The bell curve and its two parameters
A normal distribution is fixed by two numbers. The mean μ says where the peak sits, and the standard deviation σ says how wide the curve is: μ slides the whole shape left and right without changing it, σ stretches or squeezes it while the area underneath stays exactly 1. The two points where the curve changes from bending down to bending up — the inflection points — sit at μ − σ and μ + σ, which is a useful way to read σ straight off a plotted curve.
Because the area is fixed, a smaller σ means a taller peak. The height at the mean is 1/(σ√2π): 0.398942 for the standard normal, but only 0.026596 for the IQ scale with σ = 15. That height is a density, not a probability — probability is the shaded area, which is why the number you read off the vertical axis never has to be below 1.
The curve is symmetric, so the mean, the median and the mode are the same value — the 50th percentile is always μ. That is a property of this model, not of your data: if the mean and median of a real dataset differ noticeably, the data is skewed and a normal model will misprice both tails.
Finding a probability, step by step
Standardise, then look up. z = (x − μ)/σ converts your value into "how many standard deviations from the mean", and Φ(z) — the standard normal CDF — gives the area to the left. On the IQ scale (μ = 100, σ = 15): 130 becomes z = 2, so P(X > 130) = 1 − Φ(2) = 0.0227501, about 1 person in 44.
Same scale, three more answers: P(X > 115) = 0.1586553 (z = 1), P(85 < X < 115) = 0.6826895 — the middle band — and P(X > 145) = 0.0013499, roughly 1 in 741. An interval is always the difference of two left tails, Φ(z₂) − Φ(z₁), which is why the middle band and the two outer tails add to exactly 1.
Two habits prevent most mistakes. Sketch the shading before you compute — "more than" is the right tail, "at most" the left, "between" the middle. And sanity-check the sign of z: a value above the mean must give a probability below 0.5 on the right tail, and a negative z must land below the 50th percentile.
The empirical rule, with the exact numbers
"68–95–99.7" is a rounding of 0.682689, 0.954500 and 0.997300 — the probabilities of landing within ±1σ, ±2σ and ±3σ of the mean. The buttons on this page shade each band so you can read the exact figure rather than the slogan.
The outside is the part that matters in practice: 31.73% of values lie outside ±1σ (about 1 in 3), 4.55% outside ±2σ (1 in 22) and 0.27% outside ±3σ (1 in 370). A control chart with ±3σ limits therefore expects a false alarm roughly once every 370 samples — which is exactly why the limits were put there.
The rule only holds for a normal distribution, and it is worth remembering how weak the guarantee is without that assumption: Chebyshev’s inequality promises only that at least 75% of any distribution lies within ±2σ, against the normal’s 95.45%. Quoting "95% within two standard deviations" for skewed data borrows a number the data has not earned.
Going backwards: percentile to score
Type a probability instead of a value and the calculator solves x = μ + zσ. The z values worth memorising are 1.2815516 for the top 10%, 1.6448536 for the top 5%, 2.0537489 for the top 2% and 2.3263479 for the top 1%; the negatives are the same numbers mirrored.
Worked examples. Mensa’s usual threshold is the top 2% of IQ, which on the 100/15 scale is 100 + 15 × 2.0537489 = 130.8 — the reason the entry line is quoted as 130 or 132 depending on the σ of the test used. And "what SAT score puts me in the top 4%?" is 1050 + 200 × 1.7506861 = 1400.1, so 1400 is the round number behind that goal.
Careful with the wording: "top 10%" is the 90th percentile, not the 10th. Enter 0.9 with a left-tail mode, or 0.1 with a right-tail mode — both give the same score, and mixing them up is the single most common error in this direction.
Real μ and σ you can check your answer against
IQ is defined as μ = 100, σ = 15 by construction, so it is the cleanest test case: P(X > 130) = 2.28% and P(X > 145) = 0.13% are figures you can compare against published Mensa and gifted-programme thresholds.
Adult male height, from national surveys: Korea 172.5 / 5.6 cm gives P(X > 180) = 9.02%; the United States 175.3 / 7.4 cm gives P(X > 183 ≈ 6 ft) = 14.90%; Japan 170.8 / 5.8 cm gives P(X > 180) = 5.63%; the world average of 171 / 7 cm gives 9.93% above 180. The preset button on this page shows the numbers for your own country.
Exam scales are designed to be normal, which makes them good practice: SAT total 1050 / 200 puts P(X > 1400) at 4.01%, and the Korean CSAT standard score of 100 / 20 puts P(X > 130) at 6.68% and P(X > 140) at 2.28%. Percentile and standard score are two views of the same curve, which is why a standard score of 140 and "top 2%" describe the same student.
When the bell curve is the wrong model
Skewed quantities. Income, house prices, session length, response time and file size all have a floor at zero and a long right tail, so the mean sits above the median and a normal model puts probability where no data can exist. A log-normal or exponential model fits them far better — and if you are generating test data, drawing prices from a log-normal is the difference between a realistic demo and a flat one.
Counts. "How many defects in 1,000 units" or "how many calls this hour" are whole numbers with their own distributions — binomial and Poisson. A normal curve can approximate them once np ≥ 5 and n(1 − p) ≥ 5, but the approximation needs a continuity correction to be worth using, and it is never the right first model.
Bounded and multi-peaked data. Percentages and test scores hitting a ceiling, mixtures of two populations (two machines, two user segments) and heavy-tailed processes where the extreme value dominates the sum all break the model in ways that show up first in the tails — which is exactly where people use the calculator.