Uniform Distribution

What is the Uniform Distribution?

Uniform distribution is defined as the type of probability distribution where all outcomes have equal chances or are equally likely to happen and can be bifurcated into a continuous and discrete probability distribution. These are normally plotted as straight horizontal lines.

Uniform Distribution Formula

The variable can be inferred to be uniformly distributed if the density function is attributed to as displayed below: –

F(x) = 1 / (b – a)


-∞ < a <= x <= b< ∞


  • a and b are represented as parameters.
  • The symbol represents the minimum value.
  • The symbol b represents a maximum value.

The probability density function is termed as the function whose value for a given sample under a sample space has an equal likelihood of happening for any random variable. For uniform distribution function, measures of central tendenciesCentral TendenciesCentral Tendency is a statistical measure that displays the centre point of the entire Data Distribution & you can find it using 3 different measures, i.e., Mean, Median, & Mode.read more are expressed as displayed below: –

Mean = (a + b) / 2
σ = √ [(b – a) ^ 2/ 12]

Therefore, for parameters a and b, the value of any random variable x can happen at equal probability.


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For eg:
Source: Uniform Distribution (wallstreetmojo.com)

Explanation of the Uniform distribution Formula

Follow the steps to get uniform distribution.

  1. Determine Max and Min value

    Firstly, determine the maximum and minimum value.

  2. Determine the length of the interval

    Next, determine the length of the interval by deducting the minimum value from the maximum value.

  3. Determine the probability density function

    Next, determine the probability density function by dividing the unity from the interval length.

  4. Determine the mean

    Next, for the probability distributionProbability DistributionProbability distribution is the calculation that shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required. It is a mathematical function that gives results as per the possible events.read more function, determine the mean of the distribution by adding the maximum and minimum value followed by division of resulting value from two.

  5. Determine the variance of the uniform distribution

    Next, determine the variance of the uniform distribution by deducting minimum value from the maximum value further raised to the power of two and followed by the division of resulting value with twelve.

  6. Determine the standard deviation

    Next, determine the standard deviationStandard DeviationStandard deviation (SD) is a popular statistical tool represented by the Greek letter 'σ' to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data's reliability.read more of the distribution by taking the square root of the variance.

Examples of Uniform Distribution Formula (with Excel Template)

You can download this Uniform Distribution Formula Excel Template here – Uniform Distribution Formula Excel Template

Example #1

Let us take the example of an employee of company ABC. He normally takes up the services of the cab or taxi for the purpose of traveling from home and office. The duration of the wait time of the cab from the nearest pickup point ranges from zero and fifteen minutes.

Help the employee determine the probability that he would have to wait for approximately less than 8 minutes. Additionally, determine the mean and standard deviation with respect to the wait time. Determine the probability density function as displayed below wherein for a variable X; the following steps should be performed:


Use the given data for the calculation of uniform distribution.

Example 1

Calculation of the probability of the employee waiting for less than 8 minutes.

Uniform Distribution Formula Example 1.1
  • = 1 / (15 – 0)
Uniform Distribution Formula Example 1.2
  • F(x) = 0.067
Example 1.3
  • P (x < k) = base x height
  • P (x <8) = (8) x 0.067
  • P (x <8) = 0.533

Therefore, for a probability density function of 0.067, the probability that the waiting time for the individual would be less than 8 minutes is 0.533.

Calculation of mean of the distribution –

Uniform Distribution Formula Example 1.4
  • = (15 + 0) /2

Mean will be –

Example 1.5
  • Mean = 7.5 minutes.

Calculation of standard deviation of the distribution –

Uniform Distribution Formula Example 1.6
  • σ = √ [(b – a) ^ 2/ 12]
  • = √ [(15 – 0) ^ 2/ 12]
  • = √ [(15) ^ 2/ 12]
  • = √ [225 / 12]
  • = √ 18.75

Standard Deviation will be –

Example 1.7
  • σ = 4.33

Therefore, the distribution shows a mean of 7.5 minutes with a standard deviation of 4.3 minutes.

Example #2

Let us take the example of an individual that spends between 5 minutes to 15 minutes eating his lunch. For the situation, determine the mean and standard deviation.


Use the given data for the calculation of uniform distribution.

Uniform Distribution Formula Example 2

Calculation of mean of the distribution –

Example 2.1
  • =(15+0)/2

Mean will be –

Uniform Distribution Formula Example 2.2
  • Mean = 10 minutes

Calculation of standard deviation of the uniform distribution –

Uniform Distribution Formula Example 2.3
  • = √ [(15 – 5) ^ 2/ 12]
  • = √ [(10) ^ 2/ 12]
  • = √ [100 / 12]
  • = √ 8.33

Standard Deviation will be –

Example 2.4
  • σ = 2.887

Therefore, the distribution shows a mean of 10 minutes with a standard deviation of 2.887 minutes.

Example #3

Let us take the example of economics. Normally refill, and demand does not obey normal distribution. This, in turn, pushes in the usage of computational models wherein, under such a scenario, uniform distribution model proves to be extremely useful.

The normal distributionNormal DistributionNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more and other statistical models cannot be applied to limited or no availability of data. For a new product, there is the availability of limited data corresponding to the demands of the products. If this distribution model is applied under such a scenario, for lead time relative to demand of the new product, it would be far easier to determine the range that would have an equal probability of happening between the two values.

From the lead time itself and uniform distribution, more attributes can be computed, such as shortage per production cycle and cycle service level.

Relevance and Use

Uniform distribution belongs to the symmetric probability distribution. For chosen parameters or bounds, any event or experiment may have an arbitrary outcome. The parameters a and b are minimum and maximum bounds. Such intervals can be either an open interval or a closed interval.

The length of the interval is determined as the difference of maximum and minimum bounds. Determination of probabilities under uniform distribution is easy to assess as this is the most simple form. It forms the basis for hypothesis testingHypothesis TestingHypothesis Testing is the statistical tool that helps measure the probability of the correctness of the hypothesis result derived after performing the hypothesis on the sample data. It confirms whether the primary hypothesis results derived were correct.read more, cases of sampling, and is majorly used in finance.

The uniform distribution method came into the existence of the games of dices. It is basically derived from equiprobability. The game of dice always has a discrete sample space.

It is used under several experiments and computer run simulations. Due to its simpler complexity, it is easily incorporated as a computer program, which in turn is utilized in the generation of variable, which carries the equal likelihood of happening following the probability density function.

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This article has been a guide to Uniform Distribution and its definition. Here we discuss the formula for calculation of uniform distribution (probability distribution, Mean and standard deviation)  along with examples and a downloadable excel template. You can learn more from the following articles –

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