Z-Score Calculator

    Calculate Z-scores and find percentiles in normal distributions

    Z-Score Calculator

    Z = (X - μ) / σ
    Enter values to see the calculation

    About Z-Scores

    A Z-score (also called a standard score) measures how many standard deviations a value is from the mean. It enables comparison between values from different normal distributions on a standardized scale.

    •Formula

    Z = (X − μ) ÷ σ

    • –Z = Z-score (standard score)
    • –X = Raw score (the value being standardized)
    • –μ = Population mean
    • –σ = Population standard deviation

    •Interpretation

    • –Z = 0: Value equals the mean.
    • –Z > 0: Value is above the mean.
    • –Z < 0: Value is below the mean.
    • –|Z| = 1: Value is 1 standard deviation from the mean.
    • –|Z| = 2: Value is 2 standard deviations from the mean.

    •Common Z-Score Benchmarks

    • –Z = ±1.0: 68% of data falls within this range.
    • –Z = ±1.96: 95% of data falls within this range.
    • –Z = ±2.0: 95.4% of data falls within this range.
    • –Z = ±2.58: 99% of data falls within this range.
    • –Z = ±3.0: 99.7% of data falls within this range.

    •Applications

    • –Standardized test scores (SAT, GRE, IQ tests).
    • –Quality control in manufacturing.
    • –Medical diagnostics and reference ranges.
    • –Financial risk assessment.
    • –Research and data analysis.
    • –Outlier detection.

    •Advantages

    • –Allows comparison across different scales.
    • –Identifies outliers and unusual values.
    • –Standardizes data for analysis.
    • –Enables probability calculations.
    • –Facilitates hypothesis testing.

    •Requirements

    • –Data should be normally distributed.
    • –Population mean and standard deviation must be known.
    • –Observations should be independent.

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