Uniform Distribution

Basic Concepts

Asking for a random set of say 100 numbers between 1 and 10, is equivalent to creating a sample from a continuous uniform distribution, where α = 1 and β = 10 according to the following definition.

Definition 1: The continuous uniform distribution has the probability density function (pdf)

Uniform distribution pdf

where α and β are any parameters with α < β.

The corresponding cumulative distribution function (cdf) is

Uniform distribution function

The inverse cumulative distribution function is

 I(p) = α + p(β − α)

Properties

Key statistical properties are shown in Figure 1.

Key properties

Figure 1 – Statistical properties of the uniform distribution

Observation: A continuous uniform distribution in the interval (0, 1) can be expressed as a beta distribution with parameters α = 1 and β = 1.

Worksheet Functions

Real Statistics Functions: Excel doesn’t provide any functions for the uniform distribution. Instead, you can use the following functions provided by the Real Statistics Resource Pack.

UNIFORM_DIST(x, α, β, cum) = the pdf of the continuous uniform distribution f(x) at x when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

UNIFORM_INV(p, α, β) = x such that UNIFORM_DIST(x, α, β, TRUE) = p. Thus UNIFORM_INV is the inverse of the cumulative uniform distribution

Examples

Example 1: A bus arrives regularly every 20 minutes throughout the day. What is the probability that you will have to wait more than 15 minutes assuming that you arrive at a random time?

Let x = the time that you arrive in the interval a = 0 to b = 20. The random variable has a uniform distribution. Thus, the probability that you will wait at most 15 minutes is F(15) = (–a)/(ba) = (15–0)/(20–0) = .75. This means that the probability that you will need to wait more than 15 minutes is 1 – .75 = .25.

Example 2: A random sample of size 40 is taken from a population with a uniform distribution as shown in range A3:E10 of Figure 2. What is the probability that any random sample element will be less than 5?

Uniform distribution example 2

Figure 2 – Uniform distribution example

This is similar to Example 1 except that we don’t know the values of the endpoints a and b of the uniform distribution. We begin by calculating the sample mean and standard deviation (cells H3 and H4 of Figure 2). We assume that these are reasonable estimates of the population mean and standard deviation.

From Figure 1 we see that the 12.425 = 2μ = a + b and 9.809 = √12 σ = b – a. Solving these simultaneous equations for a and b, we get a = 1.308 and b = 11.117. Let x be the value of an element randomly drawn from the distribution. The probability that x < 5 can be calculated by the formula =UNIFORM_DIST(5, a, b,TRUE), which is 37.6% as shown in cell H12 of Figure 2.

Related Topics

There is also a discrete version of the uniform distribution. Related to the uniform distributions are order statistics. Click on any of the following links for more information:

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Wikipedia (2012) Continuous uniform distribution
https://en.wikipedia.org/wiki/Continuous_uniform_distribution

40 thoughts on “Uniform Distribution”

  1. Can you please mention the formula for it?

    UNIFORM_INV(p, α, β) = x such that UNIFORM_DIST(x, α, β, TRUE) = p. Thus UNIFORM_INV is the inverse of the cumulative uniform distribution

    Yours,
    Matin

    Reply
  2. Excel does not have the function for uniform distribution. I wonder why given that it should not be too difficult to put this function when it has all the other ones with more complex cases!

    Reply
  3. Hey

    If i make 200 observations and 2 columns (size = 2) of a uniform distribution between 4 and 7 and i wanted to get the mean, do i use the function average in excel or do i use:
    Mean = (α + β) / 2 ?
    regards
    Heba

    Reply
    • i mean if i wanted to get a 200 observations means so each two values from column A and B i get their mean. so i can graph the means

      Regards
      Heba

      Reply
    • Hello Heba,
      If you want to get the sample mean then use the AVERAGE function on the 200 observations.
      If you want to get the population mean when the alpha and beta parameters are known then use (α + β) / 2.
      Charles

      Reply
  4. Is it possible to sample data instances using a distribution different from the uniform distribution? If so, give an example of a probability distribution of the data instances that is different from uniform (i.e., equal probability).

    Reply
    • Hello Praveen,
      Yes, you can. E.g. to create a 10 element sample from the standard normal distribution, place the formula =NORM.S.INV(RAND()) in cell A1, highlight the range A1:A10 and press Ctrl-D.
      Charles

      Reply
  5. Hello,

    My Excel doesn’t recognize the UNIFORM_INV after installing and checking it on in Excel 2016.. What can I do? (VER() doesn’t work as well)

    Reply
  6. I’ve downloaded the Resource Pack and ticked the Realstats add-in but excel isn’t recognising UNIFORM_DIST as a formula. Any ideas/tips on what I may be doing wrong?

    Reply
  7. Charles,

    First of all thank you for porting your add-in to excel 2016. Now I have opportunity to use it. I’ve recently installed it and currently trying to learn it’s functionality . I’ve found problem with UNIFORM_DIST function.

    I make a table with two columns. First column is x values. Second column: UNIFORM_DIST(first_column_value_of_same_row,1,3,0). Third row: UNIFORM_DIST(first_column_value_of_same_row,1,3,1).
    0 0 0
    1 0.5 0
    2 0.5 0.5
    3 0.5 1
    4 0 1
    Problem: probabilities in second column don’t sum up to 1. Also the third row is made considering (α, β] interval (so α is not included – I’m not sure if that’s intentional)

    Reply
    • Artem,

      The correct form of the formula is UNIFORM_DIST(x,0,3,cum). Thus the table should look like

      x pdf cdf
      0 0.333333333 0
      1 0.333333333 0.333333333
      2 0.333333333 0.666666667
      3 0.333333333 1

      Charles

      Reply
  8. Resp. Sir
    I want to generate U(0,1),with 50 size.where p1=2*p2,p2=p3 and p4=1-(p1+p2+p3).
    Use the monte carlo method to optimize n1,n2,n3 and n4.Give early as possible.

    Reply
    • Gopal,
      You have repeated this problem to me several times now, but I am sorry to say that I still don’t understand the question well enough to give you an answer.
      Charles

      Reply
  9. Sir,I want to generate U(0,1),for sample size 50 with between 0.72 to 0.92 where p2=p3,p1=2*p2 and p4=1-(p1+p2+p3).Optimize sample size with n1,n2,n3 and n4.Is it possible in Excel 2007.Give me help

    Reply
    • You can certainly generate a sample of size 50 that follows a uniform distribution U(0,1), namely by using the RAND() formula, but I don’t understand the other constraints that you have listed.
      Charles

      Reply
      • Resp.Sir,
        I have applied Multinomial Distribution.I have to generate the probabilities p1+p2+p3<=0.8 and p1=2p2=2p3.Generate 50 random nos using U(0,1).Obtain p1,p2,p3.Then using Monte-Carlo Method and estimate n1,n2,n3.Is it possible in Excel 2007

        Reply
        • Resp.Sir,
          I have applied Multinomial Distribution.I have to generate the probabilities p1+p2+p3<=0.8 and p1=2p2=2p3.Generate 50 random nos using U(0,1).Obtain p1,p2,p3.Then using Monte-Carlo Method and estimate n1,n2,n3.Is it possible in Excel 2007
          Reply

          Reply
    • Avi,

      The following approach can be used to generate a graph of any distribution with probability density function f(x) in Excel in say the range a to b. For simplicity I will assume that a = 2 and b = 5. I will also assume that you want the granularity of the graph to be in units of size .1 (you can choose whatever granularity you like).

      Place the value of 2 (i.e. a) in cell A1. Next place the formula =A1+.1 in cell A2 (here .1 is the granularity). Next highlight the range A2:30 and press Ctrl-D. This will place the numbers 2, 2.1, 2.2, …, 4.9, 5.0 in the range A1:A31 (here 5 is b). Next, place the formula for f(x) in cell B1 where x is replaced by the cell reference A1 — for your problem you use UNIFORM_DIST(A1,2,5,FALSE). Now highlight the range B1:B31 and press Ctrl-D. Finally, highlight the range A1:B31 and create a scatter plot by choosing Insert > Charts|Scatter.

      The resulting graph will be the horizontal line y = 1/3 between 2 and 5. In general the graph will be the horizontal line y = 1/(b-a) between x = a and x = b.

      Charles

      Reply
  10. Dear Charles,

    I am trying to generate a CDF with a uniform distribution between -55 and -45 with 1000 samples. I can’t seem to get the function to work. I typed in uniform_dist(1000,-55,-45,true), and all I seem to be getting is 1’s. Am I doing something wrong?

    Regards,
    Raquel

    Reply
    • Raquel,
      Since the first argument (1000) is larger than the third argument(-45), the cdf is always 1. Generally you should choose a value for the first argument that is between the second and third arguments. E.g. uniform_dist(-51,-55,-45,true) has the value .4
      Charles

      Reply
  11. Dear Charles,

    I’ve been browsing your site in search for a solution to my question. But I am unsure what method to employ.

    I am working in Excell. My sample consists of 66 columns and 250 rows. Of the 1650o datapoints, 290 display a one; the rest are zero.The distribution of 1’s over the 66 columns is relevant to me and I wish to determine whether this distribution is statistically different from a random distribution.

    Could you please give me some pointers (binom/poisson/normal dist?; How can I efficiently perform simulations?)

    Kind regards,

    Antoine

    Reply

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