Calculate population and sample standard deviation, variance, mean, sum, and margin of error for your data set with detailed step-by-step solutions and statistical analysis.
Standard deviation is a measure of variation or dispersion in a data set. It quantifies how spread out the data points are from the mean (average). A lower standard deviation indicates that data points are closer to the mean, while a higher standard deviation shows greater variability.
Standard deviation is fundamental in statistics for understanding data distribution, making predictions, quality control, and risk assessment in various fields including finance, science, and engineering.
Used when you have data for the entire population
Used when you have data from a sample of the population
| Type | Use When | Formula | Example |
|---|---|---|---|
| Population | You have data for entire population | σ = √[Σ(xi - μ)² / N] | Test scores for all students in a class |
| Sample | You have data from a sample | s = √[Σ(xi - x̄)² / (n-1)] | Survey responses from random participants |
Step 1: Calculate the mean (average) of your data set.
Step 2: Find the difference between each data point and the mean.
Step 3: Square each difference to eliminate negative values.
Step 4: Sum all the squared differences.
Step 5: Divide by N (population) or n-1 (sample) to get variance.
Step 6: Take the square root of variance to get standard deviation.
| Field | Application | Purpose | Example |
|---|---|---|---|
| Finance | Risk Assessment | Measure investment volatility | Stock price fluctuations |
| Quality Control | Manufacturing | Monitor product consistency | Bolt diameter measurements |
| Education | Test Analysis | Evaluate score distribution | Standardized test results |
| Healthcare | Clinical Research | Analyze treatment effectiveness | Blood pressure variations |
| Weather | Climate Analysis | Study temperature patterns | Monthly rainfall data |
| Sports | Performance Analysis | Evaluate consistency | Player scoring averages |
Low Standard Deviation (< 1): Data points are close to the mean, indicating consistency and low variability.
Moderate Standard Deviation (1-3): Typical variability for many datasets, with some spread around the mean.
High Standard Deviation (> 3): Data points are widely spread, indicating high variability or inconsistency.
For normally distributed data:
Consider Context: A "high" or "low" standard deviation depends on the context and units of measurement.
Check for Outliers: Extreme values can significantly affect standard deviation. Consider identifying and analyzing outliers separately.
Sample Size Matters: Larger samples generally provide more reliable estimates of population parameters.
Units: Standard deviation has the same units as your original data, making it easier to interpret than variance.
To calculate standard deviation manually: 1) Find the mean (average) of your dataset, 2) Subtract the mean from each data point and square the result, 3) Sum all squared differences, 4) Divide by n (population) or n-1 (sample), 5) Take the square root. For dataset {2, 4, 6, 8, 10}: mean = 6, squared differences = {16, 4, 0, 4, 16}, sum = 40, divide by 5 = 8, √8 = 2.83. Our standard deviation calculator automates this process instantly. For statistical analysis, check our comprehensive statistics calculator and variance calculator for related metrics.
Population standard deviation (σ) divides by n when you have the complete dataset, while sample standard deviation (s) divides by n-1 to account for estimation uncertainty. For data {10, 20, 30, 40, 50}: population SD = 14.14, sample SD = 15.81. Use population SD when analyzing all data (e.g., all exam scores in a class). Use sample SD when your data represents a subset (e.g., 100 people surveyed from a city of 100,000). According to Statistics How To, sample SD provides unbiased estimates for larger populations. Compare with our mean calculator.
Standard deviation measures data spread: low SD means data points cluster near the mean, high SD indicates wide dispersion. For test scores with mean 75 and SD 5: most scores fall 70-80 (within 1 SD). SD of 15 means scores spread 60-90. In finance, SD of 15% on returns indicates higher volatility than 5%. The 68-95-99.7 rule applies to normal distributions: 68% of data falls within ±1 SD, 95% within ±2 SD, 99.7% within ±3 SD. Values beyond 2-3 SD are potential outliers. For quality control applications, reference NIST Statistical Methods. Identify outliers with our z-score calculator.
Use standard deviation for interpretability (same units as data) and variance for mathematical operations. If measuring heights in inches with mean 70 and variance 25, SD = 5 inches (easier to understand than 25 square inches). Variance is preferred in: ANOVA tests, regression analysis, theoretical statistics. Standard deviation is better for: describing data spread, setting control limits, comparing datasets, reporting to non-statisticians. Both measure dispersion, but SD = √variance makes it more intuitive. Calculate both metrics and understand their relationship with our statistical analysis tools.
In normal (Gaussian) distributions, standard deviation defines the bell curve's width. For normally distributed data: mean ± 1 SD contains 68.27% of data, mean ± 2 SD contains 95.45%, mean ± 3 SD contains 99.73%. Example: IQ scores (mean 100, SD 15) show 68% of people have IQ 85-115. This relationship enables probability calculations and hypothesis testing. However, not all data is normally distributed - skewed distributions have different patterns. Test normality before applying these rules. The Khan Academy statistics guide provides detailed explanations. Calculate probabilities with our probability calculator and normal distribution calculator.
There's no universal "good" standard deviation - it depends on context and measurement scale. For manufacturing (targeting 10.0mm parts): SD of 0.1mm is excellent, 0.5mm acceptable, 1.0mm concerning. For stock returns: SD of 5% indicates low volatility, 15% moderate, 30%+ high risk. In academic testing: SD of 10 points on 100-point test suggests good question discrimination, SD of 3 indicates questions too easy/hard. Compare SD to the mean: coefficient of variation (CV = SD/mean × 100%) helps assess relative variability. CV under 20% suggests low variability, 20-30% moderate, above 30% high. Evaluate data quality with our CV calculator.
Outliers dramatically increase standard deviation since SD is sensitive to extreme values. For dataset {10, 12, 11, 13, 12}: SD ≈ 1.1. Adding outlier 50: SD jumps to 14.8. This occurs because SD squares deviations, amplifying extreme differences. To handle outliers: 1) Identify them using z-scores (|z| > 3), 2) Investigate if they're data errors or genuine extremes, 3) Consider robust statistics (MAD, IQR) less sensitive to outliers, 4) Report SD with and without outliers, 5) Use trimmed means for extreme cases. Our calculator shows outlier impact clearly. For outlier detection, use our outlier detection calculator and IQR calculator.
Standard deviation can be calculated for any sample size (minimum n=2), but reliability increases with larger samples. For n<30 (small samples): use sample SD (n-1 divisor), consider t-distribution instead of z-scores, report confidence intervals, be cautious generalizing to population. For n=5: SD might vary ±30% from true population value. For n=30+: SD estimates become reliable (±10% typically). Minimum recommended: n=15-20 for basic analysis, n=30+ for hypothesis testing, n=100+ for precise population estimates. According to statistical best practices, always report sample size with SD. Calculate required sample size with our sample size calculator and perform t-tests with our t-test calculator.