Formulas for Variance, Standard Deviation, and Z-scores

Population Variance Formula

σ² = Σ(X - μ)² / N

When to use Population Variance

When you have data for the entire population and want to measure how scores spread around the population mean.

Sample Variance Formula

s² = Σ(X - M)² / (n - 1)

When to use Sample Variance

When you're working with a sample and want to estimate population variance.

Population Standard Deviation Formula

σ = √[Σ(X - μ)² / N]

Sample Standard Deviation Formula

s = √[Σ(X - M)² / (n - 1)]

Sum of Squares (SS)

SS = Σ(X - M)²

When is SS used?

It's used as a step to calculate variance and standard deviation.

Degrees of Freedom (df)

df = n - 1

Importance of Degrees of Freedom

It's used in sample-based formulas to correct bias in estimating population parameters.

Z-score Formula (individual score)

z = (X - μ) / σ

When to use Z-score for a single score

When you want to find how far one score is from the population mean using population SD.

Z-score Formula for Sample Mean

z = (M - μ) / σM

When to use Z-score for Sample Mean

When comparing a sample mean to the population mean and σ is known.

Standard Error of the Mean (SEM) Formula

σM = σ / √n

When to use SEM Formula

When working with a sampling distribution and σ is known.

Estimated Standard Error Formula

sM = s / √n

When to use Estimated Standard Error

When σ is unknown and you are estimating SEM from the sample.

t-statistic Formula

t = (M - μ) / sM

When to use t-statistic

When comparing a sample mean to the population mean and σ is unknown.