What does standard deviation tell you?

Standard deviation tells you how spread out the values in a data set are around the mean. A low value means data points cluster tightly; a high value means they are widely spread out.

What is the difference between standard deviation and variance?

Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance and is in the same units as the original data, making it easier to interpret.

When should I use sample vs. population standard deviation?

Use population (σ) when your data includes every member of the group. Use sample (s) when your data is a subset of a larger group. When in doubt on homework, use sample standard deviation.

What is a good standard deviation?

It depends on context. Compare SD to the mean: if SD divided by mean times 100 exceeds 50%, variability is high for most practical purposes.

How do you find standard deviation in Excel?

Use =STDEV.S(range) for sample standard deviation or =STDEV.P(range) for population standard deviation.

What is the relationship between standard deviation and z-score?

A z-score measures how many standard deviations a value is from the mean: z = (x - μ) / σ. Use the Z-Score Calculator to find the percentile rank of any value once you have the standard deviation.

Standard Deviation Calculator

Formula

Enter your values

Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

What Is Standard Deviation?

Standard deviation measures how spread out the values in a data set are around the mean. A low standard deviation means the values cluster tightly around the average. A high standard deviation means they are widely spread out.

Intuitive example: Two classes both average 75 on an exam. Class A has scores of 73, 74, 75, 76, 77 — tightly grouped. Class B has scores of 50, 60, 75, 90, 95 — much more spread. Class B has a higher standard deviation even though the means are equal.

Population vs. Sample Standard Deviation — Which One Should I Use?

This is the most common source of confusion. Here is the plain-language decision rule:

Population (σ): You have data for every member of the group (e.g. every student in one specific class).
Sample (s): You have data from a subset of a larger group (e.g. 30 randomly selected students from a school of 1,000).
When in doubt on a homework problem → use sample (s). Most statistics courses default to sample standard deviation unless explicitly told otherwise.

Standard Deviation Formula

The two formulas differ by one detail — whether you divide by N or N−1:

Population: σ = √[ Σ(xᵢ − μ)² / N ]

Sample: s = √[ Σ(xᵢ − x̄)² / (N − 1) ]

Symbol	What It Means
σ or s	Standard deviation (population or sample)
xᵢ	Each individual value in the data set
μ or x̄	The mean (average) of all values
N	Total number of values
N − 1	Bessel's correction — used for samples to reduce bias
Σ	Sum of all values

On Bessel's correction (N−1): Dividing by N−1 instead of N corrects for the fact that a sample tends to underestimate the true spread of the population. When N is large (30+), the difference between dividing by N and N−1 is negligible.

Standard Deviation vs. Variance

	Variance	Standard Deviation
Formula	Σ(x − μ)² / N	√Variance
Units	Squared units (e.g. cm²)	Same units as the data (e.g. cm)
Easier to interpret?	No — squared units are hard to visualise	Yes — same scale as your data
Typical use	Intermediate step in the calculation	Final reported statistic

Step-by-Step Example — How to Calculate Standard Deviation by Hand

Use the exam scores: 72, 85, 90, 68, 94, 78, 83 (7 scores). We'll calculate the sample standard deviation.

Find the mean (x̄). Add all values: 72 + 85 + 90 + 68 + 94 + 78 + 83 = 570. Divide by 7: x̄ = 570 / 7 ≈ 81.43.
Find each deviation (xᵢ − x̄). 72 − 81.43 = −9.43 | 85 − 81.43 = 3.57 | 90 − 81.43 = 8.57 | 68 − 81.43 = −13.43 | 94 − 81.43 = 12.57 | 78 − 81.43 = −3.43 | 83 − 81.43 = 1.57.
Square each deviation. 88.92 | 12.74 | 73.44 | 180.36 | 158.00 | 11.76 | 2.46.
Sum the squared deviations. 88.92 + 12.74 + 73.44 + 180.36 + 158.00 + 11.76 + 2.46 = 527.68.
Divide by N − 1. 527.68 / (7 − 1) = 527.68 / 6 = 87.95. This is the variance.
Take the square root. √87.95 ≈ 9.38. The sample standard deviation is approximately 9.38.

Why this works: Squaring the deviations removes negative signs so they don't cancel each other out. Taking the square root at the end brings the units back to the original scale.

How to Interpret Your Standard Deviation Result

Getting the number is only half the job — knowing what it means in context is the other half.

Context	Low SD Means	High SD Means
Exam scores (out of 100)	Most students scored near the class average	Scores varied widely — some very high, some very low
Lab measurements	Consistent, precise measurements	High variability — possible measurement error
Stock returns	Stable, low-risk investment	Volatile, high-risk investment

The empirical rule (68-95-99.7 rule): For data that follows a normal distribution, approximately:

68% of values fall within 1 standard deviation of the mean
95% of values fall within 2 standard deviations
99.7% of values fall within 3 standard deviations

So for our exam example (mean ≈ 81.43, SD ≈ 9.38), about 68% of scores should fall between 72.05 and 90.81.

Common Mistakes to Avoid

Mistake 1 — Using N instead of N−1 for a sampleAlways check whether your data is the full population or a sample before choosing the formula. Use N only for the full population.

Mistake 2 — Forgetting to square the deviationsSquaring removes the negative signs — without it, positive and negative deviations cancel out to near zero, making the result meaningless.

Mistake 3 — Taking the square root too earlyCalculate variance (the sum of squared deviations ÷ N or N−1) first. Only take the square root in the final step.

Mistake 4 — Mixing population and sample formulas mid-calculationPick one formula (population or sample) and use it consistently. Mixing N and N−1 in the same calculation produces an incorrect result.

Mistake 5 — Rounding intermediate stepsKeep full decimal precision through all intermediate steps. Round only the final answer. Early rounding accumulates error and changes your result.

Studying for AP Statistics? Standard deviation underlies every inference procedure — see Unit 1 for descriptive statistics, or the confidence interval & hypothesis test calculator for Units 3–4.

Frequently Asked Questions

What does standard deviation tell you?

Standard deviation tells you how spread out the values in a data set are around the mean. A low value means data points are clustered tightly together; a high value means they are widely spread out.
What is the difference between standard deviation and variance?

Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance. Standard deviation is easier to interpret because it is in the same units as the original data — variance is in squared units.
When should I use sample vs. population standard deviation?

Use population (σ) when your data includes every member of the group. Use sample (s) when your data is a subset of a larger group. When in doubt on a homework problem, use sample standard deviation.
What is a good standard deviation?

It depends on context. A standard deviation of 5 is small for test scores out of 100 but large for measurements that should be precise to within 1 unit. Compare SD to the mean: if SD/mean × 100 exceeds 50%, variability is high for most practical purposes.
How do you find standard deviation in Excel?

Use =STDEV.S(range) for sample standard deviation or =STDEV.P(range) for population standard deviation. The older =STDEV() function defaults to sample. Enter your data in a column and pass the cell range as the argument.
What is the relationship between standard deviation and z-score?

A z-score measures how many standard deviations a specific value is from the mean: z = (x − μ) / σ. Once you have the standard deviation, use the Z-Score Calculator to find the percentile rank of any value in your data set.