Standard Deviation Calculator

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Calculate standard deviation, variance, and mean for your data set

Standard Deviation Formulas

Population SD

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Sample SD

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Variance

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What is Standard Deviation?

Standard deviation (SD) is a measure of how spread out numbers are in a data set. It tells you, on average, how far each value lies from the mean. A low standard deviation means values cluster close to the mean, while a high standard deviation indicates values are spread over a wider range.

Standard deviation is the square root of variance. While variance gives us squared units, standard deviation returns to the original units of measurement, making it more interpretable. For example, if you're measuring height in centimeters, the standard deviation is also in centimeters.

In a normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3. This 68-95-99.7 rule is fundamental in statistics and quality control.

Population vs Sample Standard Deviation

🌍

Population SD (σ)

Use when you have data for the entire population. Divide by N (total count).

📊

Sample SD (s)

Use when data is a sample from a larger population. Divide by n-1 (Bessel's correction).

🔢

Variance (σ² or s²)

Standard deviation squared. Useful for statistical calculations.

📏

Coefficient of Variation

SD/Mean × 100%. Compares variability between datasets with different scales.

Standard Deviation Interpretation

What different standard deviation values mean for your data:

SD Range	Interpretation	Distribution	Action
Very Low	Values tightly clustered	Narrow bell curve	High consistency
Low	Moderate clustering	Normal distribution	Typical variation
Medium	Moderate spread	Wider distribution	Review outliers
High	Values widely spread	Flat distribution	Investigate causes
Very High	Extreme variation	Possible bimodal	Check data quality

Step-by-Step Calculation

1️⃣

Calculate the Mean

Add all values and divide by count. Mean = Σx / n. This is your center point for measuring deviation.

2️⃣

Find Deviations

Subtract the mean from each value. Some deviations will be positive, some negative. They sum to zero.

3️⃣

Square the Deviations

Square each deviation to make all values positive. This also gives more weight to larger deviations.

4️⃣

Calculate Variance & SD

Average the squared deviations (use n for population, n-1 for sample), then take the square root for SD.

Examples

Population SD of a small data set

Compute the population standard deviation of the eight values 2, 4, 4, 4, 5, 5, 7, 9. This data set is the textbook example used in most introductory statistics courses.

Resultσ = 2 (mean = 5, variance = 4)

Step 1: mean μ = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5. Step 2: squared deviations from 5 are 9, 1, 1, 1, 0, 0, 4, 16, summing to 32. Step 3: variance σ² = 32 / 8 = 4. Step 4: σ = √4 = 2. If you treat the same data as a sample, divide by n−1 = 7 instead of 8, giving s² ≈ 4.571 and s ≈ 2.138.

Frequently asked questions

When do I use population vs sample standard deviation?

Use population SD when your data includes every member of the group you're studying (e.g., all employees). Use sample SD when you've taken a subset from a larger group (e.g., surveyed 100 of 10,000 customers).

Why divide by n−1 for sample standard deviation?

This is Bessel's correction. Because the sample mean is itself estimated from the data, the sum of squared deviations around it systematically underestimates the population variance. Dividing by n−1 instead of n compensates, producing an unbiased estimator of population variance.

What's the difference between variance and standard deviation?

Variance is the average of squared deviations from the mean, so its units are the original units squared (e.g., cm²). Standard deviation is the square root of variance, returning to the original units (cm). SD is easier to interpret; variance is easier to work with algebraically.

How does the 68-95-99.7 rule work?

For data that is approximately normally distributed, about 68% of values fall within ±1 SD of the mean, about 95% fall within ±2 SD, and about 99.7% fall within ±3 SD. The rule only applies to roughly normal data — skewed or heavy-tailed distributions can deviate substantially.

Can standard deviation be negative?

No, standard deviation is always non-negative. It is the square root of variance, which is itself a sum of squared values divided by a positive count. SD equals zero only when every value in the data set is identical.

How sensitive is standard deviation to outliers?

Very. Because deviations are squared, a single extreme value can inflate SD dramatically. If your data contains outliers or heavy tails, consider robust measures of spread such as the interquartile range (IQR) or median absolute deviation (MAD) alongside SD.

How do I compare standard deviations across data sets with different scales?

Use the coefficient of variation: CV = SD / mean (often expressed as a percentage). A salary SD of $5,000 means something very different on a $30,000 mean than on a $300,000 mean. CV normalizes by the mean, so two data sets can be compared on a relative-spread basis.

What's the difference between standard deviation and standard error?

Standard deviation measures spread in your data. Standard error (SE = SD/√n) measures how precisely your sample mean estimates the population mean. SE decreases as sample size increases, while SD describes the underlying variability and does not shrink with more data.

Sources

NIST/SEMATECH e-Handbook of Statistical Methods - Measures of Scale · National Institute of Standards and Technology
Variance and Standard Deviation - Statistics and Probability · Khan Academy
Introduction to Probability and Statistics - Course 18.05 · MIT OpenCourseWare

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