Enter your data and instantly see the mean, variance, and standard deviation, with the full computation shown step by step. Supports both sample and population formulas.
Mean, variance, standard deviation
σ = √(Σ(x − μ)² / N)
Mean (μ)
18
Variance
182
Std Dev (σ)
13.490738
Step 1. Count: n = 6, Sum: 108
Step 2. Mean: μ = 108 / 6 = 18
Step 3. Σ(x − μ)² = 910
Step 4. Variance (sample): 910 / (6 − 1) = 182
Step 5. σ = √variance = 13.490738
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Standard Deviation Formula
Take each value's distance from the mean, square it, average those squared distances, then take the square root. For samples, divide by N−1 instead of N.
The mean is the average of your data. The variance measures how spread out the values are from the mean, it's the average of the squared distances. The standard deviation is the square root of the variance, expressed in the original units.
The key distinction: for an entire population, divide by N. For a sample drawn from a larger population, divide by N − 1 (Bessel's correction). The calculator handles both modes.
Worked Examples
Population vs sample, zero spread, test scores, and signed data covered.
Example 1, Small data set (population)
Data: 2, 4, 4, 4, 5, 5, 7, 9
Mean
μ = (2+4+4+4+5+5+7+9) / 8 = 40/8 = 5Squared deviations
(2−5)² + (4−5)²×3 + (5−5)²×2 + (7−5)² + (9−5)² = 9+3+0+4+16 = 32Variance (population)
σ² = 32 / 8 = 4Std deviation
σ = √4 = 2Classic textbook example. Use population formula (divide by N).
Example 2, Sample standard deviation
Sample: 10, 12, 23, 23, 16, 23, 21, 16
Mean
x̄ = 144/8 = 18Squared deviations
Σ(x − x̄)² = 192Sample variance
s² = 192 / (8 − 1) = 27.43Std deviation
s = √27.43 ≈ 5.24Sample formula divides by (n − 1), called Bessel's correction.
Example 3, Equal values
Data: 7, 7, 7, 7, 7
Mean
μ = 7Squared deviations
All (x − 7)² = 0Variance
σ² = 0Std deviation
σ = 0When all values are identical, there is zero spread.
Example 4, Test score spread
Test scores: 78, 85, 88, 92, 95
Mean
μ = 438/5 = 87.6Squared deviations
Σ(x − 87.6)² ≈ 181.2Variance (population)
σ² = 181.2 / 5 ≈ 36.24Std deviation
σ ≈ 6.02Helps interpret how varied a class's performance was.
Example 5, Negative values
Data: −5, −2, 0, 2, 5
Mean
μ = 0/5 = 0Squared deviations
25 + 4 + 0 + 4 + 25 = 58Variance (population)
σ² = 58 / 5 = 11.6Std deviation
σ = √11.6 ≈ 3.41Standard deviation is always non-negative.
Standard deviation tells you how spread out a set of values is. A small standard deviation means values are clustered near the mean, a large one means they're more variable. This calculator computes the mean, variance, and standard deviation for any dataset, with every step shown so you understand the method, not just the number.
If your data represents an entire population (every member you care about), divide by N. If it's a sample drawn from a larger population to estimate that population's variability, divide by N − 1. This adjustment (Bessel's correction) compensates for the fact that a sample underestimates true variability. Most real-world stats problems use the sample formula.
Step 1: find the mean of the data. Step 2: subtract the mean from each value and square the result. Step 3: sum those squared deviations. Step 4: divide by N (population) or N − 1 (sample) to get the variance. Step 5: take the square root to get the standard deviation. The calculator walks through these five steps with your data.
Squaring serves two purposes: it eliminates negative differences (so they don't cancel positives), and it weighs larger deviations more heavily than smaller ones. Taking the square root at the end brings the result back to the original units of measurement, making it interpretable.
Standard deviation appears in quality control, finance (volatility), test scoring, scientific measurement, A/B testing, and survey analysis. A risk-averse investor prefers assets with lower standard deviation. A teacher uses it to spot test scores that are unusually high or low. Any time you need to summarize variability in a number, standard deviation is the standard tool.
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