Choosing a distribution: three questions
First: is the outcome a count or a measurement? Counts are whole numbers and use a discrete distribution; measurements can fall anywhere on a scale and use a continuous one. "How many of the 1,000 units were defective" is a count; "how long the battery lasted" is a measurement, and picking the wrong side of that line is the most common modelling mistake.
Second, for a count: is the number of trials fixed? Twenty coin flips, 1,000 inspected units — fixed n, constant success probability, independent trials → binomial. No fixed n, just events arriving over an interval of time or space (calls per hour, typos per page, arrivals per minute) → Poisson, whose single parameter λ is both the mean and the variance.
Third, for a measurement: symmetric around an average → normal (μ, σ). Waiting time until the next event, memoryless → exponential (λ, mean 1/λ). Every value in a range equally likely → uniform (a, b). And two distributions exist for inference rather than for data: Student’s t (a sample mean when σ is unknown) and chi-square (goodness-of-fit, independence, variance).
- Quick check on the summary line: for the binomial the mean is np and the variance np(1−p); for the Poisson both are λ; for the exponential the mean is 1/λ and the median ln 2/λ (λ = 0.5 → mean 2, median 1.3863); for the uniform the mean is (a+b)/2. If the mean the page reports is nowhere near what you expected, the parameters are wrong, not the answer.
Density, cumulative probability, tail and quantile
The density f(x) of a continuous distribution is not a probability — it is a height, and it can exceed 1. A standard normal peaks at 0.398942, and an exponential with λ = 2 has density 2 at x = 0. Probability is area: only an interval has one, which is why P(X = x) is exactly 0 for every continuous distribution.
The CDF F(x) = P(X ≤ x) is the shaded area to the left, and the survival function 1 − F(x) is the right tail. The interval and outside modes combine them: P(a < X ≤ b) = F(b) − F(a), and the two-sided outside probability is F(a) + (1 − F(b)). Switching the view from density to CDF redraws the same numbers as a rising S-curve, where a probability is a height rather than an area.
The quantile is the inverse: give it 0.95 and it returns the value with 95% of the mass below it. That is where critical values come from — the 95th percentile of the standard normal is 1.6448536 and the 97.5th is 1.9599640, the two numbers behind every one- and two-tailed 5% test.
Forward and inverse in the same screen
Most calculators make you pick a direction first. Here you type in whichever box you know: put a value in the x box and the probability appears; put a probability in the p box and the value appears. The mode (left tail, right tail, interval, outside, density) applies to both directions, so "P(X ≤ ?) = 0.99" and "P(X ≤ 130) = ?" are the same screen with different fields filled.
For a discrete distribution the inverse is a step function, and the answer is the smallest k with F(k) ≥ q — not an interpolated decimal. Asking for the 0.95 quantile of a binomial with n = 20, p = 0.5 returns 14, and F(14) = 0.9793, comfortably above 0.95, because 13 would only reach 0.9423. Textbooks and software agree on this convention; a calculator that hands back 13.6 is interpolating something that does not exist.
Every field, the mode and the chosen distribution live in the query string, so the address bar is a permanent record of the calculation. Copy it into an assignment, a code review or a message and the recipient opens the identical screen — no screenshot, no "set μ to 100 and σ to 15 first".
Discrete distributions and the ≤ / < switch
In a continuous distribution P(X ≤ x) and P(X < x) are identical, because the single point contributes no area. In a discrete one they differ by exactly P(X = x), and the difference is often large: for 20 coin flips P(X ≤ 10) = 0.588099 while P(X < 10) = 0.411901 — a 17.6 percentage-point gap, which is P(X = 10) = 0.176197.
The same trap sits in the right tail. With a Poisson λ = 5, "more than 8 calls" reads as P(X ≥ 9) = 0.068094 if the caller meant strictly more, but as P(X ≥ 8) = 0.133372 if they meant 8 or more — the answer nearly doubles depending on how the sentence is read. The boundary toggle exists so you can state which one you meant instead of hoping the reader guesses.
- Practical habit: write the inequality out before you type it. "At least k" is P(X ≥ k), "more than k" is P(X ≥ k+1), "at most k" is P(X ≤ k) and "fewer than k" is P(X ≤ k−1). The label above the result restates whichever one you selected, so it can be checked at a glance.
How the seven distributions approximate each other
Binomial → Poisson when n is large and p is small. With n = 1,000 and p = 0.01, the exact binomial gives P(X ≤ 15) = 0.952129 and a Poisson with λ = np = 10 gives 0.951260 — a difference of 8.7e-4. The classic rule of thumb is n ≥ 20 with p ≤ 0.05, or n ≥ 100 with np ≤ 10.
Binomial and Poisson → normal when there is enough mass in the middle: np ≥ 5 and n(1−p) ≥ 5 for the binomial, λ ≥ 10 for the Poisson. The page checks those conditions and marks the approximation as unreliable when they fail, instead of quietly printing a number.
t → normal as the degrees of freedom grow. The two-tailed 5% critical value is 2.5706 at df = 5, 2.2281 at df = 10, 2.0423 at df = 30, 1.9840 at df = 100 and 1.9623 at df = 1,000, converging on the normal 1.9599640. That is the whole reason "n ≥ 30" appears in textbooks: past that point the two answers differ in the second decimal.
And two structural relationships worth remembering: a chi-square with k degrees of freedom is the sum of k squared standard normals (so chi-square with df = 1 has the 5% critical value 3.8415 = 1.9599640²), and the exponential is the waiting time between Poisson events — the same process seen from the time axis instead of the count axis.
Presets, and why they differ by country
Thirteen presets load a real scenario in one click: IQ (μ = 100, σ = 15), adult male height, SAT scores (1050 / 200), Korean CSAT standard scores (100 / 20), the standard normal, 20 coin flips, a 1% defect rate over 1,000 units, the continuity-correction demo, a call centre at λ = 5, a wait time at λ = 0.5, a t test at df = 10, a chi-square goodness-of-fit at df = 10 and a fair die.
Height is one concept with different national numbers, so the button shows yours: 172.5 / 5.6 cm in Korea (Size Korea), 175.3 / 7.4 cm in the United States (NHANES 2015–2018), 170.8 / 5.8 cm in Japan (国民健康・栄養調査) and 171 / 7 cm as the world average (NCD-RisC 2020) everywhere else. Exams are different institutions that both exist, so both buttons stay and only the order changes — the local exam first.
The country decides which buttons appear, never what a click writes. A preset only sets plain parameter text (μ, σ, x), so a link shared from Seoul reproduces identically in Chicago — the shared URL carries numbers, not a locale.
Taking the same calculation to scipy, R or Excel
Every result comes with the equivalent one-liner. The normal CDF is norm.cdf(x, loc=μ, scale=σ) in scipy, pnorm(x, μ, σ) in R and =NORM.DIST(x, μ, σ, TRUE) in Excel; the inverse is norm.ppf / qnorm / =NORM.INV. The pattern holds across the family: p* / q* in R, .cdf / .ppf in scipy.
Two translation traps are worth naming. Excel’s cumulative flag must be TRUE for a probability and FALSE for a density — the same function name does both, and forgetting it produces a plausible-looking wrong number. And scipy’s exponential takes scale = 1/λ, not λ, so expon.cdf(3, scale=2) is the λ = 0.5 case; passing 0.5 there silently models a four-times-faster process.
Under the hood this page uses the Hart/West rational approximation for the normal CDF, Acklam plus a Halley step for its inverse, and continued-fraction expansions of the incomplete gamma and beta functions for the rest — the same algorithms the libraries use, which is why the snippets and the screen agree to the decimals shown.