Standard Deviation Calculator

Essential Statistics Formulas

A comprehensive reference guide for statistical formulas with examples and explanations. Click to copy any formula to your clipboard.

Mean (Average)

x̄ = Σx / n

Sum of all values divided by the number of values

Example:

For data: 2, 4, 6, 8, 10 Mean = (2+4+6+8+10)/5 = 30/5 = 6

When to use:

Use when data is normally distributed without outliers

Median

M = middle value (odd n) or average of two middle values (even n)

The middle value when data is sorted

Example:

Odd n: 1, 3, 5, 7, 9 → Median = 5 Even n: 1, 3, 5, 7 → Median = (3+5)/2 = 4

When to use:

Use when data has outliers or is skewed

Mode

Mode = most frequent value

The value that appears most often

Example:

For data: 1, 2, 2, 3, 3, 3, 4 Mode = 3 (appears 3 times)

When to use:

Use for categorical data or finding the most common value

Range

R = max - min

Difference between largest and smallest values

Example:

For data: 2, 5, 8, 12, 15 Range = 15 - 2 = 13

When to use:

Quick measure of spread, but sensitive to outliers

Population Variance

σ² = Σ(x - μ)² / n

Average squared deviation from the mean

Example:

Data: 2, 4, 6 (μ = 4) σ² = [(2-4)² + (4-4)² + (6-4)²]/3 = [4+0+4]/3 = 2.67

When to use:

Use when you have the entire population

Sample Variance

s² = Σ(x - x̄)² / (n-1)

Unbiased estimate of population variance

Example:

Data: 2, 4, 6 (x̄ = 4) s² = [(2-4)² + (4-4)² + (6-4)²]/(3-1) = 8/2 = 4

When to use:

Use when working with a sample

Population Standard Deviation

σ = √[Σ(x - μ)² / n]

Square root of population variance

Example:

If σ² = 2.67, then σ = √2.67 = 1.63

When to use:

Measure of spread for entire population

Sample Standard Deviation

s = √[Σ(x - x̄)² / (n-1)]

Square root of sample variance

Example:

If s² = 4, then s = √4 = 2

When to use:

Measure of spread for a sample

Coefficient of Variation

CV = (σ / μ) × 100%

Relative measure of variability

Example:

If σ = 2 and μ = 10 CV = (2/10) × 100% = 20%

When to use:

Compare variability between datasets with different units

Interquartile Range

IQR = Q₃ - Q₁

Range of the middle 50% of data

Example:

If Q₁ = 25 and Q₃ = 75 IQR = 75 - 25 = 50

When to use:

Robust measure of spread, not affected by outliers

Quick Reference Guide

Greek Letters in Statistics

μ (mu)Population mean
σ (sigma)Population standard deviation
σ² (sigma squared)Population variance
ρ (rho)Population correlation
α (alpha)Significance level
β (beta)Type II error probability

Common Statistical Symbols

x̄ (x-bar)Sample mean
sSample standard deviation
s²Sample variance
nSample size
Σ (capital sigma)Sum of values
H₀ / H₁Null / Alternative hypothesis

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