AP Stats Standard Deviation Calculator

Standard deviation tells you how spread out your data are around the mean. A small standard deviation means the values cluster close to the mean; a large one means they are widely scattered. Use this calculator to find both the sample and population standard deviation — and to check which one your AP Statistics question is asking for.

What Is Standard Deviation?

Standard deviation measures how spread out values are around the mean. A small standard deviation means the values cluster close to the mean, and a large one means they are widely scattered. It is the square root of the variance, which puts it back in the original units of the data, such as points or centimeters.

How to Use This Calculator

1. Type your numbers into the box, separated by commas — for example, 2, 4, 4, 4, 5, 5, 7, 9.
2. Click Calculate.
3. Read the mean, count, and both standard deviations. Compare the sample and population values, then pick the one that matches your question.
4. Stuck? Click Try the worked example to load a ready-made data set and see exactly how the answer is found.

You need at least 2 numbers. The sample standard deviation is undefined for a single value because it divides by n − 1, which would be zero.

How to Calculate Standard Deviation

First find the mean, then find how far each value is from that mean, square those distances, average them, and take the square root. Squaring stops positive and negative distances from cancelling each other out.

Sample standard deviation: s = √[ Σ(x − x̄)² / (n − 1) ]

Population standard deviation: σ = √[ Σ(x − μ)² / n ]

The only difference is the denominator: n − 1 for a sample, n for a population. That is why the sample value is always a little larger than the population value for the same data.

Worked example: 2, 4, 4, 4, 5, 5, 7, 9

1. Count: n = 8.
2. Mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5.
3. Squared deviations: (2−5)² = 9, (4−5)² = 1, (4−5)² = 1, (4−5)² = 1, (5−5)² = 0, (5−5)² = 0, (7−5)² = 4, (9−5)² = 16. Sum = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
4. Population variance: 32 ÷ 8 = 4, so population SD σ = √4 = 2.
5. Sample variance: 32 ÷ 7 ≈ 4.5714, so sample SD s = √4.5714 ≈ 2.1381.

So for this data set: mean = 5, population SD = 2, and sample SD ≈ 2.1381. Try it in the calculator above to confirm.

Tips and Common Mistakes

• Do not mix up sample and population. This is the most common AP Stats error with standard deviation. If the problem says “a sample of students,” use s (n − 1). If it says “all students in the class” and the class is the whole group of interest, use σ (n).
• Check your data entry. A missing comma (for example, typing 45 instead of 4, 5) changes the mean and every squared deviation.
• Do not forget to take the square root. The average of the squared deviations is the variance, not the standard deviation. The final step is always √variance.
• Watch for outliers. Standard deviation is not resistant to outliers: one very large or very small value inflates it a lot. If the distribution is strongly skewed, also consider the IQR.
• Keep the units. Standard deviation is in the same units as the original data. If the data are in points, the standard deviation is in points — not “points squared” (that is the variance).
• Do not round too early. Carry full precision through the squared deviations and round only the final answer, usually to 2–4 decimal places.

Frequently Asked Questions

What is standard deviation in AP Statistics?

It is a measure of spread: the typical distance of data values from the mean. A standard deviation of 0 means every value is identical; larger values mean the data are more spread out.

What is the difference between sample and population standard deviation?

Sample standard deviation (s) divides the sum of squared deviations by n − 1 and is used when the data are a sample from a larger population. Population standard deviation (σ) divides by n and is used only when the data are the entire population. Dividing by n − 1 corrects for the fact that a sample tends to underestimate the population spread.

When should I use Sx vs. σx on my TI-84?

On a TI-84, run 1-Var Stats and look at the results: Sx is the sample standard deviation and σx is the population standard deviation. In AP Statistics, you will almost always want Sx, because the data are usually a sample. Use σx only when the problem explicitly treats the data as the whole population.

Can standard deviation be negative?

No. It is a square root of a sum of squares, so it is always zero or positive. If you get a negative value, you made an arithmetic or calculator error — check for a misplaced minus sign or a wrong data entry.

Why is standard deviation squared in the calculation, then square-rooted?

Simply averaging the deviations from the mean would always give zero, because values above and below the mean cancel out. Squaring makes every deviation positive, and the final square root brings the result back to the original units.

How does standard deviation relate to variance?

Variance is the standard deviation squared (s² or σ²). Variance is useful in formulas, but standard deviation is easier to interpret because it is in the same units as the data.

What if I have only one number?

Spread is meaningless for a single value, and the sample formula divides by n − 1 = 0. Enter at least 2 numbers to get a standard deviation.

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