Top 10 Types of Distribution in Statistics With Formulas

In schools, we study statistics as a part of mathematics, but it becomes a separate subject when we take it at an advanced level. Statistics has its own importance, and many students make their career in it. It is essential in almost every field, such as banking, finance, medicine, weather forecasting, etc. But several students consider it a complex subject. Statistics include several topics and terms that are difficult to understand. Here, we are going to discuss one of the statistical topics, which is types of distribution in Statistics.

A distribution in statistics is a parameterized mathematical function. There are several kinds of distribution in statistics, and each book has listed them with their properties. In this blog, you will find out all the useful information on different types of distribution in statistics with examples. But first, we learn what distribution is in statistics.

What is distribution in statistics?

Distribution in statistics is a function that represents the possible values for a variable and how frequently they happen. 

Let’s take an example of a dice. It has six surfaces that are numbered from 1 to 6. We roll the dice. What is the probability of obtaining 1?

It is one out of six, thus one-sixth, right? Then, what is the probability of obtaining 2? Again – one-sixth. Okay!

This is same for 3, 4, 5, and 6.

Now, what is the probability of obtaining a 7? Obviously, it is impossible to get a 7 while rolling a die.

Consequently, the probability is 0.

Different types of distribution in statistics With Examples

Normal Distribution

On other occasions, it is referred to as the bell curve since it produces a shape of a bell when graphed. It occurs naturally in several cases; for example, the normal distribution can be seen in tests such as GRE and SAT. Furthermore, there are several groups that follow the normal distribution pattern. Because of this, it is widely used in statistics, business, and government bodies like the FDA: 

• Measurement errors.
• Points on a test.
• Salaries.
• Height of people.
• Blood pressure.
• IQ scores.

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Properties of a normal distribution

• The curve remains symmetric at the center.
• The area under the curve is 1.
• The mean, median, and mode are always equal.
• Exactly half value is on the left of the center and the other on the right.

T- Distribution

It is one of the most important distributions in statistics. It is also known as Student’s t- distribution, which is the probability distribution. That is used to estimate the parameters of the population when the given sample size is small. 

Properties of t-distribution

• Like normal distribution, the student distribution has bell-shaped and symmetric with zero mean.
• The range of student distribution from – ∞ to ∞ (infinity).
• The t distribution’s shape changes with the degree of freedom.
• The variance is always more than one, and it can be represented when the degree of freedom V>=3 and given: Var (t) = [v/v -2].
• It is not packed that much at the center but higher at trials; therefore, its shape is like platykurtic.
• The dispersion of t distribution is much more than the normal distribution.
• The sample “n” is regarded as having a normal distribution as its size grows. In this case, the sample size is bigger than n >= 30.

Uniform distribution

The basic form of a continuous distribution is known as uniform distribution. It has the constant probability that it forms a rectangular distribution. And it implies that each value has the same length of distribution. Which has an equal probability of occurrence. In contrast, this function belongs to the type of maximum entropy probability distributions.

Characteristics of uniform distribution

• The density function combines to unity.
• Each of its input functions has equal weightage.
• The uniform function’s mean is given by:

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• The variance of the uniform distribution is given by:

Bernoulli distribution

A Bernoulli distribution is a kind of discrete probability distribution- a random trial that has two results. There is a special case that has the value n = 1, for example, a single coin toss.

Characteristics of Bernoulli distribution

• The number of trials that have to be performed under a single experiment must be predefined.
• Each trial must have two outcomes that are success or failure.
• The success’ probability in each experiment should be the same.
• The experiment should be independent of each other, which means the outcome of one trial is not affected by the other’s trial outcome.

Properties 

The expected value of the randomly selected variable is given by E(x) = p and can be derived:

E(x) = 0*(1-p) + 1 * p = p

The variance of the Bernoulli variable is given by p*(1-p) and is given as:

Var(X) = p – p2 = p*(1-p)

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Binomial distribution

A probability distribution concludes the value that takes one of two independent values under a set of assumptions or parameters. Besides, the binomial distribution’s assumptions must have a single result with the same probability of success. And that trail must be independent of each other.

Properties of a binomial distribution

• When an experiment has independent trails, each of them has two results: success and failure.

• Bi-parametric distribution is another name for the binomial distribution. Because it is categorised using the n and p parameters.This has a mean value of μ = np.
• The variance of the binomial distribution is σ² = npq.
• It is possible to state that the variance must be less than the mean value since the values of p and q are always less than or equal to 1. npq < np

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Poisson distribution

It is a tool that is used to predict a certain probability of the event when you know the value of a certain event. The Poisson distribution provides us the probability of an applied number of events that happen in a fixed period of time.

Properties of Poisson distribution

• The expected value and variance of the random variable are equivalents to λ.
• The absolute deviation is associated with the mean is given by:

• The coefficient of variance is, whereas the index dispersion is.
• The Poisson distribution’s expected value can be broken down by taking the product of exposure and intensity.
• The mean of the Poisson distribution is given by “m”.

Exponential distribution

It is also known as a negative exponential distribution that represents the time between the trails in a Poisson process. The relationship between the exponential distribution and Poisson distribution. 

Some of the formulas of it

• An exponential random variable’s expected value is given by:

• An exponential random variable’s variance value is given by:

• The exponential random variable’s moment generating function is given by:

• An exponential random variable’s characteristic function is given by:

Beta distribution

It is the family of continuous probability distributions that are set under the interval [0,1], which is expressed by alpha and beta. Additionally, this concept is applied to models with unclear success probabilities for random experiments. It also offers a powerful tool with the basic statistics that can compute the confidence level of completion time.

Properties of beta distribution

There are a few properties that can satisfy these distributions are:

The terms to measure the central tendency are:

• Mean
• Harmonic Mean
• Mode
• Median
• Geometric Mean

The terms to measure the statistics dispersion are:

• Geometric variance and covariance
• Variance
• Mean absolute deviation around the mean
• Mean absolute difference

Beta-binomial distribution

It is the simplest Bayesian model that is widely used in intelligence testing, epidemiology, and marketing. A distribution is said to be beta-binomial if the probability of success is p, and the shape of the beat binomial parameter is α > 0 and β > 0. 

The parametric shape can be defined as the success probability:

• A distribution can approach a binomial distribution for the larger value of α and β.
• The value of discrete uniform distribution equals the distribution from 0 to n, if the value of both α and β is equal to 1.
• For the value of n = 1, the beta-binomial distribution is the same value as that of Bernoulli distribution.

The major difference between a beta-distribution and binomial distribution is that p is always fixed for a set of trials in a binomial distribution, whereas the p for beta-binomial is not fixed, and it changes from trial to trial.

Log-normal distribution

If the log to the power is normally distributed, then the variable is taken as lognormally distributed. Or we can say that ln(x) is normally distributed and that the variable x is assumed to have a log-normal distribution.

Properties of the log-normal distribution

• An average person can learn useful information about what to expect from a repeated trial number by examining the expected value or mean of distribution.
• The median of a log-normal distribution is another consideration of central tendency, and it is useful for outliers that help the means to lead.
• The distribution’s mode is a value with the highest probability of occurring.
• How to spread out the information can be measured by the variance. The square root of the variance and the standard deviation are useful as these have the same unit of the data.

These values are much easier to measure for a continuous probability distribution. This blog has discussed many statistical distribution types along with examples and characteristics. 

Conclusion

This blog has discussed many statistical distribution types along with examples and characteristics. Besides, this can help the students to understand the complicated terms of statistics. Therefore, you need to read this blog carefully so that you can understand each term easily. The distribution in statistics is also necessary to write an assignment during their academic studies.
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