Standard Deviation Calculator

Standard deviation measures how spread out values are around the mean (average). A small standard deviation means values cluster tightly together; a large one means they are spread widely apart.

Population standard deviation formula: σ = √(Σ(xᵢ − μ)² / N)

Sample standard deviation formula: s = √(Σ(xᵢ − x̄)² / (n − 1))

Use population formula when you have all possible data. Use sample formula when your data is a sample from a larger population (the “n−1” corrects for bias).

Step-by-step worked example: Data: 4, 7, 13, 2, 1, 7

Step 1. Find the mean: (4+7+13+2+1+7)/6 = 34/6 = 5.67

Step 2. Find each value’s distance from the mean, squared: (4−5.67)² = 2.79 (7−5.67)² = 1.77 (13−5.67)² = 53.73 (2−5.67)² = 13.47 (1−5.67)² = 21.81 (7−5.67)² = 1.77

Step 3. Average those squares (population): 95.34 / 6 = 15.89

Step 4. Take the square root: √15.89 = 3.99

Working with the unrounded mean of 5.6667 throughout gives 15.8889 and σ = 3.9861, which is what the calculator reports. The two agree to two decimals, and that gap is worth noticing: rounding the mean early is the most common source of a slightly-off answer in hand calculations.

The sample standard deviation for the same data divides by 5 rather than 6, giving s = 4.3665. It is always the larger of the two, and the gap shrinks as the data set grows.

What standard deviation tells you: In a normal distribution (bell curve):

• 68% of values fall within ±1 standard deviation of the mean
• 95% within ±2 standard deviations
• 99.7% within ±3 standard deviations

Practical uses:

• Finance: stock volatility (higher σ = riskier investment)
• Quality control: product consistency
• Medicine: identifying outliers in patient data
• Education: grade distributions and standardised test scoring
• Sports: consistency of athlete performance