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🎯 Z-Score Calculator

Calculate how many standard deviations a value is from the mean. First calculate SD in the Calculator tab, or enter values manually.

Value (x)

Mean (μ)

Std Dev (σ)

📋 Z-Score Reference Table

Z-Score	Percentile	Interpretation

📐 Standard Deviation Formulas

Population Standard Deviation (σ)

Use when you have data for the ENTIRE population.

σ = √(Σ(xᵢ - μ)² / N)

Where:
μ = population mean = Σx / N
N = total number of values
Σ(xᵢ - μ)² = sum of squared deviations

Sample Standard Deviation (s)

Use when your data is a SAMPLE from a larger population. Uses (n-1) — Bessel's correction.

s = √(Σ(xᵢ - x̄)² / (n-1))

Why n-1? → Bessel's correction makes sample SD
an unbiased estimator of population SD.
With n-1, sample SD is slightly larger → accounts
for the fact that a sample underestimates spread.

Step-by-Step Process

1. Find mean: x̄ = Σx / n
2. Find deviations: (xᵢ - x̄) for each value
3. Square each: (xᵢ - x̄)²
4. Sum squares: Σ(xᵢ - x̄)²
5. Divide by n (pop) or n-1 (sample) → Variance
6. Take square root → Standard Deviation

Z-Score Formula

Z = (x - μ) / σ

Z tells you how many standard deviations
a value is above (+) or below (-) the mean.

|Z| < 1 → within 1 SD (common)
|Z| 1-2 → somewhat unusual
|Z| 2-3 → unusual (only 4.5% of data)
|Z| > 3 → very unusual (only 0.27% of data)

The Empirical Rule (68-95-99.7)

In a normal distribution:
68.27% of data falls within μ ± 1σ
95.45% of data falls within μ ± 2σ
99.73% of data falls within μ ± 3σ

This is why anything beyond 3σ is called
an "outlier" — only 0.27% chance naturally!

Coefficient of Variation (CV)

CV = (σ / |μ|) × 100%

CV measures relative variability.
Low CV → data is tightly clustered
High CV → data is widely spread

Useful for comparing variability across
datasets with different units or scales.

❓ Frequently Asked Questions

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📊 Standard Deviation Calculator

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📋 Full Statistical Summary

🔔 Normal Distribution (Bell Curve)

📝 Step-by-Step Calculation Table

📐 Calculation Steps