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Normal Distribution & Z-Score Calculator

Enter a mean and standard deviation to instantly compute PDF/CDF, quantiles, interval probabilities, and z-scores, with step-by-step explanations and a shaded normal curve.

The default sample asks “In a test with mean 100 and standard deviation 15, what proportion scores at or below 120?” Switch modes to keep only the fields you need, and toggle one- or two-tailed views with a single click.

Other languages: ja | en | zh-CN

How to use (3 steps)

  1. Enter the mean and standard deviation, then choose what you want to find (PDF/CDF, interval, percentile, or z-score). The default values use a test with mean 100 and σ = 15.
  2. Fill in x, the interval [a, b], the percentile p, or the z-score z, and pick left, right, or two-tailed if you are in CDF mode.
  3. Click “Calculate” to see the numeric result, working steps, and a shaded normal curve. You can copy the result text or a shareable URL for homework or lesson notes.
Mode
Tail

By default we show “P(X ≤ 120)” for a test with mean 100 and σ = 15. You can also set a = 85 and b = 115 to explore the classic “within 1σ of the mean” interval.

Results

Value: —

Steps

    Probabilities are shown both as a 0–1 value and as a percentage. Use these as an educational guide; final decisions should rely on appropriate professional or academic judgement.

    Normal curve preview

    We plot the range from −4σ to +4σ and highlight the chosen tail or interval in blue, so you can see at a glance which part of the distribution the probability refers to.

    FAQ

    What is a normal distribution?

    A normal distribution models a continuous variable that clusters around a mean μ with spread measured by the standard deviation σ. The familiar bell-shaped curve is symmetric: values close to μ are common, while very large or very small values are rare.

    What do the mean μ and standard deviation σ represent?

    The mean μ is the central or average value, and the standard deviation σ describes how spread out the data are around μ. A small σ means most observations lie close to μ; a large σ means observations are more dispersed.

    What is a z-score?

    A z-score shows how many standard deviations a value x is above or below the mean: z = (x − μ)/σ. For example, z = 1 means “one σ above the mean”, and z = −2 means “two σ below the mean”. This makes different normal distributions comparable.

    When should I use left-tail, right-tail, or two-tailed probabilities?

    Left-tail covers P(X ≤ x), right-tail covers P(X ≥ x), and two-tailed combines both ends where |X−μ| is at least |x−μ|. Match the choice to your hypothesis test or instructional example.

    How accurate is the inverse CDF (quantile) result?

    We rely on Peter Acklam’s rational approximation with a single Newton step. The typical absolute error stays near 10−6, which is more than adequate for teaching, homework, and exam preparation.

    Related

    How it’s calculated

    • Standardises to z = (x − μ) / σ and evaluates the normal PDF f(x) = 1/(σ√(2π))·exp(−(x−μ)²/(2σ²)) and CDF Φ(z) = 0.5·(1 + erf(z / √2)).
    • Tail and interval probabilities reuse Φ: left P(X ≤ x), right 1 − Φ(z), two-tailed 2·min(Φ(z), 1 − Φ(z)), and intervals Φ((b−μ)/σ) − Φ((a−μ)/σ).
    • Quantiles use Peter Acklam’s inverse-normal approximation with one Newton refinement (typical error ≈ 10⁻⁶); everything runs in your browser and the shareable URL only stores μ, σ, mode, and inputs.