Standard Deviation Explained Simply with Real Examples

What Standard Deviation Actually Tells You

Standard deviation is the single number that summarizes how spread out a dataset is around its mean. Two datasets with the same mean can have wildly different standard deviations — and that difference tells you whether the mean is a useful summary or a misleading one.

Consider two hypothetical portfolios, each with a 7% average annual return over 30 years:

• Portfolio A: returns ranged from 5% to 9%, standard deviation 1.2%
• Portfolio B: returns ranged from −15% to +35%, standard deviation 14%

Same average. Wildly different experience. Portfolio A is a low-volatility bond proxy; Portfolio B is a concentrated equity strategy. An investor who knows only the mean would treat them as identical; an investor who knows the standard deviation would recognize them as fundamentally different risk profiles.

The standard deviation converts the spread of a dataset into the same units as the mean. Returns in percent have standard deviations in percent. Heights in inches have standard deviations in inches. This unit-preserving property is what makes standard deviation so much more interpretable than its cousin, the variance, which is in squared units.

The Two Formulas: Population and Sample

There are two standard deviation formulas, and using the wrong one produces a (typically) small but systematic error.

Population Standard Deviation (σ)

Used when your dataset is the entire population of interest — every possible observation you care about. The formula:

σ = √[ (1/N) × Σ(xi − μ)² ]

Where:

• N = total number of observations in the population
• xi = each individual observation
• μ = population mean

Sample Standard Deviation (s)

Used when your dataset is a sample drawn from a larger population, and you want to estimate the population standard deviation from that sample. The formula:

s = √[ (1/(N−1)) × Σ(xi − x̄)² ]

Where:

• N = number of observations in the sample
• xi = each individual observation
• x̄ = sample mean

The only difference is the denominator: N for population, N − 1 for sample. That single change is called Bessel's correction, after the German astronomer Friedrich Bessel, and it has a precise statistical justification.

Why Bessel's Correction Matters

When you compute standard deviation from a sample, you use the sample mean (x̄) — not the population mean (μ) — as the reference point. The sample mean is always closer to the sample data points than the population mean is, by definition: the sample mean is the value that minimizes the sum of squared deviations from the sample. Using the sample mean as the reference therefore systematically understates the spread of the underlying population.

Bessel's correction divides by N − 1 instead of N, which inflates the standard deviation estimate by a factor of √(N / (N − 1)). For large N, the correction is negligible: a sample of 1,000 produces a correction factor of √(1000/999) ≈ 1.0005, a 0.05% increase. For small N, the correction is substantial: a sample of 5 produces a correction factor of √(5/4) ≈ 1.118, an 11.8% increase.

This is why spreadsheet functions distinguish between =STDEV.P(range) for population standard deviation and =STDEV.S(range) for sample standard deviation. Using STDEV.P on a sample understates the population spread; using STDEV.S on a true population slightly overstates it. The right choice depends on whether your data is the entire population or a sample.

Worked Example: Two Test Score Datasets

Suppose two classes take the same exam. Class A scores: 78, 82, 85, 87, 88, 90, 92, 93, 95, 100. Class B scores: 60, 70, 80, 85, 90, 92, 95, 97, 99, 100.

Both have a mean of 89. Computing the sample standard deviation:

• Class A: Σ(xi − x̄)² = (78−89)² + (82−89)² + ... + (100−89)² = 121 + 49 + 16 + 4 + 1 + 1 + 9 + 16 + 36 + 121 = 374. Divide by N − 1 = 9: 41.56. Square root: s ≈ 6.45.
• Class B: Σ(xi − x̄)² = (60−89)² + (70−89)² + ... + (100−89)² = 841 + 361 + 81 + 16 + 1 + 9 + 36 + 64 + 100 + 121 = 1,630. Divide by N − 1 = 9: 181.11. Square root: s ≈ 13.46.

Class B's standard deviation is roughly twice Class A's, despite identical means. Class A's scores cluster tightly around the mean; Class B's spread widely. The mean is a more informative summary for Class A than for Class B.

Had we used the population formula (dividing by N = 10 instead of N − 1 = 9), Class A's standard deviation would have come out as approximately 6.12 instead of 6.45 — a roughly 5% underestimate of the population spread, exactly as Bessel's correction predicts for N = 10.

The Empirical Rule (68-95-99.7)

For datasets that follow a normal distribution (the bell curve), the empirical rule states:

• Approximately 68% of observations fall within ±1 standard deviation of the mean
• Approximately 95% of observations fall within ±2 standard deviations of the mean
• Approximately 99.7% of observations fall within ±3 standard deviations of the mean

This is why a 3-sigma event in a normal distribution is considered "rare" — only 0.3% of observations fall outside that range. In Class A above, 3 standard deviations is approximately ±19.4 points, so a score outside the 70-108 range would be a 3-sigma event. The empirical rule would predict about 0.03 × 10 = 0.3 observations outside that range; in practice, none of our 10 observations fall outside.

The empirical rule applies only to normal distributions. For non-normal distributions — particularly skewed or heavy-tailed distributions — the rule systematically understates the frequency of extreme observations. This is why financial risk models that assume normality underestimate the probability of market crashes, which are far more common in practice than the 4-sigma events they would be under a true normal distribution.

When Standard Deviation Misleads

• Skewed distributions. For right-skewed data (such as income distributions), the mean exceeds the median, and standard deviation does not symmetrically describe the spread. The interquartile range (IQR) is a more robust measure of spread for skewed data.
• Outliers. A single extreme observation can dramatically inflate the standard deviation. In Class B above, removing the 60 score drops the standard deviation from 13.46 to 10.68 — a 21% reduction from removing one observation. Robust statistics such as median absolute deviation (MAD) are less sensitive to outliers.
• Non-normal distributions. The 68-95-99.7 empirical rule fails for distributions that are not approximately normal. Always visualize the distribution (histogram, density plot) before applying normality-based rules.
• Time series with trend or autocorrelation. Computing standard deviation across a time series with a strong trend (such as a stock price series) produces a number that conflates volatility with drift. Returns are typically de-trended (by computing period-over-period changes) before standard deviation is computed.

Variance versus Standard Deviation

Variance is standard deviation squared (σ² or s²). The two are mathematically interchangeable: take a square root of variance to get standard deviation; square standard deviation to get variance. The practical difference is units. Variance is in squared units (squared dollars, squared percent, squared inches), which are not directly interpretable. Standard deviation is in the same units as the original data, which is why standard deviation is the more commonly reported figure.

Mathematically, the variance is more convenient for theoretical work because variances add across independent random variables (the variance of a sum equals the sum of the variances). Standard deviations do not add — the standard deviation of a sum is the square root of the sum of variances. This is why portfolio variance, not standard deviation, is the natural object for portfolio theory.

Conclusion

Standard deviation is the spread metric of choice for approximately normal distributions, because it shares the units of the original data and obeys the 68-95-99.7 empirical rule. The choice between population (divide by N) and sample (divide by N − 1) formulas depends on whether the data is the entire population or a sample drawn from one. Bessel's correction matters most for small samples, where its 11.8% inflation factor at N = 5 dwarfs measurement error. For skewed, heavy-tailed, or outlier-contaminated data, robust alternatives such as the interquartile range or median absolute deviation provide more reliable spread summaries.

Compute your own standard deviation with our Standard Deviation Calculator, which reports both population and sample values side by side.

Written by the HT99 Tools Editorial Team. See our disclaimer for full terms.