Standard Deviation Calculator
Enter a list of numbers to instantly calculate standard deviation, variance, mean, and more. Choose between population (σ) and sample (s) formulas. Full step-by-step breakdown included.
Last updated: March 2026
How to Use the Standard Deviation Calculator
- Enter your numbers in the text area, separated by commas, spaces, or line breaks. Example:
2, 4, 4, 4, 5, 5, 7, 9 - Select the calculation type: Population SD (σ, divides by N) if you have all data, or Sample SD (s, divides by N−1) if you have a subset.
- View results instantly — standard deviation, variance, mean, count, sum, and sum of squares update as you type.
- Review the step-by-step breakdown below the calculator to see every deviation and squared difference.
What Standard Deviation Measures
Standard deviation (SD) quantifies how spread out the values in a data set are around the mean. A small SD means the values are tightly clustered; a large SD means they are widely dispersed.
Consider two sets of exam scores: Class A = {48, 49, 50, 51, 52} and Class B = {20, 35, 50, 65, 80}. Both have a mean of 50, but Class A has SD ≈ 1.6 (very consistent) while Class B has SD ≈ 22.4 (highly variable).
The Standard Deviation Formula
The two formulas differ only in the denominator:
- Population SD: σ = √[ Σ(xᵢ − μ)² / N ]
- Sample SD: s = √[ Σ(xᵢ − x̄)² / (N − 1) ]
The N−1 denominator in the sample formula is called Bessel's correction. It compensates for the fact that a sample mean tends to be closer to the sample values than the true population mean, which would otherwise underestimate variability.
Variance vs. Standard Deviation
Variance (σ²) is the squared version of standard deviation. While variance is mathematically convenient (it adds across independent variables), it is expressed in squared units (e.g., kg²) which are hard to interpret. Standard deviation brings the measure back to the original units (e.g., kg), making it the preferred measure of spread in most practical contexts.
The Normal Distribution and the 68-95-99.7 Rule
In a perfectly normal (bell-shaped) distribution, standard deviation defines how data is distributed around the mean:
- 68% of values fall within 1 SD of the mean (between μ−σ and μ+σ)
- 95% of values fall within 2 SDs of the mean
- 99.7% of values fall within 3 SDs of the mean
This rule, known as the empirical rule, is fundamental in quality control, grading on a curve, medical testing, and financial risk assessment. A value more than 3 SDs from the mean is considered a rare outlier (occurs less than 0.3% of the time in a normal distribution).
Z-Scores: Standard Deviations as a Universal Scale
A z-score converts any data point to a standard scale by measuring how many SDs it is from the mean: z = (x − μ) / σ. This allows comparison across different datasets. A student scoring 85 on a test with mean 70, SD 10 has z = 1.5 — they outperformed 93% of students.
Standard Deviation in Finance
In investing, standard deviation measures volatility — the risk of a security's price fluctuating. A stock with monthly returns averaging 1% and SD 2% is less risky than one with the same average but SD 15%. Portfolio diversification reduces the combined SD even when individual assets are volatile, because not all assets move in the same direction simultaneously.
How to Calculate Standard Deviation If...
What is the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?
The population standard deviation of {2, 4, 4, 4, 5, 5, 7, 9} is exactly 2. Mean = 40/8 = 5. Squared deviations: 9+1+1+1+0+0+4+16 = 32. Variance = 32/8 = 4. SD = √4 = 2. This is a classic textbook example.
What is the standard deviation of 10, 20, 30, 40, 50?
The population SD is approximately 14.14. Mean = 30. Squared deviations: 400+100+0+100+400 = 1000. Variance = 1000/5 = 200. SD = √200 ≈ 14.14.
What is the standard deviation of a single number?
The standard deviation of a single number is always 0. With only one value, there is no spread — the value equals the mean, so all deviations are zero.
What is the standard deviation of all identical values?
If all values are the same, the standard deviation is 0. For example, {5, 5, 5, 5} has mean 5, all deviations are 0, so variance = 0 and SD = 0.
How do I find standard deviation if I only have mean and variance?
Standard deviation equals the square root of variance. If variance = 25, then SD = √25 = 5. If variance = 7.5, then SD = √7.5 ≈ 2.739.
What is the standard deviation of test scores 75, 82, 90, 68, 95?
The sample SD of {68, 75, 82, 90, 95} is approximately 10.82. Mean = 82. Sample SS = 196+49+0+64+169 = 478. Variance = 478/4 = 119.5. SD = √119.5 ≈ 10.93.
How does adding a constant to all values affect standard deviation?
Adding or subtracting a constant to every value does NOT change the standard deviation. It shifts the mean but keeps all relative distances the same. For example, {5, 10, 15} and {15, 20, 25} have identical SDs.
How does multiplying all values by a constant affect standard deviation?
Multiplying every value by k multiplies the standard deviation by |k|. If {2, 4, 6} has SD = 1.633, then {4, 8, 12} (multiplied by 2) has SD = 3.266.
What is the standard deviation of 1 through 10?
The population SD of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is approximately 2.872. Mean = 5.5. This is a useful benchmark for evenly spaced consecutive integers.
What does a negative standard deviation mean?
Standard deviation cannot be negative. It is the square root of variance (sum of squared values divided by count), which is always non-negative. If you get a negative result, there is an error in the calculation.
People Also Search For
How to calculate standard deviation by hand step by step
To calculate SD by hand: (1) find the mean, (2) subtract the mean from each value, (3) square each result, (4) sum the squares, (5) divide by N (population) or N−1 (sample), (6) take the square root. For {2, 4, 6}: mean = 4. Deviations: −2, 0, 2. Squared: 4, 0, 4. Sum = 8. Variance = 8/3 ≈ 2.667. SD ≈ 1.633.
Standard deviation calculator with mean and variance shown
This calculator displays standard deviation, variance, mean, count, sum, and sum of squares simultaneously. It also shows the complete step-by-step calculation including each individual deviation and squared deviation, making it ideal for checking homework or understanding the process.
Population vs sample standard deviation — which should I use?
Use population SD when your data contains every member of the group; use sample SD when your data is a random subset. In practice, most surveys and experiments use sample SD because you rarely measure an entire population. Excel uses STDEV.S (sample) by default; STDEV.P is for population.
What is a good standard deviation for test scores?
A standard deviation of 10–15 points on a 100-point test is typical and healthy. An SD under 5 suggests the test was too easy (everyone scored similarly). An SD over 25 may indicate the test was too hard or the class was very mixed in ability. Teachers use SD to evaluate whether a test discriminates well between students.
How to interpret standard deviation in a dataset
Compare the SD to the mean using the coefficient of variation (CV = SD / mean × 100%). A CV under 10% means low variability; 10–30% is moderate; over 30% is high variability. For example, SD = 5 with mean = 50 gives CV = 10% (consistent data), but SD = 5 with mean = 6 gives CV = 83% (very variable).
Standard deviation vs standard error — what is the difference?
Standard deviation measures spread in your data; standard error (SE) measures uncertainty in your estimate of the mean. SE = SD / √N. As sample size increases, SE decreases (your estimate of the mean becomes more precise) but SD stays about the same (data spread doesn't change with sample size).
Frequently Asked Questions
What is the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?
The population SD is exactly 2. Mean = 5, sum of squared deviations = 32, variance = 4, SD = √4 = 2. This is a commonly cited textbook example.
What is the difference between population and sample standard deviation?
Population SD (σ) divides by N; sample SD (s) divides by N−1 (Bessel's correction). Use population SD when you have all the data; use sample SD when your data is a random subset of a larger population.
What does variance mean in statistics?
Variance is the average of squared differences from the mean. It is the square of the standard deviation. Variance is expressed in squared units (e.g., m²), while SD is in the original units (e.g., m), making SD more interpretable.
Can standard deviation be negative?
No. Standard deviation is always zero or positive. It is the square root of variance, which is a sum of squared values divided by a positive count — always ≥ 0. SD = 0 only when all values are identical.
What does a standard deviation of 0 mean?
An SD of 0 means all values in the dataset are identical. There is no variation — every data point equals the mean.
What data formats does the calculator accept?
The calculator accepts numbers separated by commas, spaces, or new lines. Negative numbers and decimals are supported. Non-numeric values are automatically ignored.
What is the 68-95-99.7 rule?
In a normal distribution, 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This is called the empirical rule and is used in quality control, standardized testing, and risk analysis.
How do I calculate a z-score?
z = (x − μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation. A z-score tells you how many standard deviations a value is from the mean. Positive z = above mean; negative z = below mean.
Is my data private?
Yes — all calculations run entirely in your browser. No data you enter is transmitted to any server or stored anywhere.
How many numbers can I enter?
There is no hard limit. The calculator handles large datasets efficiently in the browser. For very large datasets (thousands of values), performance depends on your device.
How is standard deviation used in finance?
In finance, SD measures investment volatility (risk). A higher SD means more unpredictable returns. Fund managers use SD to assess and compare the riskiness of portfolios, stocks, and bonds.