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Cumulative Distribution Function CDF

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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.

Stephanie Glen. "Cumulative Distribution Function CDF" From Elementary Statistics for the rest of us!

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