Normal Distribution & Z-Score Calculator – CDF, PDF, quantiles

Q: 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.

Q: 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.

Q: 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.

Q: 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.

Q: 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.

How to use (3 steps)

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.
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.
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.

Common tasks (examples)

Tail probability: find P(X ≤ x), P(X ≥ x), or a two-tailed probability → use CDF and switch Left/Right/Two-tailed.
Interval probability: find P(a < X < b) → use Interval probability.
Percentile / cutoff score: find x for a percentile p (e.g., 0.95) → use Inverse CDF (quantile).
Z-score: convert between x and z for a given mean μ and σ → use Z-score.
If you only have raw data: compute mean and standard deviation first with the Mean and Standard Deviation Calculator or Descriptive Statistics.

Interpretation (and when to use a normal model)

CDF gives the probability P(X ≤ x). For a right tail, use P(X ≥ x) = 1 − CDF. Two-tailed means “as extreme or more extreme than x” on both sides of the mean.
PDF is a density, not a probability. Probabilities come from areas under the curve (intervals), not from the height at a single point.
Z-score standardises your value: z = (x − μ)/σ. This converts any normal distribution to the standard normal, so you can compare results across different μ and σ.
When it fits: the normal distribution is a good model for many measurement errors and approximately symmetric, unimodal data. Averages of many small effects often look more normal than the raw components.
Limits: if your variable is discrete, strongly skewed, bounded (e.g., 0–1), or has heavy tails/outliers, a normal approximation can be misleading.

Worked examples

Test scores: μ = 100, σ = 15, x = 120 → z ≈ 1.33, so P(X ≤ 120) ≈ 0.909 (90.9%).
95th percentile cutoff: μ = 100, σ = 15 → x ≈ 100 + 1.645×15 ≈ 124.7 for p = 0.95.

References

Mode

Mean μ

Std. deviation σ

x

Lower bound a

Upper bound b

Percentile p

z

Tail

Left Right Two-tailed

Embed this calculator

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.

Teacher mode

Show z = (x − μ) / σ and Φ(−z) = 1 − Φ(z) side by side to contrast one- and two-tailed probabilities and symmetry around the mean.
Derive interval probability as Φ((b−μ)/σ) − Φ((a−μ)/σ) and write out each z-bound to help students follow the calculation step by step.
Use x = μ + σΦ⁻¹(p) for quantiles as a direct replacement for printed Z tables in slides, notes, and worked examples.
Typical approximation error is around 10⁻⁶ for common percentiles; adjust the rounding to match the level and marking scheme you are teaching.

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.

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.