What Is Standard Deviation?
The two formulas above handle two different situations. Use σ (sigma) when your dataset contains every member of the group you're describing. Use s when your data is a sample drawn from a larger population. The difference between them is the denominator: N for population, n−1 for sample. That small change in the denominator matters a great deal with small samples — see Section 5 for a full comparison.
Standard deviation was formalized by Karl Pearson in 1893, building on work by Francis Galton. Today it appears across descriptive statistics, hypothesis testing, finance, engineering, and machine learning — wherever you need to quantify how much values spread around an average.
- Symbol: σ (population) or s (sample)
- Units: Same as the original data — if data is in grams, SD is in grams
- Always: Zero or positive; negative standard deviation is impossible
- Zero SD: Every value in the dataset is identical
- Empirical rule: For normal distributions, ≈68% of values fall within 1σ of the mean, ≈95% within 2σ, ≈99.7% within 3σ
- Related: Variance = SD² | Standard Error = SD / √n
The Standard Deviation Formula
Population Standard Deviation
When your dataset contains every member of the group being described — all students in a single class, all products from one production run, all days in a calendar year — use the population formula:
σ = population standard deviationΣ = sum over all valuesxᵢ = each individual valueμ = population meanN = total number of valuesSample Standard Deviation
When your data is a subset drawn from a larger population — survey responses from 500 of 10,000 customers, blood samples from 30 of thousands of patients — use the sample formula. The denominator changes from N to n−1 (called Bessel's correction). This adjustment compensates for the fact that a sample tends to underestimate how spread out the full population really is.
s = sample standard deviationΣ = sum over all valuesxᵢ = each individual valuex̄ = sample meann−1 = degrees of freedomHow to Calculate Standard Deviation: The SPREAD Method
Every standard deviation calculation follows the same five operations. The SPREAD framework below names each step so the process is easy to remember. You'll see this exact sequence in each worked example that follows.
To calculate standard deviation: (1) Find the mean. (2) Subtract the mean from each value to get deviations. (3) Square each deviation. (4) Average the squared deviations — divide by N for population or n−1 for sample — to get the variance. (5) Take the square root of the variance. The result is the standard deviation.
Standard Deviation Examples with Step-by-Step Solutions
The five examples below each show the complete calculation. Follow each SPREAD step with the numbers to see exactly where every figure comes from.
Example 1: Student Exam Scores (Population SD)
A teacher records scores for all 8 students in a class: 72, 85, 90, 68, 78, 92, 84, 71. Find the standard deviation.
Because this dataset covers every student in the class — the entire population — we use population standard deviation (σ).
Find the mean (μ).
Sum: 72 + 85 + 90 + 68 + 78 + 92 + 84 + 71 = 640
Mean: 640 ÷ 8 = 80
Calculate deviations (xᵢ − 80).
Score, Deviation, Squared Deviation
Sum the squared deviations.
64 + 25 + 100 + 144 + 4 + 144 + 16 + 81 = 578
Divide by N to get variance.
Variance (σ²) = 578 ÷ 8 = 72.25
Take the square root.
σ = √72.25 = 8.50
✓ Population SD = 8.50 points. The typical student scored about 8.5 points away from the class average of 80. Scores ranged from 68 to 92 — a spread of 24 points — and the SD confirms there was meaningful variation across the class.
Example 2: Employee Salaries (Sample SD)
An HR analyst surveys 6 of the company's 200 employees to estimate salary spread. Salaries ($k): 45, 52, 48, 61, 57, 55.
These 6 employees are a sample from the full 200. Use sample standard deviation (s) with n−1 in the denominator.
Find the sample mean (x̄).
Sum: 45 + 52 + 48 + 61 + 57 + 55 = 318
Mean: 318 ÷ 6 = 53
Compute deviations and square them.
Salary ($k), Deviation, Squared
Sum squared deviations: 64 + 1 + 25 + 64 + 16 + 4 = 174
Divide by n−1 = 5: Sample variance = 174 ÷ 5 = 34.8
Take the square root: s = √34.8 ≈ 5.90
✓ Sample SD ≈ $5,900. The analyst estimates that salaries across the full company spread about $5,900 above and below the average of $53,000. The range in this sample ($45k–$61k) spans $16k, and the SD captures the typical distance from center.
Example 3: Monthly Stock Returns (Finance)
A stock posts these monthly returns (%) over 8 months: 2, −1, 4, −3, 1, 5, −2, 2. Calculate the standard deviation of returns.
This 8-month window is a sample of the stock's ongoing return history. Investors use return SD as a measure of volatility.
Mean return (x̄): (2 − 1 + 4 − 3 + 1 + 5 − 2 + 2) ÷ 8 = 8 ÷ 8 = 1.0%
Return (%), Deviation, Squared
Sum squared deviations: 1 + 4 + 9 + 16 + 0 + 16 + 9 + 1 = 56
Sample variance: 56 ÷ (8−1) = 56 ÷ 7 = 8.0
Sample SD: s = √8 ≈ 2.83%
✓ Monthly return SD ≈ 2.83%. The stock averages a 1% gain per month, but any given month could plausibly range from about −1.8% to +3.8% (within one SD). The months of −3% and +5% sit about 1.4 to 1.8 SDs from the mean — not extreme, but worth watching. This volatility figure feeds directly into portfolio risk models and the Sharpe ratio.
Example 4: Basketball Points Per Game (Sports)
A scout tracks a player's points across all 10 games of a playoff tournament: 22, 18, 31, 25, 19, 27, 24, 30, 21, 23. How consistent was this player?
All 10 tournament games represent the complete dataset for that tournament — population standard deviation applies.
Mean: (22+18+31+25+19+27+24+30+21+23) ÷ 10 = 240 ÷ 10 = 24 points
Points, Deviation, Squared
Sum squared deviations: 4+36+49+1+25+9+0+36+9+1 = 170
Population variance: 170 ÷ 10 = 17.0
Population SD: σ = √17 ≈ 4.12 points
✓ σ = 4.12 points. With a mean of 24 and SD of 4.12, the player was fairly consistent — most games fell between 20 and 28 points. The high of 31 sits roughly 1.7 SDs above average; the low of 18 sits 1.5 SDs below. No extreme outlier games. Scouts compare this SD against other candidates to find the most reliable scorer.
Example 5: Coffee Package Weights — Manufacturing QC
A factory labels coffee packages as 250g. A quality inspector weighs 10 packages from one production run: 248, 252, 250, 251, 249, 250, 252, 248, 251, 249.
This batch of 10 is being treated as the entire production run under inspection. We calculate σ to characterize line consistency.
Mean: (248+252+250+251+249+250+252+248+251+249) ÷ 10 = 2500 ÷ 10 = 250g
Weight (g), Deviation, Squared
Sum: 4+4+0+1+1+0+4+4+1+1 = 20
Population variance: 20 ÷ 10 = 2.0 g²
σ = √2 ≈ 1.41g
✓ σ = 1.41g. Every package in this run sits within 2g of the 250g target, and the typical deviation is just 1.41g. Regulators generally allow a ±4.5% tolerance on packaged goods (about ±11g here), so this line is well within spec. If σ were 5g or more, the line would need adjustment before packages drifted outside tolerance.
Real-Life Standard Deviation Examples by Domain
The following fifteen examples show standard deviation at work across everyday situations. Each gives you the dataset, the SD result, and what that number means in context.
Healthcare Examples
Body Temperature
Normal human body temperature averages 37.0°C with a population SD of roughly 0.4°C. A reading of 38.5°C sits more than 3 SDs above the mean — clinically significant. This is exactly why doctors use 38.0°C as the fever threshold: it sits 2.5 SDs from average, placing it in the top 0.6% of healthy readings.
Hospital Wait Times
Two emergency departments both average a 40-minute wait. Department A has σ = 5 min; Department B has σ = 22 min. In Department A, nearly every patient waits between 30 and 50 minutes. In Department B, some wait 10 minutes and others wait 90. Same average, very different patient experience — SD tells the difference.
Blood Pressure in Clinical Trials
A clinical trial tests whether a new drug reduces systolic blood pressure. Before the drug, 30 patients have mean BP = 145 mmHg, s = 12 mmHg. After treatment, mean = 133 mmHg, s = 10 mmHg. The sample SDs are used to calculate the standard error and run a paired t-test to determine whether the 12-point drop is statistically real.
Newborn Birth Weights
US full-term newborn birth weights average about 3,400g with σ ≈ 500g. A baby born at 2,200g sits 2.4 SDs below the mean — well into the low-birth-weight category (below 2,500g, roughly 1.8 SDs below average). Neonatal teams use these reference SDs to flag which newborns need additional monitoring from the first hour of life.
Business and Finance Examples
Investment Portfolio Risk
The S&P 500 has averaged roughly 10% annual return since 1928, with annual return SD near 20%. This means in about 2 of every 3 years, returns land between −10% and +30%. A bond fund might return 4% with SD of 3%, far less volatile. Portfolio managers balance these SDs against each other to hit a target risk level. See our guide on portfolio diversification statistics for more.
Customer Satisfaction Ratings
Two competing coffee shops both score 4.1 out of 5 on a review platform. Shop A has SD = 0.3, Shop B has SD = 1.4. Shop A's ratings cluster tightly — nearly everyone gives 4 or 5 stars. Shop B gets plenty of 5-star reviews, but also a substantial number of 1-star reviews. Identical means, opposite experiences.
Monthly Sales Revenue
A retail store's monthly revenue over a year: $82k, $91k, $78k, $105k, $88k, $94k, $76k, $110k, $93k, $85k, $99k, $112k. Mean = $92.8k, s = $11.4k. That $11.4k SD tells the finance team how wide a buffer to maintain in operating cash — targeting 2 SDs of coverage means holding about $23k in reserve against a bad month.
Loan Default Rates
A lender analyzes 12 months of regional default rates: mean = 3.2%, s = 0.9%. When the monthly rate hits 5.0%, that's 2 SDs above the average — a statistical signal that something unusual is happening in the market. Credit risk teams set alert thresholds at 1.5 or 2 SDs above the historical mean as an early warning system.
Education and Research Examples
IQ Scores
IQ tests are deliberately scaled to μ = 100 and σ = 15. This is not a coincidence — the tests are calibrated so that score spread fits this distribution. A score of 130 sits exactly 2 SDs above the mean, placing someone in the top 2.3% of the population. MENSA's entry cutoff is 130 (or equivalent on other validated instruments).
SAT Score Distribution
SAT Math scores historically average around 528 with SD near 116. A student scoring 760 is roughly 2 SDs above the mean — a score that most selective universities consider exceptional. SD is built into how admissions offices read test data: a score 1 SD above average (about 644) usually qualifies as competitive, while 2 SDs above (about 760) is considered high-achieving.
Research Measurement Error
A chemistry lab measures the concentration of a solution 8 times to establish baseline accuracy: 10.2, 10.4, 10.1, 10.5, 10.3, 10.2, 10.6, 10.3 mg/dL. Mean = 10.325, s = 0.168. The standard deviation of repeated measurements is the instrument's precision. Labs report results as mean ± SD — for example, 10.3 ± 0.2 mg/dL — so readers know how reproducible the number is.
Sports Examples
Marathon Finish Times
At a major city marathon, the mean finish time for all 5,000 participants might be 4:22:00 with σ = 48 minutes. That 48-minute SD tells race organizers when to schedule post-race services — the bulk of finishers (about 68%) will cross between 3:34 and 5:10. Elite runners finishing around 2:10 sit roughly 13 SDs below the population mean, which is one reason their performance looks so far removed from the everyday runner's experience.
NFL Quarterback Passer Ratings
In a typical NFL season, quarterback passer ratings average around 90 with SD ≈ 12. A QB rating of 115 sits about 2.1 SDs above average — genuinely elite. A rating of 66 sits 2 SDs below — a quarterback who's unlikely to start the following week. Teams use SD-based benchmarks like these to define "replacement level" — the performance you can reasonably expect from a free agent pickup.
Environmental Examples
Daily High Temperatures
San Diego's daily high temperatures over a year have a mean near 70°F and SD of about 8°F — modest variation throughout the seasons. Chicago's mean might be similar across a full year, but its SD exceeds 20°F because of brutally cold winters and hot summers. Two cities, similar annual averages, dramatically different climates. The SD captures what the mean cannot.
Crop Yield Variability
A wheat farmer tracks yield (bushels/acre) across 15 fields: mean = 62 bu/ac, s = 8 bu/ac. Fields yielding under 46 bu/ac or over 78 bu/ac fall more than 2 SDs from average and warrant investigation — possible soil issues, irrigation problems, or disease pressure. The SD also feeds into the farmer's crop insurance calculations, since policies are priced based on historical yield variability in the region.
Population vs Sample Standard Deviation
This is the most common source of confusion in standard deviation problems. The short rule: if you have data on everyone in the group, use population SD. If you're working with a subset, use sample SD.
| Feature | Population SD (σ) | Sample SD (s) |
|---|---|---|
| When to use | You have every member of the group | Your data is a subset of a larger group |
| Denominator | N (total population size) | n−1 (degrees of freedom) |
| Symbol | σ (sigma) | s |
| Variance symbol | σ² | s² |
| Excel formula | =STDEV.P( ) | =STDEV.S( ) |
| Python (NumPy) | np.std(data) | np.std(data, ddof=1) |
| R | sqrt(mean((x-mean(x))^2)) | sd(x) — R defaults to n−1 |
| Typical use | Entire class, full production batch, census data | Survey sample, clinical trial participants, market research |
Why n−1 for Samples?
When you take a sample, you're calculating deviations from the sample mean — not the true population mean. The sample mean is itself an estimate, and it sits slightly closer to the sample values than the true population mean does. Dividing by n instead of n−1 produces an answer that's consistently too small (a biased estimator). Switching to n−1 corrects for this bias — the resulting s² is an unbiased estimator of the population variance σ². This adjustment is called Bessel's correction, after the mathematician Friedrich Bessel.
The practical effect is largest with small samples. With n = 5, dividing by 4 instead of 5 increases your SD estimate by about 12%. With n = 30, the difference is only 1.7%. With n = 100, it becomes negligible.
Which Standard Deviation Should You Use?
Standard Deviation Comparison Tables
Standard Deviation vs Variance
| Feature | Standard Deviation | Variance |
|---|---|---|
| Symbol | σ or s | σ² or s² |
| Formula | √[Σ(x−μ)² / N] | Σ(x−μ)² / N |
| Units | Same as original data | Squared units (e.g., cm²) |
| Interpretation | Typical distance from the mean | Average squared distance from the mean |
| Better for | Communicating spread to an audience | Calculations (ANOVA, regression, confidence intervals) |
| Always positive? | Yes (≥ 0) | Yes (≥ 0) |
| Zero when | All values are identical | All values are identical |
Standard Deviation vs Standard Error
| Feature | Standard Deviation (s) | Standard Error (SE) |
|---|---|---|
| What it measures | Spread of individual data points | Precision of the sample mean as an estimate |
| Formula | s = √[Σ(x−x̄)²/(n−1)] | SE = s / √n |
| Changes with sample size? | Not dramatically | Decreases as n grows (SE → 0 as n → ∞) |
| Use case | Describing data variability | Building confidence intervals, hypothesis tests |
| Reports alongside | Mean, when describing a dataset | Mean, in research papers reporting estimates |
High SD vs Low SD
| Context | Low Standard Deviation | High Standard Deviation |
|---|---|---|
| Student scores | Class scored similarly (narrow range) | Wide gap between top and bottom students |
| Manufacturing | Consistent, tight production tolerances | Inconsistent output; higher defect risk |
| Finance | Stable, lower-risk investment returns | Volatile, higher-risk investment |
| Healthcare | Predictable patient outcomes | Highly variable patient responses |
| Weather | Mild climate, few temperature extremes | Large swings between seasons |
| Sports | Consistent performer; reliable output | Boom-or-bust player; unpredictable |
Standard Deviation vs Mean Absolute Deviation
| Feature | Standard Deviation (σ) | Mean Absolute Deviation (MAD) |
|---|---|---|
| Formula | √[Σ(x−μ)² / N] | Σ|x−μ| / N |
| Squaring? | Yes — amplifies outliers | No — treats all deviations equally |
| Outlier sensitivity | Higher | Lower |
| Mathematical properties | Supports algebra; used in calculus-based statistics | Simpler; harder to use in advanced formulas |
| Used in | Virtually all standard statistical methods | Robust statistics, some financial risk models |
Standard Deviation in Excel, Python, and R
Excel
Excel offers separate functions for population and sample standard deviation. The older STDEV function (still available) calculates sample SD; use the newer, clearer versions:
The plain STDEV function calculates sample SD (n−1). If you want population SD, you must use STDEV.P. Many students and analysts accidentally use population SD functions on sample data, or vice versa, because the default function name doesn't make this distinction clear.
Python (NumPy)
R
data <- c(72, 85, 90, 68, 78, 92, 84, 71)
# R's sd() uses n-1 (sample SD) by default:
sd(data) # 9.10
# Population SD (R has no built-in — calculate manually):
sqrt(mean((data - mean(data))^2)) # 8.50
# Or using variance for population SD:
pop_var <- mean((data - mean(data))^2)
pop_sd <- sqrt(pop_var)
For more on running statistical calculations in software, the standard deviation calculator on Statistics Fundamentals handles both population and sample calculations with a step-by-step breakdown, and the standard deviation visualizer shows the result on an interactive bell curve.
Common Mistakes with Standard Deviation
| Mistake | Wrong | Correct |
|---|---|---|
| Using N instead of n−1 on sample data | Dividing by n for a survey sample | Divide by n−1 to get an unbiased estimate of population σ |
| Forgetting the square root | Reporting variance (72.25) as the SD | Take the square root of variance to get SD in original units |
| Confusing SD with standard error | "The mean is 80 ± 8.5" (reporting SD as if it were error in the mean) | SE = s / √n. Use SE when reporting precision of the mean |
| Treating SD as the maximum possible deviation | "No value can be more than one SD from the mean" | SD is a typical distance. Many values sit more than 1 SD away |
| Comparing SDs from different scales | Concluding heights (SD = 7 cm) vary less than weights (SD = 12 kg) | Use coefficient of variation (CV = SD/mean × 100%) to compare across different units |
| Ignoring outliers that inflate SD | Reporting SD without noting one extreme value is pulling it upward | Check for outliers first. Report SD with and without the outlier if appropriate |
| Applying the empirical rule to non-normal data | Assuming 68% of values fall within 1 SD for a skewed dataset | The 68-95-99.7 rule only applies to approximately normal distributions |
When you need a number that an audience can read alongside the data — a research paper, a class report, a business slide — use standard deviation. When you're building a mathematical model or running ANOVA, regression, or variance decomposition calculations, work with variance. The two are always interconvertible: σ² = variance, σ = √variance.
Standard Deviation Calculator
Enter your dataset below to calculate standard deviation with a complete step-by-step breakdown. Separate values with commas. Toggle between population and sample as needed.
Standard Deviation Calculator
Entity and Formula Glossary
| Term / Symbol | Definition | Formula / Notes |
|---|---|---|
| Standard Deviation (σ, s) | Average distance of data points from the mean | √[Σ(x−μ)²/N] or √[Σ(x−x̄)²/(n−1)] |
| Variance (σ², s²) | Squared standard deviation; average squared deviation from mean | Σ(x−μ)²/N or Σ(x−x̄)²/(n−1) |
| Mean (μ, x̄) | Arithmetic average of all values | Σxᵢ / N |
| Population | The entire group being studied | Use N and σ |
| Sample | A subset drawn from a larger population | Use n−1 and s |
| Standard Error (SE) | SD of sampling distribution of the mean | SE = s / √n |
| Z-score | How many SDs a value sits from the mean | z = (x − μ) / σ |
| Normal Distribution | Bell-shaped distribution where 68/95/99.7% rule applies | N(μ, σ²) |
| Coefficient of Variation | SD as a percentage of the mean; allows cross-scale comparison | CV = (σ/μ) × 100% |
| Bessel's Correction | Using n−1 instead of n to get unbiased sample variance | Reduces bias in small-sample estimates |
| Dispersion | General term for how spread out data is | SD, variance, IQR, range are all measures |
| Empirical Rule | 68-95-99.7 rule for normal distributions | Approx. 68% within 1σ, 95% within 2σ, 99.7% within 3σ |
| Outlier | Value far from the rest; often defined as >2 or 3 SDs from mean | See outliers guide |
Frequently Asked Questions
Standard deviation measures how far data points typically sit from the average. A standard deviation of 10 on a test means the typical student scored about 10 points away from the class mean — some higher, some lower. A standard deviation of 2 means scores clustered much more tightly. The smaller the SD, the more consistent the data.
Population SD (σ) divides by N and applies when your dataset covers every member of the group. Sample SD (s) divides by n−1 and applies when your data is a subset. The n−1 denominator corrects for a systematic bias: samples tend to underestimate how spread out the full population really is. With large samples (n > 30), the difference between the two becomes small — less than 2%.
No. The formula squares all deviations before averaging them, and then takes a square root. Squaring removes all negative values. The result is always zero or positive. It equals exactly zero only when every value in the dataset is identical — for example, if every student scored exactly 80 on a test.
A high standard deviation means data points spread widely around the mean. Whether a standard deviation is "high" depends entirely on context. An SD of 5 is extremely high for body temperature (where normal variation is less than 1°C) but modest for annual rainfall in millimeters. Always compare SD to the mean and to other datasets in the same domain, not to an absolute number.
Variance is standard deviation squared (variance = σ²), and standard deviation is the square root of variance (σ = √variance). Variance works in squared units, which makes it hard to interpret in real-world terms — the variance of heights measured in centimeters comes out in cm². Standard deviation restores the original unit. Both measure spread, but standard deviation is the one you'd quote in a report or explain to someone unfamiliar with statistics. See the full comparison at standard deviation vs variance.
For any approximately normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three. A class with mean 70 and SD 10 will have roughly 68% of students scoring between 60 and 80, and 95% scoring between 50 and 90. This rule does not apply to skewed or heavily non-normal distributions. Learn more at the empirical rule guide.
In portfolio management, standard deviation of investment returns is the primary measure of risk. A stock with annual return SD of 25% is more volatile — and riskier — than one with SD of 8%. The Sharpe ratio divides excess return (return above risk-free rate) by SD to produce a risk-adjusted performance score. Value at Risk (VaR) models use historical return SD to estimate how much a portfolio might lose in a bad month or year. See our statistics in risk management article for examples.
Squaring has mathematical advantages over absolute values. Squared deviations are differentiable everywhere (absolute value has a corner at zero), which makes them compatible with calculus-based optimization used in regression. Squared deviations also decompose cleanly — the variance of a sum of independent variables equals the sum of their individual variances — a property that absolute deviations don't share. Mean absolute deviation (MAD) is a valid alternative that's more robust to outliers, but it doesn't support the algebraic manipulations that power ANOVA, regression, and the central limit theorem.
Follow the SPREAD steps above: (1) Find the mean. (2) Subtract the mean from each value. (3) Square each result. (4) Add them all up. (5) Divide by N (or n−1). (6) Take the square root. For small datasets (5–10 values), this is manageable by hand. The worked example on student exam scores above shows every arithmetic step. For larger datasets, use Excel's =STDEV.S() or Python's np.std(data, ddof=1).
A z-score tells you how many standard deviations a particular value sits above or below the mean: z = (x − μ) / σ. A score of 90 in a class with mean 80 and SD 8.50 gives z = (90 − 80) / 8.50 ≈ 1.18. This means 90 sits about 1.18 standard deviations above the class average. Z-scores let you compare values across datasets with different means and units. Learn more in the z-score guide.
Sources and Further Reading
All calculations in this guide follow the formulas given in NIST's Engineering Statistics Handbook and OpenStax Introductory Statistics — two of the most widely cited open references in applied statistics education.
- NIST Engineering Statistics Handbook — Measures of Scale. National Institute of Standards and Technology. itl.nist.gov
- OpenStax Introductory Statistics — Ch. 2: Descriptive Statistics. Rice University. openstax.org
- NumPy Documentation — numpy.std. NumPy Developers. numpy.org
- Penn State STAT 200 — Elementary Statistics. Penn State Eberly College of Science. online.stat.psu.edu
- Pearson, K. (1893) — "Contributions to the Mathematical Theory of Evolution." Philosophical Transactions of the Royal Society A, 185, 71–110. Introduced the term "standard deviation."
- Khan Academy — Statistics and Probability: Variance and Standard Deviation. khanacademy.org