Statistics How To

Cumulative Distribution Function CDF

Probability Distributions > Cumulative Distribution Function

What is a Cumulative Distribution Function (CDF)?

The cumulative distribution function gives you the cumulative probability associated with a function. It is a similar concept to a cumulative frequency table. With a table, the frequency is the amount of times a particular number or item happens. The cumulative frequency is the total counts up to a certain number:
cumulative frequency

The cumulative distribution function works in the same way, except with probability.

cumulative distribution function

The yellow area represents the probability of a dog being above 11 pounds.

You can use the CDF to figure out probabilities above a certain value, below a certain value, or between two values. For example, if you had a CDF that showed weights of cats, you can use it to figure out:

  • The probability of a cat weighing more than 11 pounds.
  • The probability of a cat weighing less than 11 pounds.
  • The probability of a cat weighing between 11 and 15 pounds.

In the case of the above scenario, it would be important for, say, a veterinary pharmaceutical company knowing the probability of cats weighing a certain amount in order to produce the right volume of medications that cater to certain weights.

Cumulative Distribution Functions in Elementary Statistics

The cumulative distribution function gives the cumulative value from negative infinity up to a random variable X and is defined by the following notation:
F(x) = P(X≤x).
This concept is used extensively in elementary statistics, especially with z-scores. The z-table works from the idea that a score found on the table shows the probability of a random variable falling to the left of the score (some tables also show the area to some z-score to the right of the mean). The normal distribution, the basis of z-scores, is a cumulative distribution function:

empirical rule

Standard normal distribution showing standard deviations. Image credit: University of Virginia.


Need help with a homework or test question? Chegg offers 30 minutes of free tutoring, so you can try them out before committing to a subscription. Click here for more details.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments? Need to post a correction? Please post on our Facebook page.
Cumulative Distribution Function CDF was last modified: October 14th, 2017 by Stephanie