Z-Score Calculator

📚 How It Works

What is a Z-Score?

A z-score (standard score) measures how many standard deviations a value is from the mean. It allows comparison of values from different normal distributions.

Z-Score Formula

z = (X - μ) / σ

Where:

• X = individual value
• μ = population mean
• σ = population standard deviation

Reverse Formula (Find Value from Z)

X = μ + (z × σ)

Interpreting Z-Scores

• z = 0: Value equals the mean (50th percentile)
• z = +1: Value is 1 standard deviation above mean (84th percentile)
• z = -1: Value is 1 standard deviation below mean (16th percentile)
• z = +2: Value is 2 standard deviations above mean (97.7th percentile)
• z = -2: Value is 2 standard deviations below mean (2.3rd percentile)
• |z| > 3: Unusual/extreme value (outlier)

68-95-99.7 Rule

For normal distributions:

• ~68% of data falls within z = ±1 (1 standard deviation)
• ~95% of data falls within z = ±2 (2 standard deviations)
• ~99.7% of data falls within z = ±3 (3 standard deviations)

Z-Score to Percentile

The percentile tells you what percentage of values fall below your z-score. For example:

• z = 0 → 50th percentile (half below, half above)
• z = 1 → 84.13th percentile (84.13% of values are below)
• z = -1 → 15.87th percentile (only 15.87% of values are below)

Real-World Applications

• Education: Standardized test scores (SAT, GRE, IQ tests)
• Medicine: Growth charts, bone density, lab results
• Finance: Stock returns, portfolio analysis
• Quality Control: Identifying defects and outliers
• Sports: Player statistics, performance analysis
• Research: Hypothesis testing, identifying significant results

Example Calculation

Question: A student scores 85 on a test. The class mean is 75 with standard deviation 10. What's their z-score?

Solution:

z = (85 - 75) / 10 = 10 / 10 = 1.0

Interpretation: The student scored 1 standard deviation above the mean, placing them at the 84.13th percentile. They scored better than approximately 84% of the class.

Common Z-Score Benchmarks

Z-Score	Percentile	Common Use
-1.96	2.5%	95% confidence interval (lower)
-1.645	5%	90% confidence interval (lower)
0	50%	Mean/median
+0.674	75%	Third quartile (Q3)
+1.645	95%	90% confidence interval (upper)
+1.96	97.5%	95% confidence interval (upper)
+2.576	99.5%	99% confidence interval (upper)

Important Notes

• Z-scores assume a normal distribution
• Works best with large samples (n > 30)
• Positive z = above mean, negative z = below mean
• Standard normal distribution has μ = 0, σ = 1