Probability · Distributions
Normal Distribution Calculator
The normal distribution — the bell curve — is where much of statistics lives: heights, measurement errors, test scores, sample means. Give it a mean μ and a standard deviation σ and it tells you how likely any range of values is. This finds the probability below, above or between values, or runs it backwards to the value at a given percentile — each with the z-score, the percentile, and the curve drawn and shaded, plus an honest note on where the bell curve fits the world and where it quietly does not.
μ sets where the curve is centred and σ how wide it is. A z-score is just how many σ a value sits from μ: z = (x − μ) ∕ σ.
Result
In plain English
The normal curve is a recipe for how probability is spread: most of it bunched near the mean, thinning symmetrically as you move away, with the famous 68-95-99.7 split inside one, two and three standard deviations. Because the shape is fixed, two numbers — where it is centred (μ) and how spread out it is (σ) — pin down the chance of landing in any range. The area under the curve over a stretch of values is that probability.
- normal distribution
- The symmetric bell-shaped distribution N(μ, σ²), set by its mean μ and standard deviation σ. The total area under the curve is 1.
- z-score
- z = (x − μ) ∕ σ, the number of standard deviations a value lies from the mean. It converts any normal problem to the single standard normal, N(0, 1).
- area = probability
- The probability that X falls in a range is the area under the curve over that range. P(X < x) is everything to the left; P(a < X < b) is the slice between.
- percentile (the inverse)
- Run it backwards: the value with p% of the distribution below it is μ + σ·z, where z is the standard-normal value at that percentile (1.645 for the 95th, 1.96 for the 97.5th).
- a single point has probability 0
- For a continuous distribution only ranges have probability; the chance of exactly any value is zero, which is why these are all “less than” or “between,” never “equal to.”
- thin tails
- The normal’s tails shrink very fast, so it treats far-out values as near-impossible. Many real quantities (returns, floods, insurance losses) have fatter tails, where the bell curve badly understates the extremes.
Frequently asked
How do you find a probability from the normal distribution?
Convert the value to a z-score, z = (x − μ) ∕ σ, then read the area under the standard normal curve. P(X < x) is the cumulative area to the left of z; P(X > x) is 1 minus that; P(a < X < b) is the area at b minus the area at a. For μ = 100, σ = 15 and x = 115, z = 1, and the area to the left is about 0.8413 — so roughly 84% of the distribution lies below 115. This calculator does the conversion and the lookup for you and shades the matching region.
How do I find the value at a given percentile?
That is the inverse normal: x = μ + σ·z, where z is the standard-normal score for that percentile (found from the inverse CDF, sometimes called the quantile or probit). For the 90th percentile z ≈ 1.2816, so with μ = 100 and σ = 15 the value is 100 + 15 × 1.2816 ≈ 119.2 — 90% of the distribution falls below it. Common z values: 1.645 for the 95th percentile, 1.96 for the 97.5th, 2.326 for the 99th. Pick the “Value at percentile” mode above.
What is the 68-95-99.7 rule?
The empirical rule: for a normal distribution about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. So with μ = 100 and σ = 15, roughly 68% lie between 85 and 115, and 95% between 70 and 130. The exact figures are 68.27%, 95.45% and 99.73%. It is a quick mental check — and a reminder that values beyond three σ are genuinely rare if the data really are normal.
When is it wrong to assume a normal distribution?
Whenever the data are skewed, bounded, heavy-tailed or multi-peaked. Incomes and house prices are right-skewed; counts and proportions are bounded; financial returns, natural disasters and insurance losses have fat tails where extreme events are far more common than the bell curve allows. Assuming normality there understates risk badly — the 2008 crisis is partly a story of normal-curve models treating once-in-a-century moves as once-in-the-age-of-the-universe. Check with a histogram or a Q-Q plot before trusting these probabilities in the tails.