Standard Deviation Calculator
Enter your numbers to calculate population and sample standard deviation, variance, mean, and more. See every calculation step explained.
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4, 8, 6, 5, 3
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4 8 6 5 3
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4; 8, 6 5 3
Enter your numbers in any format above and click Calculate to get both population (σ) and sample (s) standard deviation, variance, mean, and a full step-by-step breakdown.
Understanding Standard Deviation
Standard deviation measures the amount of variation or dispersion in a dataset. It tells you, on average, how far each data point is from the mean. A small standard deviation indicates that data points tend to be close to the mean, while a large standard deviation indicates the data is spread over a wider range of values.
Population Standard Deviation (σ)
σ = √(Σ(xᵢ − μ)² / N)
Used when you have data for the entire population. Divides by N (total number of values). For example, if you measure the height of every student in a class, use population standard deviation since you have the complete set.
Sample Standard Deviation (s)
s = √(Σ(xᵢ − x̄)² / (n − 1))
Used when data represents a sample from a larger population. Divides by n−1 (Bessel's correction) to give an unbiased estimate. For example, if you survey 100 people out of a city of 100,000, use sample standard deviation.
Variance (σ² or s²)
Variance = Σ(xᵢ − μ)² / N
Variance is the average of squared differences from the mean and equals the standard deviation squared. It measures spread in squared units. While less intuitive than standard deviation, variance has useful mathematical properties in advanced statistics.
The 68-95-99.7 Rule
μ ± 1σ ≈ 68% | μ ± 2σ ≈ 95% | μ ± 3σ ≈ 99.7%
For normally distributed data, approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This empirical rule helps quickly interpret how unusual a data point is.
Applications of Standard Deviation
Standard deviation is one of the most widely used statistical measures across industries. It quantifies uncertainty, risk, and variability — critical concepts in science, finance, manufacturing, and education.
Finance & Investing
- • Stock volatility measurement
- • Portfolio risk assessment
- • Sharpe ratio calculation
- • Value at Risk (VaR) models
- • Options pricing (Black-Scholes)
Science & Research
- • Experiment result analysis
- • Measurement uncertainty
- • Error bars on graphs
- • Clinical trial data analysis
- • Quality control in labs
Education & Testing
- • Exam score analysis
- • Grading curves
- • Student performance comparison
- • Standardized test scoring
- • Z-score and percentile ranking
Population vs. Sample: Which to Use?
Choosing the correct formula depends on whether your data represents the entire population or a sample drawn from it. Using the wrong formula introduces bias into your results.
Use Population (σ) When:
- • You have data for every member of the group
- • All test scores in a class (not estimating for other classes)
- • All products in a single production batch
- • Census data for an entire country
- • Complete historical stock prices for a specific period
Use Sample (s) When:
- • Your data is a subset of a larger group
- • A survey of 500 voters out of millions
- • Quality testing a few items from a factory run
- • Clinical trial with a limited number of patients
- • Any time you want to generalize to a broader population
Frequently Asked Questions
What is standard deviation?
Standard deviation is a measure of how spread out numbers are from their average (mean). A low standard deviation means data points are close to the mean, while a high standard deviation means they are spread over a wide range. It is calculated as the square root of variance. For example, the dataset {2, 4, 4, 4, 5, 5, 7, 9} has a mean of 5 and a population standard deviation of about 2.0.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by N (the total count), while sample standard deviation (s) divides by N−1. Use population standard deviation when your data includes every member of the group you are studying. Use sample standard deviation when your data is a subset (sample) drawn from a larger population. Dividing by N−1 (called Bessel's correction) produces an unbiased estimate of the population variance from a sample.
How do you calculate standard deviation step by step?
Step 1: Find the mean (average) of the data. Step 2: Subtract the mean from each data point to get the deviation. Step 3: Square each deviation. Step 4: Sum all squared deviations. Step 5: Divide by N for population variance or N−1 for sample variance. Step 6: Take the square root to get the standard deviation. For example, for {2, 4, 6}: mean = 4, deviations = {−2, 0, 2}, squared = {4, 0, 4}, sum = 8, population variance = 8/3 ≈ 2.667, population σ ≈ 1.633.
What is variance and how does it relate to standard deviation?
Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. While variance measures spread in squared units, standard deviation is expressed in the same units as the original data, making it easier to interpret. For example, if test scores have a variance of 25 points², the standard deviation is 5 points. Both metrics describe data spread, but standard deviation is more commonly reported.
When should I use standard deviation vs. other measures of spread?
Use standard deviation when your data follows a roughly normal (bell-shaped) distribution. For skewed data or data with outliers, the interquartile range (IQR) or median absolute deviation (MAD) may be better choices since they are less affected by extreme values. Standard deviation is ideal for symmetrical distributions and is required for many statistical tests including Z-tests, t-tests, and confidence intervals.
What does standard deviation tell you about a normal distribution?
In a normal distribution, about 68% of data falls within ±1 standard deviation of the mean, about 95% within ±2 standard deviations, and about 99.7% within ±3 standard deviations. This is known as the 68-95-99.7 rule (or empirical rule). For instance, if the mean height is 170 cm with σ = 10 cm, roughly 68% of people are between 160 cm and 180 cm, and 95% are between 150 cm and 190 cm.
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