Standard Deviation

Measure how spread out data is from the mean—both sample and population calculations.

Last updated: March 2026

Calculator

Dataset (comma or space separated)

Enter at least 2 numeric values

Sample s

5.2372

(n−1 divisor)

Population σ

4.8990

(n divisor)

Sample Variance (s²)27.4286

Population Variance (σ²)24.0000

Mean18.0000

Sum144

Standard Error1.8516

Count (n)8

What is Standard Deviation?

Standard deviation (σ for population, s for sample) measures how spread out data is from the average (mean). A small standard deviation means data clusters tightly around the mean. A large standard deviation means data is scattered far from the mean. It's the square root of variance and always expressed in the same units as the original data.

There are two formulas: population SD (if you have all data) uses divisor n; sample SD (if you have a subset) uses divisor n−1 (Bessel's correction). The n−1 adjustment in sample SD corrects for bias and provides an unbiased estimate of population SD. Use sample SD when analyzing a subset and want to estimate the population's spread.

Standard deviation is fundamental to statistics: it's used in confidence intervals (±1.96σ for 95% CI in normal distribution), hypothesis testing, and understanding data quality. In the 68-95-99.7 rule, approximately 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean (for normal distributions).

How to Calculate

Step-by-Step Process

Step 1: Calculate the mean (average) of all values

Step 2: Find deviation: subtract mean from each value

Step 3: Square each deviation

Step 4: Sum all squared deviations

Step 5: Divide by n (population) or n−1 (sample) → variance

Step 6: Take square root → standard deviation

Key Formulas

Population Variance:

σ² = Σ(x − μ)² / n

Sample Variance (Unbiased):

s² = Σ(x − x̄)² / (n − 1)

Population Std Dev:

σ = √(σ²)

Sample Std Dev:

s = √(s²)

Standard Error (SE):

SE = s / √n (sampling distribution spread)

Sample vs Population: When to Use Which?

Sample (s, use n−1)

You have: subset of data

Goal: estimate population SD

Example: 30 student scores from 500

Population (σ, use n)

You have: ALL data

Goal: describe existing dataset

Example: all 500 student scores

Real-World Example

Test Scores: Comparing Two Classes

Class A:

Scores: 72, 74, 76, 78, 80, 82, 84, 86 (consistent)

Class B:

Scores: 50, 60, 70, 80, 82, 85, 88, 95 (varied)

Analysis:

Both class averages: ≈ 79

• Class A mean = 79, Sample SD ≈ 5.0

• Class B mean = 79, Sample SD ≈ 14.4

Interpretation:

Class A has tightly clustered scores (low SD = consistent performance). Class B has highly varied scores (high SD = some struggling, some excelling). Same average masks different class dynamics!

Insight:

SD reveals data spread. Class A teacher can confidently say "most students have similar capabilities." Class B teacher needs differentiated instruction for diverse learner levels.

Frequently Asked Questions

Why n−1 instead of n in sample formula?

Using n underestimates true population SD (bias). Using n−1 (Bessel's correction) corrects this bias. As n increases, the difference shrinks. For n > 30, sample and population SD are very similar.

What's the relationship between variance and SD?

SD = √(Variance). Variance is SD squared. Both measure spread, but SD is in original units (easier to interpret). Example: if scores have SD = 10, variance = 100.

Is standard deviation always positive?

Yes! SD is always positive or zero (only zero if all values are identical). This is because we square deviations, making them all non-negative, then take the square root.

How does SD relate to the 68-95-99.7 rule?

For normal distributions: 68% of data is within ±1SD, 95% within ±2SD, 99.7% within ±3SD. This rule breaks down for non-normal data, but works well for approximately normal distributions.

Can I compare SDs from different datasets?

Not directly if they have different means or units. Use Coefficient of Variation (CV = SD/mean × 100%) to compare spread of datasets with different averages or units.

What if my dataset has outliers?

Large outliers inflate SD significantly. If outliers are errors, remove them. If real, report SD but also mention median and interquartile range (IQR) as robust alternatives to mean and SD.

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Squared standard deviation.

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