Sample variance (s²) measures the spread of sample data by averaging the squared distances from the sample mean, dividing by n-1 instead of n. Its square root, the sample standard deviation s, replaces the unknown σ in standard error and t-interval calculations on the AP Statistics exam.
Sample variance, written s², tells you how spread out your sample data is around the sample mean x̄. You find it by taking each data point's distance from x̄, squaring it, adding all those squared distances up, and dividing by n-1 (not n). That n-1 detail matters. Dividing by n would systematically underestimate the true population variance, because data points are, on average, closer to their own sample mean than to the actual population mean. Dividing by n-1 corrects for that and gives an unbiased estimate.
In practice on the AP exam, you almost never report variance itself. You take its square root to get the sample standard deviation s, and s is what shows up in formulas. Since you basically never know the population standard deviation σ in real life, s steps in as its stand-in. That swap is exactly why the t-distribution exists. Using s instead of σ adds extra uncertainty, so the sampling distribution gets fatter tails, and the t-distribution with n-1 degrees of freedom accounts for it.
Sample variance lives in Topic 7.2 (Constructing a Confidence Interval for a Population Mean) in Unit 7: Inference for Quantitative Data: Means. It backs up several learning objectives at once. AP Stats 7.2.A says that when s is used instead of σ, the resulting distribution is a t-distribution with more area in the tails. AP Stats 7.2.D defines the standard error as SE = s/√n, where s is the square root of the sample variance. And AP Stats 7.2.E builds the whole interval, x̄ ± t*(s/√n), on top of that. So every one-sample t-interval you compute is quietly running on sample variance. Notice the n-1 thread, too. The same n-1 that makes s² unbiased is also the degrees of freedom for your t-distribution. That is not a coincidence; it is the exam's way of reminding you that estimating the mean from your own data costs you one degree of freedom.
Keep studying AP Statistics Unit 7
Standard Deviation (Units 1 & 7)
Sample standard deviation s is just the square root of sample variance. You meet s as a descriptive statistic back in Unit 1, then in Unit 7 it gets promoted to inference, replacing the unknown σ inside the standard error formula.
Confidence Interval (Unit 7)
The one-sample t-interval x̄ ± t*(s/√n) only exists because we use sample variance to estimate spread. If we magically knew σ, we would use a z-interval instead. Sample variance is the reason the answer is t, not z.
Margin of Error (Unit 7)
Margin of error equals t* times s/√n. A bigger sample variance means a bigger s, which means a wider interval. Spread in your data translates directly into uncertainty in your estimate.
Population Variance (Units 1 & 7)
Population variance σ² divides by N and describes the whole population. Sample variance s² divides by n-1 and estimates σ² from a sample. The entire point of s² is that you almost never get to see σ².
You will rarely see a question asking you to report sample variance by itself. Instead, it hides inside almost every Unit 7 problem. A typical question hands you a small dataset, like the differences in test scores (5, 8, -2, 4, 7, 2) for 6 students using a new study method, and asks for a 95% confidence interval for the mean improvement. To answer, you compute x̄ and s (your calculator's 1-Var Stats gives both), find t* with n-1 = 5 degrees of freedom, and build x̄ ± t*(s/√n). Two traps to watch. First, calculators report both s (sample, divides by n-1) and σ (population, divides by n). Grab s. Second, if a problem hands you the variance instead of the standard deviation, take the square root before plugging into SE = s/√n. The interval formulas are not printed on the formula sheet, but you can rebuild them from the test statistic and standard error formulas that are.
Population variance (σ²) describes the true spread of an entire population and divides the sum of squared deviations by N. Sample variance (s²) estimates that spread from a sample and divides by n-1. The n-1 is a deliberate correction. Because each data point tends to sit closer to its own sample mean than to the true population mean, dividing by n would lowball the variance. Dividing by n-1 fixes the bias. On the AP exam, you essentially never know σ², which is why s² (through s) appears in every one-sample t-procedure.
Sample variance s² measures spread by summing squared deviations from the sample mean and dividing by n-1, which makes it an unbiased estimate of the population variance.
Its square root, the sample standard deviation s, replaces the unknown σ in the standard error formula SE = s/√n for every one-sample t-procedure.
Using s instead of σ is exactly why you use a t-distribution with n-1 degrees of freedom, which has heavier tails than the normal distribution.
The same n-1 from the variance formula becomes the degrees of freedom for your t-interval, and as degrees of freedom increase the t-distribution looks more normal.
On your calculator, 1-Var Stats shows both s (divides by n-1) and σ (divides by n); for inference from a sample, always use s.
A larger sample variance produces a larger standard error and therefore a wider confidence interval, so more spread in the data means less precision in your estimate.
Sample variance (s²) measures how spread out sample data is around the sample mean. You square each deviation from x̄, sum them, and divide by n-1. Its square root is the sample standard deviation s, which you use in t-intervals like x̄ ± t*(s/√n).
Data points are, on average, closer to their own sample mean than to the true population mean, so dividing by n would underestimate the population variance. Dividing by n-1 corrects this bias and makes s² an unbiased estimator of σ².
No. Sample standard deviation s is the square root of sample variance s². The AP formulas (like SE = s/√n) use s, so if a problem gives you the variance, take the square root before plugging in.
Population variance σ² uses every value in the population and divides by N; sample variance s² uses only sample data and divides by n-1 to estimate σ². On the AP exam you almost never know σ², which is why inference for means uses s and the t-distribution.
Not really. Your calculator's 1-Var Stats gives you s directly, and the formula sheet provides the standard error formulas you build intervals from. What you do need to know is why n-1 is used and that using s instead of σ means using a t-distribution with n-1 degrees of freedom.