Calculate Standard Score with Step-by-Step Derivation
Value (x)
Mean (μ)
Std Dev (σ)
Results
Step-by-Step Derivation
Z Score Formula
z = (x - μ) / σ
Where: x = value, μ = population mean, σ = population standard deviation
Z-score standardizes any normal distribution to the standard normal distribution with μ=0 and σ=1.
⚠σ must be positive. Z-score assumes data follows normal distribution.
What is Z Score?
The z-score (standard score) is a measure of how many standard deviations an element is from the mean. It allows comparison of scores from different normal distributions.
Positive Z
Value is above the mean
Negative Z
Value is below the mean
Z = 0
Value equals the mean
68-95-99.7 Rule
Percentages within ±1, ±2, ±3 σ
💡 Example: x=85, μ=75, σ=10. z=(85-75)/10=1. This means 85 is 1 standard deviation above the mean.
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