Standard Deviation Formula

📖 What is Standard Deviation?

Standard deviation measures how spread out data values are from the mean. A small standard deviation means values are clustered close together; a large one means they are spread far apart.

σ = √( Σ(xᵢ − μ)² / n )

🔤 What Each Symbol Means

σ

Standard deviation (sigma)The measure of spread we're calculating

μ

Mean (mu)The average of all data values

xᵢ

Each data valueEvery individual value in the dataset

n

Number of valuesTotal count of data points (use n-1 for sample std dev)

📝 Step-by-Step Example

Find the standard deviation of {2, 4, 4, 4, 5, 5, 7, 9}.

1

Find the meanμ = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5

2

Subtract mean from each value, then square(2-5)²=9, (4-5)²=1, (4-5)²=1, (4-5)²=1, (5-5)²=0, (5-5)²=0, (7-5)²=4, (9-5)²=16

3

Sum the squared differences9+1+1+1+0+0+4+16 = 32

4

Divide by n, then take square rootσ = √(32/8) = √4

✓

Answer: σ = 2

💡 Population vs Sample

Population std dev (σ):

Divide by n — use when you have ALL data

Sample std dev (s):

Divide by (n−1) — use when you have a subset of data

Standard Deviation

📖 What is Standard Deviation?

🔤 What Each Symbol Means

📝 Step-by-Step Example

💡 Population vs Sample

🔗 Related

🎯 Quick Fact