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:

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

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:

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