Distribution explorer

The basic five and when each applies

• Normal: continuous measurements from a stable process, such as diameters, weights and fill volumes. Symmetric, described by its mean and standard deviation. About 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
• Binomial: the number of defective items in a sample of n, when each item has the same probability p of being defective, independently. Pass or fail data.
• Poisson: the number of defects or events in a fixed amount of product, time or area, when they occur independently at an average rate λ. Defects per unit, calls per hour. Its mean and variance are both λ.
• Uniform: every value between two limits equally likely.
• Exponential: the time between independent random events, such as time between failures at a constant failure rate. Its mean and standard deviation are equal.

Defectives or defects?

The question that decides between binomial and Poisson. A defective is a unit that fails; each unit is counted once whatever is wrong with it, so binomial. A defect is a single nonconformity; one unit can have several, so Poisson.

Reading the probabilities

Enter a value and the tool shades the probability of falling at or below it. Enter an upper value as well and it shades the probability between the two. For a normal distribution it also gives the z-score: how many standard deviations the value is from the mean, the number used with a standard normal table.

For whole-number distributions, "at or below 3" includes 3 and "at or above 3" includes 3; the two do not add to 1. The tool shows both.

More distributions for life data, small lots and sampling

• Weibull (shape β, scale η): time or cycles to failure. β below 1 means a falling failure rate (early-life failures), β = 1 is the exponential, β above 1 a rising rate (wear-out). By life η, 63.2% have failed. P(X > x) is the reliability at x.
• Lognormal (μ and σ of ln X): a positive, right-skewed quantity whose logarithm is normal, such as repair times. Its mean is e^{μ + σ²/2}, not e^μ.
• Hypergeometric (lot size N, D defectives in the lot, sample n): defectives in a sample drawn without replacement. When the sample is under about 10% of the lot, the binomial with p = D/N is close enough.
• Multinomial: more than two outcomes per trial (grade A, grade B, scrap). Enter each category's probability and count; the tool gives the probability of exactly those counts, n!/(x₁!…x_k!) × p₁^x₁…p_k^x_k.

The sampling distributions: t, chi-square and F

These three describe statistics rather than individual items. Student's t with ν degrees of freedom is the mean standardized with s instead of σ; it has heavier tails than the normal and approaches it as ν grows. Chi-square with k degrees of freedom is (n − 1)s²/σ² for normal data, and the statistic of the goodness-of-fit and contingency-table tests. F with ν₁ and ν₂ degrees of freedom is the ratio of two independent variance estimates, used to compare variances and in ANOVA. Enter a lower-tail probability in the percentile box to get a critical value: 0.975 with t on 10 df gives 2.228, the value in a t table for a two-sided test at α = 0.05.

On the exam

CSSYB IV.E.1 covers the basic distribution types. The CQE asks you to define and distinguish the continuous distributions (normal, uniform, exponential, lognormal, Weibull, Student's t and F; VI.C.1) and the discrete ones (binomial, Poisson, hypergeometric and multinomial; VI.C.2). The CSSBB asks you to use the normal, Poisson, binomial, chi-square, t and F distributions (V.E.2) and to identify the hypergeometric, exponential, lognormal and Weibull (V.E.3). CQPA III.A.2 lists normal, binomial, Poisson and Weibull. Expect to choose the right distribution for a described situation, look up a critical value, and say how a skewed or bimodal shape changes how the data should be read (a bimodal histogram often means two processes or two populations are mixed).

Common questions

Which probability distribution should I use?

Measurements that vary around a center: normal. Defectives in a sample: binomial, or hypergeometric when you sample a large share of a small lot without replacement. Defects or events counted per unit or per hour: Poisson. Time to failure: exponential or Weibull. The t, chi-square and F distributions are for tests and confidence intervals.

What is the difference between binomial and Poisson?

The binomial counts defective units out of n, each either good or bad, with a fixed chance p. The Poisson counts defects or events where there is no fixed maximum, at an average rate λ. When n is large and p is small, the binomial is close to a Poisson with λ = np.

What does the Weibull shape parameter tell you?

A shape β below 1 means a falling failure rate (early-life failures), β = 1 means a constant rate (the exponential distribution), and β above 1 means a rising rate (wear-out).

Is there a simpler probability calculator?

Yes. The probability calculator handles combinations, permutations, “at least one” and series or parallel reliability.

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