What is Standard Deviation? Definition, Formula, Examples

Symbol: σ (population) or s (sample) ·
Range: 0 to ∞ (non-negative) ·
Formula: σ = √(Σ(x-μ)²/N) for population ·
Key property (normal distribution): 68% of data within 1σ, 95% within 2σ, 99.7% within 3σ

Four key facts about standard deviation, one pattern: it is the universal measure of spread that bridges raw math and real-world decisions.

What is a standard deviation?

Definition and formula

• Standard deviation is a measure of how spread out numbers are in a dataset. It answers a simple question: on average, how far does each data point sit from the mean? The National Library of Medicine (NIH) defines it as the measure of how dispersed data is relative to the mean (source).
• The population standard deviation (σ) uses all members of a group. The formula is σ = √(Σ(x-μ)² / N), where μ is the population mean and N is the population size (Khan Academy).

Population vs sample standard deviation

• The sample standard deviation (s) estimates the population parameter from a subset of data. It uses n-1 in the denominator — a correction called Bessel’s correction — to produce a less biased estimate (Outlier).
• Why n-1? Because using n would make the sample standard deviation systematically too small, especially for small sample sizes. The adjustment compensates for the fact that the sample mean itself is an estimate.

Relationship with variance

• Variance is the average of squared differences from the mean. Standard deviation is simply the square root of variance (Math is Fun). Think of variance as the squared unit; standard deviation brings it back to the original unit of measurement.
• For population data: variance = σ². For sample data: variance = s². Taking the square root converts variance into a number you can directly compare to the data.

The implication: standard deviation gives you dispersion in the same units as your data, making it far more intuitive than variance for everyday use.

How do I calculate standard deviation?

1. Step 1: Find the mean — Add up all data points and divide by the count. For a dataset of 5 numbers — say, 5, 5, 9, 9, 9, 10, 5, 10, 10 — the sum is 72 over 9 values, giving a mean of 8.0 (Khan Academy).
2. Step 2: Calculate deviations and squares — Subtract the mean from each data point, then square the result. For the value 5: deviation = 5 – 8 = -3, squared = 9. For 9: deviation = 1, squared = 1 (Khan Academy).
3. Step 3: Sum and divide (variance) — Sum all squared deviations. For the 9 values: 9+9+9+1+1+1+9+1+1 = 41. For population variance, divide by N = 9: 41/9 ≈ 4.56. For sample variance, divide by n-1 = 8: 41/8 ≈ 5.13 (Outlier).
4. Step 4: Take square root — The square root of the variance gives standard deviation. For the population: √4.56 ≈ 2.14. For the sample: √5.13 ≈ 2.27 (Outlier).

The catch: the difference between population and sample SD can be significant for small datasets — always ask whether you have the full population or just a sample.

What does 1.5 standard deviation mean?

68-95-99.7 rule

• In a normal distribution, roughly 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three (Wikipedia).
• A value 1.5 standard deviations above the mean sits at the 93.3rd percentile in a normal distribution — meaning only about 6.7% of data lies above it.

Z-scores and probability

• A z-score tells you how many standard deviations a data point is from the mean. A z-score of 1.5 means the value is 1.5σ above the mean. For a normal distribution, this corresponds to approximately 87% of data falling below that point.
• Z = (x – μ) / σ. If mean = 50 and σ = 10, a value of 65 gives z = 1.5 — that student’s test score is 1.5 standard deviations above the class average.

Practical meaning of SD values

• A standard deviation of 0.5 on a test with mean 70 means most scores cluster between 69.5 and 70.5 — very tight. On a scale of 0-100 with mean 50, σ = 20 means scores are spread widely, from roughly 10 to 90.
• The interpretation of “large” SD is always relative to the scale. For human height in inches (mean ≈ 67, σ ≈ 4), a σ of 4 is moderate. For IQ scores (mean = 100, σ = 15), a σ of 15 is the standard.

The pattern: standard deviation provides a universal scale for interpreting variability, but the meaning always depends on the context of the data.

What is standard deviation in statistics?

Role of SD in descriptive statistics

• Standard deviation is a cornerstone of descriptive statistics — it tells you the spread and is used alongside the mean to summarize a distribution (National Library of Medicine (NIH)).
• In statistical inference, SD appears in confidence intervals, hypothesis tests (t-tests, ANOVA), and effect size calculations. A small standard error (SD/√n) means more precise estimates.

SD in finance: risk measurement

• In finance, standard deviation is the most common measure of volatility and risk. A stock with σ = 30% annualized is considered more volatile — and riskier — than one with σ = 15% (Investopedia).
• Investors use SD to compare the risk of different assets. For example, the S&P 500 Index historically shows annualized SD around 15-20% — a baseline for equity market risk. Individual stocks like AAPL can have higher SD due to company-specific factors.

SD in physics: error analysis

• Physicists use standard deviation to quantify measurement uncertainty. If you measure the speed of light 100 times, the SD tells you how precise your measurement setup is.
• The standard deviation of repeated measurements is called the standard error of the mean, used to report results as “value ± uncertainty.”

What this means: standard deviation is the same mathematical tool, but its interpretation changes drastically by field — a “good” SD in one context might be “bad” in another.

What is the standard deviation of 5 5 9 9 9 10 5 10 10?

Step-by-step solution

• Dataset: 5, 5, 9, 9, 9, 10, 5, 10, 10 (n=9). Mean = (5+5+9+9+9+10+5+10+10) / 9 = 72/9 = 8.0.
• Deviations: (5-8)²=9, (5-8)²=9, (9-8)²=1, (9-8)²=1, (9-8)²=1, (10-8)²=4, (5-8)²=9, (10-8)²=4, (10-8)²=4. Sum = 9+9+1+1+1+4+9+4+4 = 42.
• Population variance = 42/9 ≈ 4.67. Population SD = √4.67 ≈ 2.16.
• Sample variance = 42/8 = 5.25. Sample SD = √5.25 ≈ 2.29.

Verify with calculator

• Most scientific calculators and spreadsheet software (Excel: STDEV.P or STDEV.S) can compute this in seconds. The values match the manual calculation.

Interpretation of result

• A population SD of ~2.16 means that, on average, each data point is about 2.16 units away from the mean of 8.0.
• Given the scale (values range from 5 to 10), this is moderate dispersion — the data aren’t tightly clustered (which would give SD near 0) nor extremely spread (which would push SD above ~3).

The pattern across all these examples: standard deviation is always a contextual measure. Its power lies not in isolated numbers but in how it helps you compare dispersion across different datasets.

“Standard deviation is a measure of the amount of variation of the values of a variable about its mean.”

Wikipedia

“Standard deviation measures how far values in a dataset typically deviate from the mean. It is calculated as the square root of the variance.”

Investopedia

For the student learning statistics, the investor assessing portfolio risk, or the physicist reporting experimental error, the standard deviation is the same tool with different stakes. The key takeaway: never interpret an SD value without understanding the context of the data it describes — a “high” SD in one field may be entirely ordinary in another.

A comprehensive overview of standard deviation definition and formula provides the definition, formula, and examples in one place.

Frequently asked questions