# 9. Probability Distributions

Here you can find the most common statistical probability distributions, which you can calculate online. In addition you can see calculated tables of the probability distributions.

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### The Normal Distribution

The normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. The normal distribution is useful because of the central limit theorem, which states that averages of random variables independently drawn from independent distributions become normally distributed when the number of values is sufficiently large.

The simplest case of a normal distribution is known as the standard normal distribution, which has a mean of zero and a standard deviation of one. The standard normal distribution is used to estimate probabilities.

**Normal distribution: Calculate online** (calculate p from Z). Here is a **table of the normal distribution** (one-sided probabilities).

**Inverse normal distribution: Calculate online** (calculate Z from p).

### Student’s t-distribution

In probability and statistics, Student’s t-distribution (or simply the t-distribution) is a continuous probability distribution that arises in estimating the mean of a normally distributed variable in the population, when the sample size is small and the standard deviation in the populations is unknown. t-distributions describe samples drawn from a full population; accordingly, the t-distribution for each sample size is different, and the larger the sample, the closer the t-distribution approaches the normal distribution.

**Student’s t-distribution: Calculate online** (calculate p from t). Here is a **table of Student’s t-distibution**.

**Inverse Student’s t-distribution: Calculate online** (calculate t from p).

### The F-distribution

In probability theory and statistics, the F-distribution is a continuous probability distribution that is used most notably in the analysis of variance, e.g. in a F-test. Any F-test can be regarded as a comparison of two variances. Most commonly it is used to assess the ratio of two sample variances. The F-test in one-way analysis of variance is used to assess whether the values of a quantitative variable within three or more groups differ from each other.

**F-distribution: Calculate online** (calculate p from F). Here is a **table of the F-distribution**.

**Inverse F-distribution: Calculate online** (calculate F from p).

### The Chi-square (χ²) distribution

In probability theory and statistics, the chi-square (χ²) distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squared distribution is used primarily in hypothesis testing e.g. in contingency tables, in testing goodness of fit of observed data to hypothetical distributions and in the log-rank test in survival analysis.

**Chi-square distribution: Calculate online** (calculate p from χ²). Here is a **table of the Chi-square distribution**.

**Inverse Chi-square distribution: Calculate online** (calculate χ² from p).

### The Binomial Distribution

The binomial distribution is a discrete probability distribution of the number of successes r in a sequence of n independent yes/no experiments, each of which yields success with probability p. The binomial distribution is the basis for the popular binomial significance test. The binomial distribution is frequently used to model the number of successes r in a sample of size n drawn with replacement from a population of size N.

**Binomial Distribution: Calculate online**. Here is a **table of the Binomial distribution**.

### The Poisson Distribution

The Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

**Poisson Distribution: Calculate online**. Here is a **table of the Poisson distribution**.