Normal Distribution Calculator

IQ scores follow a normal distribution with a mean of 100 and a standard deviation of 15. A score of 125 sits at the 95.22nd percentile — 4.78% of test-takers score higher. Normal Distribution Calculator runs that arithmetic for any μ and σ you give it: z-score, CDF (the left-tail probability), its mirror the right tail, the probability density at x, and the percentile rank.

The same calculation appears in manufacturing quality control (how many units will fall outside a tolerance?), exam grading (what percentage of students scored above the cutoff?), and hypothesis testing (where does a result land on the null distribution?). The optional interval panel computes P(a < X < b) for any two bounds, useful for questions like "what fraction of products weigh between 497 g and 503 g?"

Worked Example: IQ Scores at z = 1.667

1. Press Load Sample or enter Mean = 100, Std Dev = 15, and Query Value = 125.
2. The z-score appears as 1.6667 — the score is 1.667 standard deviations above the mean.
3. CDF reads 95.22%, so 95.22% of the population scores 125 or below. P(X > 125) = 4.78%.
4. To find the probability of scoring between 90 and 110, fill Lower bound = 90 and Upper bound = 110. The interval panel shows P(90 < X < 110) ≈ 49.50% — just under half the population lands in this central band.

The Formulas Behind the Output

The z-score standardises the query value: z = (x − μ) / σ. The CDF is the integral of the standard normal PDF from −∞ to z — computed here with the Abramowitz & Stegun rational approximation of erf, which keeps the maximum error below 1.5 × 10⁻⁷ without any external library. The probability density f(x) = (1 / (σ√2π)) × e^(−(x−μ)² / (2σ²)) gives the relative likelihood at exactly x; it is not itself a probability (it can exceed 1 for narrow distributions).

Precision and Known Limits

• The approximation is accurate to 7 decimal places, sufficient for every classroom and engineering use case on this platform. For tails beyond z = ±8 the underlying probability is below 6 × 10⁻¹⁶ and the CDF rounds to exactly 0 or 1.
• Standard deviation must be strictly positive; entering 0 triggers the validation message "Standard deviation (σ) must be a positive number."
• The interval bounds are independent of the query value. Enter any a and b without needing x to lie between them. If b ≤ a, the tool refuses with "Upper bound must be greater than lower bound."
• The Empirical Rule panel always shows the exact 68.27 / 95.45 / 99.73 percentages derived from the same erf approximation, so they are consistent with the CDF results above rather than the textbook rounded figures.