Standard Deviation Calculator
Comprehensive statistics calculator that calculates standard deviation, variance, mean, median, mode, range, sum, and count for data sets. Enter numbers separated by spaces, commas, or semicolons and choose between sample standard deviation (n-1 denominator) for sample data or population standard deviation (n denominator) for complete populations. The calculator provides detailed formulas, interpretation guidelines, and visual breakdowns of results. Perfect for students, researchers, analysts, or anyone working with statistical data. All calculations happen instantly in your browser with complete privacy—no data is stored or transmitted.
Standard Deviation Calculator
How it works: This calculator calculates comprehensive statistics for your data set, including standard deviation, variance, mean, median, mode, and more. Enter numbers separated by spaces, commas, or semicolons. Choose between sample standard deviation (n-1 denominator) for sample data or population standard deviation (n denominator) for complete populations. Standard deviation measures how spread out the data is from the mean.
What Is a Standard Deviation Calculator?
A standard deviation calculator computes the spread or dispersion of a data set around its mean. Along with the mean, standard deviation (SD) is one of the two most fundamental descriptive statistics. A low SD means data points cluster tightly around the average; a high SD means they are spread out. SD is the square root of variance, and it is expressed in the same units as the data — making it much more interpretable than variance alone.
Standard deviation has applications across almost every field: grading on a curve (education), portfolio risk (finance), quality control (manufacturing), clinical trial analysis (medicine), and A/B test significance (tech). This calculator handles both population SD (σ) and sample SD (s), and outputs mean, median, mode, range, and variance alongside SD.
How to Calculate Standard Deviation
- Enter your data values, separated by commas or spaces.
- Select Population if your data is the entire group; select Sample if your data is a subset (most real-world analyses use Sample SD, which divides by n−1).
- The calculator computes: Mean → Deviations from mean → Squared deviations → Variance → √Variance = SD.
- Interpret: roughly 68% of data falls within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD (the empirical rule, assuming normal distribution).
Worked Example: Test Scores
Data: 70, 75, 80, 85, 90 (5 student test scores)
Mean = (70+75+80+85+90) ÷ 5 = 80
Squared deviations: (70−80)²=100, (75−80)²=25, (80−80)²=0, (85−80)²=25, (90−80)²=100
Population Variance = (100+25+0+25+100) ÷ 5 = 50
Population SD = √50 = 7.07
Sample SD (÷ by n−1=4) = √(250/4) = √62.5 = 7.91
Interpretation: scores are typically within ±7 points of the 80-point mean. A student scoring 93+ is more than 1 SD above average (roughly top ~16%).
Interpreting Standard Deviation
| Range | % of Normal Data Within | Example Interpretation (Mean=100, SD=15) |
|---|---|---|
| Mean ± 1 SD | ~68.3% | 85–115 contains most of the data |
| Mean ± 2 SD | ~95.4% | 70–130 covers nearly all typical values |
| Mean ± 3 SD | ~99.7% | 55–145 — values outside are rare outliers |
| Mean ± 4 SD | ~99.994% | 40–160 — extremely unusual to see values outside |
The Empirical Rule applies to normally distributed data. Skewed distributions may deviate significantly from these percentages.
Key Concepts: Population vs. Sample, Variance, CV
Population SD (σ) vs. Sample SD (s). Population SD divides by n — use when you have every data point in the group (e.g., all students in one class). Sample SD divides by n−1 (Bessel's correction) — use when your data is a sample drawn from a larger population (e.g., 100 patients from millions). Sample SD is an unbiased estimator of the true population SD. Most real-world use cases call for sample SD.
Variance is simply SD squared (SD²). While variance is mathematically useful (variances add for independent variables), it is in squared units — making it hard to interpret directly. Always report SD for human-readable results.
Coefficient of Variation (CV) = (SD ÷ Mean) × 100%. CV allows comparing spread across datasets with different units or scales. A stock with Mean=$100, SD=$15 has CV=15%. Another with Mean=$1000, SD=$100 has CV=10% — the second is relatively less volatile despite a higher absolute SD.
Tips for Statistical Analysis
Check for outliers before computing SD. SD is heavily influenced by outliers because deviations are squared. A single extreme value can inflate SD dramatically. First, look at your data distribution — if it's heavily skewed, consider using median absolute deviation (MAD) instead, which is more robust to outliers.
Use sample SD (n−1) for almost all real-world work. Unless you literally have data from every member of the population, use the sample formula. The difference is small for large datasets but can be significant when n is small (below 30). Statistical software like Excel uses STDEV (sample) by default and STDEVP for population.
Normal distribution assumptions. The empirical rule (68-95-99.7) only applies to normally distributed data. Before applying it, check your data for normality using histograms or Q-Q plots. Income data, stock returns, and many biological measurements are not perfectly normal — use non-parametric methods when in doubt.
Frequently Asked Questions
What is standard deviation?
Standard deviation (SD) is a measure of how spread out data values are around their mean. A low SD means values are close to the mean; a high SD means they are spread out. Formula: SD = √(Σ(xᵢ − x̄)² / n) for population, or √(Σ(xᵢ − x̄)² / (n−1)) for a sample.
What is the difference between standard deviation and variance?
Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance, bringing the result back to the original units. Variance = SD². SD is preferred for reporting because it is interpretable in the same scale as the data.
When should I use population vs. sample standard deviation?
Use population SD (÷n) only when you have data for every member of the group. In virtually all practical research, surveys, and experiments, you have a sample — so use sample SD (÷n−1). This correction (Bessel's correction) reduces bias when estimating the true population variability from a sample.
What does a high standard deviation mean?
A high SD means the data points are widely scattered around the mean — high variability. In finance, high SD = higher risk/volatility. In manufacturing, high SD = inconsistent quality. In test scores, high SD = wide range of performance. Whether high SD is good or bad depends on the context.
What is the empirical rule (68-95-99.7 rule)?
For normally distributed data: ~68% of values fall within ±1 SD of the mean, ~95% within ±2 SD, and ~99.7% within ±3 SD. This is used in quality control (Six Sigma: 6 SDs from mean), IQ score interpretation, and statistical hypothesis testing to identify unusual values.
What is a z-score?
A z-score measures how many standard deviations a data point is from the mean: z = (x − mean) / SD. A z-score of 2 means the value is 2 SDs above average. Z-scores standardize data for comparison across different scales and are used in hypothesis testing, percentile calculation, and anomaly detection.
What is the coefficient of variation?
The Coefficient of Variation (CV) = (SD / Mean) × 100%. It expresses SD as a percentage of the mean, allowing comparison across datasets with different units or scales. Example: Stock A has SD=$5 on a $50 mean (CV=10%); Stock B has SD=$20 on a $500 mean (CV=4%). Stock B is relatively less volatile despite a higher absolute SD.
How do I calculate standard deviation in Excel?
Excel uses =STDEV(range) for sample SD and =STDEVP(range) for population SD. Modern Excel also supports =STDEV.S(range) and =STDEV.P(range). For the mean, use =AVERAGE(range). For variance: =VAR(range) (sample) or =VARP(range) (population).
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