Standard Deviation
Square root of the variance; typical spread.
Formula
Variables
Example
For 2, 4, 6: σ = √2.67 ≈ 1.63.
How it works
Standard deviation measures how far data values typically sit from the mean, using the average of their squared distances and then taking the square root to bring the result back to the original units. Squaring ensures that positive and negative deviations do not cancel out, and the square root makes the spread easier to interpret in the same scale as the data.
When to use it
Use this when you want a single number that summarizes how consistent or variable a set of measurements is, such as test scores, lab results, or daily temperatures. It is especially useful for comparing the spread of two data sets that share the same units.
Worked example, step by step
Data: 3, 7, 7, 9 Step 1: Find the mean: \u0304x = (3 + 7 + 7 + 9) / 4 = 26 / 4 = 6.5 Step 2: Find each deviation from the mean: 3 - 6.5 = -3.5, 7 - 6.5 = 0.5, 7 - 6.5 = 0.5, 9 - 6.5 = 2.5 Step 3: Square the deviations: (-3.5)^2 = 12.25, (0.5)^2 = 0.25, (0.5)^2 = 0.25, (2.5)^2 = 6.25 Step 4: Add the squared deviations: 12.25 + 0.25 + 0.25 + 6.25 = 19 Step 5: Divide by n = 4: 19 / 4 = 4.75 Step 6: Take the square root: \u03c3 = \u221a4.75 \u2248 2.18 Final answer: the standard deviation is about 2.18.
Common mistakes to avoid
- Using n - 1 instead of n when the formula given is for the full population
- Forgetting to square the deviations before averaging them
- Computing the standard deviation from the raw values instead of first finding the mean
Did You Know?
Standard deviation is the "typical" distance from the mean — and its units match the data, unlike variance.
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More in Statistics & Probability
View allArithmetic Mean
BasicThe average of a data set.
Median
BasicThe middle value separating the higher and lower halves.
Variance
IntermediateAverage squared deviation from the mean.
Z-Score
IntermediateNumber of standard deviations a value is from the mean.
Bayes' Theorem
IntermediateUpdates a probability given new evidence.
Combinations
IntermediateNumber of ways to choose r items from n (order irrelevant).