard deviation (σ) is a parameter that describes the spread of every value in the population, and it divides by N. Sample standard deviation (s) is a statistic computed from sample data, divides by n − 1, and varies from sample to sample. Quick test: if the number came from measuring everyone, it's σ; if it came from a sample, it's s. On the calculator, that's the difference between σx and Sx in your 1-Var Stats output, and picking the wrong one is a classic point-loser.
Sample standard deviation (s) measures the typical distance of sample values from the sample mean, in the same units as the data.
The formula divides by n − 1 instead of n, which corrects for the fact that samples tend to underestimate population spread.
Because s is a statistic, it varies randomly from sample to sample, and Topic 5.1 asks you to recognize whether that variation looks random or suspicious.
When σ is unknown, which is nearly always on the AP exam, you substitute s and use t-procedures instead of z-procedures.
Memorize the symbol pairs: s and x̄ are sample statistics, σ and μ are the population parameters they estimate.
It's the statistic s that measures how spread out a sample's values are around the sample mean x̄, calculated as s = √[Σ(xᵢ − x̄)² / (n − 1)]. It estimates the unknown population standard deviation σ.
Sample standard deviation (s) is computed from sample data, divides by n − 1, and varies between samples. Population standard deviation (σ) is a fixed parameter for the entire population and divides by N. On your calculator's 1-Var Stats, they appear as Sx and σx.
Deviations are measured from x̄, which sits in the middle of your own sample, so the raw spread comes out a bit too small. Dividing by n − 1 inflates the estimate just enough to make it an unbiased estimator of the population variance.
No, that's expected. Sample statistics vary randomly from sample to sample, and LO 5.1.A is literally about asking whether observed variation in statistics is random or suggests something else. Four income samples giving s values of $12,400 to $15,800 can all be perfectly normal.
Use s whenever σ is unknown, which covers nearly every inference problem on the exam. Using s instead of σ is exactly why you run a t-test or t-interval rather than a z-procedure for means and slopes.