What is the cumulative distribution function (CDF)?

The cumulative distribution function (CDF) calculates the cumulative probability for a given x-value. Use the CDF to determine the probability that a random observation that is taken from the population will be less than or equal to a certain value. You can also use this information to determine the probability that an observation will be greater than a certain value, or between two values.

Example of using the CDF to evaluate fill weights

For example, soda can fill weights follow a normal distribution with a mean of 12 ounces and a standard deviation of 0.25 ounces. The probability density function (PDF) describes the likelihood of possible values of fill weight. The CDF provides the cumulative probability for each x-value.

The CDF for fill weights at any specific point is equal to the shaded area under the PDF curve to the left of that point.

Use the CDF to determine the probability that a randomly chosen can of soda has a fill weight that is less than 11.5 ounces, greater than 12.5 ounces, or between 11.5 and 12.5 ounces.

The probability that a randomly chosen can of soda has a fill weight that is less than or equal to 11.5 ounces is the CDF at 11.5, or approximately 0.023.

The probability that a randomly chosen can of soda has a fill weight that is greater than 12.5 ounces is 1 minus the CDF at 12.5 (0.977), or approximately 0.023.

The probability of a randomly chosen can of soda having a fill weight between 11.5 ounces and 12.5 ounces is the CDF at 12.5 minus the CDF at 11.5 or approximately 0.954.

Use the CDF to calculate p-values

In order to calculate a p-value for an F-test, you must first calculate the cumulative distribution function (CDF). The p-value is 1 – CDF

Suppose you perform a multiple linear regression analysis with the following degrees of freedom: DF (Regression) = 3; DF (Error) = 25; and the F-statistic = 2.44.

Calculate a p-value for the F-test

  1. Open the cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Cumulative Distribution Function
  2. From Form of input, select A single value.
  3. From Value, enter 2.44.
  4. From Distribution, select F.
  5. In Numerator degrees of freedom, enter 3.
  6. In Denominator degrees of freedom, enter 25.
  7. In Noncentrality parameter, enter 0.
  8. Click OK.

The cumulative probability is 0.912050. Subtract this value from 1 to get the calculated p-value.

The calculated p-value is 0.08795. Using the 0.05 cutoff value, you would not conclude statistical significance because 0.08795 is not less than 0.05.
Note

This example is for an F-distribution; however, you can use a similar method for other distributions.

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