How to Calculate Standard Deviation: Step-by-Step With Worked Examples

What Is Standard Deviation?

Standard deviation is a measure of how spread out numbers are from their average (mean) value. A low standard deviation means data points cluster closely around the mean, while a high standard deviation means they are dispersed over a wider range.

Introduced by English statistician Karl Pearson in 1894, standard deviation (represented by the Greek letter sigma for populations and the letter "s" for samples) has become the most widely used measure of statistical dispersion. It is fundamental to virtually every field that uses data: the National Institute of Standards and Technology (NIST) maintains reference datasets specifically for validating standard deviation calculations in scientific software.

In practical terms, standard deviation answers the question: "How much do individual measurements typically differ from the average?" If the average test score in a class is 75 and the standard deviation is 10, most scores fall between 65 and 85. If the standard deviation is 3, most scores cluster between 72 and 78. The concept appears everywhere -- from quality control in manufacturing (Six Sigma methodology) to investment risk analysis (portfolio volatility) to medical research (clinical trial endpoints).

The Standard Deviation Formula

There are two versions of the formula depending on whether you are working with an entire population or a sample drawn from a larger population.

The only difference between the two formulas is the denominator: N for population, n-1 for sample. The n-1 adjustment (called Bessel's correction) compensates for the fact that a sample's mean is always slightly closer to the sample's data points than the true population mean would be, which would otherwise cause the standard deviation to be systematically underestimated.

Step-by-Step Calculation: Worked Example

Let us calculate the sample standard deviation for this dataset of five test scores: 72, 85, 90, 68, 95.

Step 1: Find the mean (average).

Mean = (72 + 85 + 90 + 68 + 95) / 5 = 410 / 5 = 82

Step 2: Subtract the mean from each data point (find deviations).

Step 3: Sum the squared deviations.

Sum = 100 + 9 + 64 + 196 + 169 = 538

Step 4: Divide by n-1 (for sample) to get the variance.

Variance = 538 / (5 - 1) = 538 / 4 = 134.5

Step 5: Take the square root to get standard deviation.

Standard deviation = sqrt(134.5) = 11.60

This means the test scores typically deviate from the mean of 82 by about 11.6 points. You can verify this calculation instantly with our standard deviation calculator, which shows each step.

Population vs. Sample: When to Use Which

Choosing the correct formula depends on whether your dataset represents the complete group of interest or a subset of it.

In practice, sample standard deviation is used far more frequently because it is rare to have data for an entire population. When in doubt, use sample standard deviation (dividing by n-1). For large datasets (n greater than 30), the difference between the two formulas becomes negligible -- for n=100, the population SD is only about 0.5% lower than the sample SD.

The Empirical Rule (68-95-99.7)

For data that follows a normal (bell-shaped) distribution, standard deviation has a powerful interpretive property known as the empirical rule or the 68-95-99.7 rule.

• 68% of data falls within 1 standard deviation of the mean (mean +/- 1 SD)
• 95% of data falls within 2 standard deviations (mean +/- 2 SD)
• 99.7% of data falls within 3 standard deviations (mean +/- 3 SD)

Example: Adult male height in the U.S. has a mean of approximately 69.1 inches (5 feet 9 inches) with a standard deviation of 2.9 inches, according to CDC anthropometric data. Applying the empirical rule:

• 68% of men are between 66.2 and 72.0 inches (5'6" to 6'0")
• 95% of men are between 63.3 and 74.9 inches (5'3" to 6'3")
• 99.7% of men are between 60.4 and 77.8 inches (5'0" to 6'6")

Any value more than 3 standard deviations from the mean is considered an extreme outlier, occurring less than 0.3% of the time. This principle is the foundation of Six Sigma quality control, which aims for defect rates below 3.4 per million opportunities (equivalent to being more than 6 standard deviations from the mean). Use our z-score calculator to determine how many standard deviations a specific value is from the mean.

Key Terms You Should Know

Standard deviation is closely related to several other statistical measures. Understanding these connections helps you choose the right tool for your analysis.

• Variance: The square of the standard deviation. Variance (sigma-squared for populations, s-squared for samples) is mathematically simpler to work with in formulas but harder to interpret because it is in squared units. If heights are measured in inches, variance is in square inches. Standard deviation converts back to the original units by taking the square root.
• Mean (average): The central value around which standard deviation measures spread. Calculated by summing all values and dividing by the count. Our mean, median, mode calculator computes all three measures of central tendency.
• Z-score: The number of standard deviations a specific data point is from the mean. Calculated as z = (x - mean) / SD. A z-score of 2.0 means the value is 2 standard deviations above the mean. Z-scores allow comparison across different datasets and scales.
• Standard error: The standard deviation of a sampling distribution, calculated as SD / sqrt(n). It measures how much a sample mean is likely to vary from the true population mean. Smaller standard errors indicate more precise estimates.
• Coefficient of variation (CV): Standard deviation expressed as a percentage of the mean (CV = SD/mean x 100%). Allows comparison of variability between datasets with different scales or units. A salary dataset with CV of 25% has more relative spread than one with CV of 10%.

Real-World Applications

Standard deviation is not just an academic exercise -- it drives decisions worth billions of dollars across multiple industries.

• Finance and investing: Standard deviation of returns is the primary measure of investment risk. The S&P 500's historical annualized standard deviation is approximately 15%, meaning annual returns typically range from about -5% to +25% (one SD from the long-term average of ~10%). Portfolio managers use standard deviation to construct diversified portfolios that minimize risk for a given return target. According to Morningstar, funds with higher standard deviations have experienced steeper drawdowns during market downturns.
• Manufacturing quality control: The Six Sigma methodology, developed at Motorola and popularized by GE, uses standard deviation to define acceptable quality levels. A "six sigma" process produces fewer than 3.4 defects per million units -- meaning the process is controlled to within 6 standard deviations of the target.
• Scientific research: Standard deviation is used to calculate confidence intervals and p-values. In medical research, a treatment is considered statistically significant if the effect exceeds approximately 2 standard errors (p less than 0.05). In particle physics, the discovery standard is 5 sigma -- the Higgs boson was confirmed in 2012 when CERN detected it at 5.9 sigma.
• Weather forecasting: Temperature variability at a location is expressed using standard deviation. A city with an average January high of 35 degrees F and SD of 5 degrees has predictable winters, while one with SD of 15 degrees has wildly variable conditions. Use our probability calculator for related statistical computations.

Second Worked Example: Investment Returns

Let us calculate the population standard deviation for five years of annual returns for a mutual fund: 12%, -5%, 18%, 7%, 3%.

Step 1: Mean = (12 + (-5) + 18 + 7 + 3) / 5 = 35 / 5 = 7%

Step 2: Sum of squared deviations = 25 + 144 + 121 + 0 + 16 = 306

Step 3: Variance = 306 / 5 = 61.2 (population, since these are all 5 years of the fund's existence)

Step 4: Standard deviation = sqrt(61.2) = 7.82%

This fund has an average return of 7% with a standard deviation of 7.82%. Applying the empirical rule, about 68% of future annual returns would be expected to fall between -0.82% and 14.82% (assuming returns are normally distributed). This level of volatility is moderate -- slightly below the S&P 500's historical standard deviation of approximately 15%. For portfolio analysis, you can use our correlation calculator to measure how different investments move relative to each other.

Common Mistakes to Avoid

Even experienced analysts make errors with standard deviation. Watch out for these pitfalls.

• Using the wrong formula: The most common error is using population SD (dividing by N) when sample SD (dividing by N-1) is appropriate. For small samples, this matters significantly. With n=5, the sample SD is about 12% larger than the population SD.
• Confusing standard deviation with variance: Variance is the standard deviation squared. If a report says "variance is 25," the standard deviation is 5, not 25. Always check which measure is being reported.
• Applying the empirical rule to non-normal data: The 68-95-99.7 rule only works for approximately normally distributed data. Skewed distributions (like income or home prices) require different approaches like Chebyshev's theorem, which guarantees at least 75% of data within 2 SD regardless of distribution shape.
• Comparing SDs across different scales: A standard deviation of 5 means very different things for a dataset with mean 10 versus mean 10,000. Use the coefficient of variation (SD/mean) to compare variability across different scales.
• Ignoring outliers: Standard deviation is sensitive to extreme values because deviations are squared. A single outlier can dramatically inflate the result. Always examine your data for outliers before interpreting the SD.