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Z-Score Calculator

Calculate a Z-score, percentile, and probability for one value or standardize every value in a dataset.

Z-Score Calculator
Z-Score Results

Z-Score

1.000

(85 - 75) / 10

Moderate
Between 1-2 standard deviations

Percentile

84.1%

Better than 84.1% of values

P(X ≤ 85) = 84.1%

P(X ≤ 85)

84.13%

Area to the left

P(X > 85)

15.87%

Area to the right

Standard Ranges

Within 1σ (68% of data):65.0 to 85.0
Within 2σ (95% of data):55.0 to 95.0
Within 3σ (99.7% of data):45.0 to 105.0

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What is Z-Score Calculator?

A Z-score (also called a standard score) tells you how many standard deviations a value is from the mean. It standardizes values from any normal distribution, allowing you to compare values from different datasets and determine how typical or unusual a value is.

Z-Score Formula

  • Z = (X - μ) / σ
  • Z: Z-score (standard score)
  • X: The value you're analyzing
  • μ: Population mean
  • σ: Population standard deviation

Interpreting Z-Scores

  • Z = 0: Value equals the mean
  • Z > 0: Value is above the mean
  • Z < 0: Value is below the mean
  • |Z| < 1: Typical (within 1 standard deviation)
  • |Z| ≥ 2: Unusual (rare occurrence)
  • |Z| ≥ 3: Extremely rare (outlier)

Practical Applications

  • Education: Compare test scores across different exams or classes
  • Psychology: Interpret IQ scores, personality assessments
  • Medicine: Evaluate lab results, growth charts, vital signs
  • Business: Analyze sales performance, quality control, market research
  • Sports: Compare athlete performance across different metrics
  • Quality Control: Identify defective products or processes

Z-Score and Percentiles

  • Z-scores can be converted to percentiles using the standard normal distribution
  • A Z-score of 0 corresponds to the 50th percentile
  • A Z-score of 1 corresponds to approximately the 84th percentile
  • A Z-score of -1 corresponds to approximately the 16th percentile

Assumptions and Limitations

  • Most useful when data follows a normal (bell-shaped) distribution
  • Requires knowing or estimating the population mean and standard deviation
  • Less meaningful for severely skewed or non-continuous data
  • Outliers can significantly affect the mean and standard deviation

Worked Z-Score Example

A test score of 85 from a distribution with mean 75 and standard deviation 5 has:

Z = (85 - 75) / 5 = 2

The score is two standard deviations above the mean. Under a normal-distribution interpretation, that is approximately the 97.7th percentile.

Method reference: NIST Engineering Statistics Handbook: Z-scores and outlier detection.



FAQ - Z-Score Calculator

A Z-score of 1.5 means the value is 1.5 standard deviations above the mean. This puts it in approximately the 93rd percentile, meaning it's higher than about 93% of all values in the distribution.