Normal Distribution Calculator — PDF, CDF & Inverse

The normal distribution is a bell-shaped, symmetric, continuous probability distribution defined by two parameters:

• ›μ (mean) — determines where the bell is centred on the number line
• ›σ (standard deviation) — determines how wide or narrow the bell is; larger σ = flatter, wider curve

Its central importance comes from the Central Limit Theorem: regardless of the original distribution, the average of a large sample will be approximately normally distributed. This is why IQ scores, measurement errors, heights, and countless natural phenomena follow the normal distribution.

• ›PDF f(x) — the height of the bell curve at x. Not a probability on its own. Useful for comparing relative likelihoods of different values.
• ›CDF Φ(x) = P(X ≤ x) — the area under the bell curve from −∞ to x. This IS a probability and is what this calculator uses for all queries.

Example: for N(0,1), f(0) ≈ 0.399 (the peak height) but Φ(0) = 0.5 (50% of values lie below 0). The PDF value 0.399 is not a probability — it only makes sense as part of an integral. You use P(a ≤ X ≤ b) = Φ(b) − Φ(a) to get actual probabilities.

Use the P(a ≤ X ≤ b) mode:

• ›Enter μ and σ for your distribution
• ›Enter a (left boundary) and b (right boundary) — a must be less than b
• ›The result is Φ((b−μ)/σ) − Φ((a−μ)/σ)
• ›The bell curve shades the region between a and b automatically

Example: for heights N(175.3, 7²), P(170 ≤ X ≤ 180) = Φ((180−175.3)/7) − Φ((170−175.3)/7) ≈ Φ(0.671) − Φ(−0.757) ≈ 0.749 − 0.225 ≈ 52.5%.

For any normal distribution N(μ, σ²), the exact probabilities are:

• ›P(μ − σ ≤ X ≤ μ + σ) = 68.2689% — about 2 in 3 values
• ›P(μ − 2σ ≤ X ≤ μ + 2σ) = 95.4500% — about 1 in 22 outside
• ›P(μ − 3σ ≤ X ≤ μ + 3σ) = 99.7300% — about 1 in 370 outside
• ›P(μ − 4σ ≤ X ≤ μ + 4σ) = 99.9937% — about 1 in 15,787 outside

In manufacturing, a "Six Sigma" process targets ±6σ, which corresponds to 3.4 defects per million — well beyond the standard empirical rule range.

The inverse CDF Φ⁻¹(p) is the value x such that P(X ≤ x) = p.

• ›Φ⁻¹(0.5) = μ (the median — 50% lies below)
• ›Φ⁻¹(0.975) ≈ μ + 1.96σ (95% CI upper critical value)
• ›Φ⁻¹(0.99) ≈ μ + 2.326σ (99th percentile)
• ›Φ⁻¹(0.001) ≈ μ − 3.09σ (0.1th percentile)

This calculator uses the Acklam rational approximation for maximum accuracy, correctly handling extreme probabilities close to 0 or 1 that simple polynomial approximations struggle with.

The normal distribution is appropriate when:

• ›The data is continuous (not integer counts)
• ›The histogram is roughly bell-shaped and symmetric
• ›Mean ≈ median (skewness near zero)
• ›Extreme values are rare (thin tails)
• ›You are analysing sample means (Central Limit Theorem applies)

Prefer other distributions for:

• ›Count data with a known upper limit → Binomial
• ›Rare event counts with no upper limit → Poisson
• ›Strictly positive, right-skewed data → Log-normal or Gamma
• ›Heavy-tailed data with outliers → t-distribution or Cauchy

There is no need to manually standardise your data:

• ›Enter μ and σ directly — the calculator converts to z-scores internally
• ›For the standard normal, enter μ = 0, σ = 1
• ›The z-score for your query value is shown in the output for cross-checking against z-tables
• ›All probabilities are returned in the original x scale
• ›The inverse mode outputs x directly: x = μ + σ × Φ⁻¹(p)

The erf-based CDF implementation is accurate to 1.5 × 10⁻⁷ (Abramowitz & Stegun formula 7.1.26), and the inverse CDF (Acklam approximation) is accurate to approximately nine significant figures across the full (0, 1) domain.

Yes — inputs are automatically persisted to your browser's localStorage:

• ›μ, σ, a, b, and p are saved on every change
• ›The selected query mode is preserved between visits
• ›All data stays in your browser — nothing is sent to any server
• ›Inputs are restored the next time you open the page

Click Reset All to clear both the form and the saved localStorage data.