Types of Distributions in Statistics: Guide

Types of distributions in statistics are grouped first by what the variable is: a discrete distribution assigns probabilities to countable values (0, 1, 2, …) and is described by a probability mass function (pmf), while a continuous distribution spreads probability over an interval and is described by a probability density function (pdf). The ones most courses teach first are the binomial and Poisson (discrete) and the normal, uniform and exponential (continuous). This page tells you which one fits which kind of data, gives the formulas with every symbol explained, and works one binomial, one Poisson and one normal example through to the arithmetic.

Types of distributions in statistics: discrete vs continuous

Statlect, a probability and statistics reference, divides univariate distributions into discrete (Bernoulli, binomial, Poisson, geometric) and continuous (uniform, exponential, normal, Student’s t, chi-square, gamma, beta, F, log-normal), and lists multivariate families separately (multinomial, multivariate normal, Dirichlet, Wishart). The NIST/SEMATECH e-Handbook of Statistical Methods catalogues a longer set in its Gallery of Distributions, including Cauchy, Weibull, lognormal and extreme value distributions alongside binomial and Poisson.

There is no official count of “the 4 types of distributions”. When a course says four, it usually means the commonly taught set: normal, binomial, Poisson and uniform. Treat the number as a teaching convention, not a classification.

Which distribution fits which data: a decision table

Start from what you counted or measured. Each row below follows the use statements in the NIST Gallery of Distributions and the descriptions on Statlect.

What you have	Distribution	Type	Source description
Number of successes in a fixed number of independent two-outcome trials	Binomial	Discrete	NIST: used when there are exactly two mutually exclusive outcomes of a trial
Number of events in a given time interval	Poisson	Discrete	NIST: used to model the number of events occurring within a given time interval
Number of trials until the first success	Geometric	Discrete	Statlect
Waiting time or time to failure at a constant failure rate	Exponential	Continuous	NIST: primarily used in reliability applications; constant failure rate
Equal probability on equal-length intervals	Uniform	Continuous	Statlect
Symmetric, bell-shaped measurements described by a mean and spread	Normal	Continuous	NIST: the most important distribution in statistics

If you already have data and need to decide which shape it follows, see how to identify the distribution of your data, which covers plots and tests.

Formulas with every symbol explained

Binomial distribution

P(x; p, n) = C(n, x) · p^x · (1 − p)^n−x, for x = 0, 1, …, n

n: the fixed number of trials
x: the number of successes
p: the probability of success on each trial
C(n, x): the number of ways to choose x successes from n trials, n! / (x!(n − x)!)
Mean = np; standard deviation = √(np(1 − p)) (NIST)

Poisson distribution

p(x; λ) = e^−λ · λ^x / x!, for x = 0, 1, 2, …

λ: the average number of events per interval (must be positive)
x: the observed number of events in one interval
e: Euler’s number, about 2.71828
Mean = λ; standard deviation = √λ (NIST)

Normal distribution

f(x) = exp(−(x − μ)² / (2σ²)) / (σ√(2π))

μ: the location parameter, which is the mean
σ: the scale parameter, which is the standard deviation (variance = σ²)

Exponential distribution

In the NIST form, f(x) = (1/β) · e^{−(x − μ)/β} for x ≥ μ and β > 0, where μ is a location (shift) parameter and β is the scale parameter. With μ = 0 the mean equals β and so does the standard deviation; the rate is 1/β. Software often asks for the rate rather than β, a frequent source of errors.

Uniform distribution

Every interval of the same length within the allowed range has the same probability. Its two parameters are the lower and upper limits of that range.

Worked examples with the arithmetic

Binomial: exactly 3 heads in 10 fair coin flips

Here n = 10, x = 3, p = 0.5. C(10, 3) = 10! / (3! · 7!) = 120. Then P(X = 3) = 120 × 0.5³ × 0.5⁷ = 120 × 0.5¹⁰ = 120 / 1024 = 0.1172. The mean is np = 5 and the standard deviation is √(10 × 0.5 × 0.5) = √2.5 = 1.58. The cumulative probability of 3 or fewer heads is 0.1719.

Poisson: exactly 2 events when the average is 4 per interval

Here λ = 4, x = 2. e⁻⁴ = 0.018316. P(X = 2) = 0.018316 × 4² / 2! = 0.018316 × 16 / 2 = 0.1465. The mean is 4 and the standard deviation is √4 = 2. The probability of 2 or fewer events is 0.2381.

Normal: a score of 115 when μ = 100 and σ = 15

The standardized value is z = (115 − 100) / 15 = 1.00. The cumulative probability P(X ≤ 115) = Φ(1.00) = 0.8413, so about 84.1% of values fall at or below 115. Within one standard deviation of the mean (85 to 115) lies 0.6827 of the distribution, and within two (70 to 130) lies 0.9545.

Interpretation: reading the result

Output	What it tells you
pmf value (binomial, Poisson)	Probability of exactly that count, for example 0.1172 for 3 heads in 10 flips
Cumulative probability	Probability of that value or lower, for example Φ(1.00) = 0.8413
pdf value (continuous)	A density, not a probability. Probabilities for continuous variables come from areas under the curve, so any single exact value has probability zero
Mean and standard deviation	Centre and spread of the model, computed from the parameters (np and √(np(1 − p)); λ and √λ; μ and σ)

Assumptions and common mistakes

Binomial needs a fixed n and the same p on every trial. If the number of trials is not fixed, or p changes, the binomial does not describe the count.
Poisson uses one parameter for both mean and variance. Both equal λ, so count data whose spread is far larger than its mean do not fit; see Poisson regression analysis for the modelling consequences.
Reading a pdf height as a probability. Only areas are probabilities.
Passing the wrong exponential parameter. Check whether your software expects the rate (1/β) or the scale (β).
Assuming normality because the data are numbers. Test it; if it fails, consider nonparametric statistics or compare parametric and non-parametric tests.
Heavy-tailed data. Some data are better described by families such as the power law, the Rayleigh or the generalized Pareto distribution than by the normal.

How to report a distribution (APA style)

State the model, its parameters and the probability you computed, with statistical symbols in italics and probabilities without a leading zero:

The number of heads was modelled as binomial (n = 10, p = .50); P(X = 3) = .117.
Weekly incident counts were modelled as Poisson (λ = 4); P(X ≤ 2) = .238.
Scores were assumed normally distributed (M = 100, SD = 15); P(X ≤ 115) = .841.

How to calculate it in Excel, R or Python

Excel

=BINOM.DIST(3, 10, 0.5, FALSE)   → 0.1172
=POISSON.DIST(2, 4, FALSE)        → 0.1465
=NORM.DIST(115, 100, 15, TRUE)    → 0.8413

R

dbinom(3, size = 10, prob = 0.5)  # 0.1171875
dpois(2, lambda = 4)              # 0.1465251
pnorm(115, mean = 100, sd = 15)   # 0.8413447

Python (SciPy)

from scipy import stats
stats.binom.pmf(3, 10, 0.5)       # 0.1171875
stats.poisson.pmf(2, 4)           # 0.1465251
stats.norm.cdf(115, loc=100, scale=15)  # 0.8413447

In the Excel functions, the last argument (FALSE or TRUE) switches between the pmf and the cumulative probability.

FAQ

What are the main types of distributions with examples?

Binomial (heads in 10 flips), Poisson (arrivals per hour), geometric (flips until the first head), normal (measurement error), uniform (a random number between two limits) and exponential (time to failure at a constant failure rate).

What are the types of distributions in probability?

Probability distributions are discrete (pmf), continuous (pdf) or multivariate (more than one variable at once, such as the multinomial and multivariate normal; see multivariate normal distribution).

What are the types of distributions in research?

Researchers choose by the measurement: counts of events lead to Poisson or binomial models, continuous measurements often to the normal, and waiting times to the exponential. The choice determines which tests and models are valid; the t distribution versus the normal is a common example.

Is there a formula for each type?

Yes. Every distribution has a pmf or pdf; the ones for binomial, Poisson, normal and exponential are given above.

Sources

NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6 Gallery of Distributions: itl.nist.gov/div898/handbook/eda/section3/eda366.htm (read 2026-11-07), with the normal, exponential, binomial and Poisson sub-pages.
Statlect, Probability distributions: statlect.com/probability-distributions (read 2026-11-07).

Types of Distributions in Statistics: Discrete, Continuous and Which to Use