< Probability and statistics definitions < Frequency curve

## What is a frequency curve?

A **frequency curve** (sometimes called a *smoothed frequency polygon*) relates the magnitude of a variable to its frequency of occurrence. It is an estimate of the cumulative distribution of that variable in the population and is created from sample data.

Frequency curves are smooth curves corresponding to the limiting case of a histogram calculated for a frequency distribution of a continuous distribution — as the sample size gets very large. As the number of data points in the histogram increases, the bars on the histogram become thinner, and the frequency curve will begin to look like a smooth curve. This smooth curve is called the probability density function (pdf).

- The pdf gives the probability a random variable will take on a particular value.
- It is always non-negative, and it integrates to 1 (i.e., the area under the frequency curve = 1).
- The pdf can be used to calculate the probability that a random variable will have a certain range of values.

Frequency curves are useful when you have large amounts of data in a sample. The idea is that tiny slices can still contain large numbers of data points — enough so that the tiny slices are meaningful for analysis. On the other hand, if you have a small sample (say, less than 30), it’s unlikely that a frequency curve — and the resulting pdf — will give you easier analysis than if you simply used the histogram from which the pdf was created from.

## Creating a frequency curve

To create the best possible frequency curve, choose one that matches your data from a known probability distribution such as the Gumbel, Normal, Log-normal, Exponential, Weibull, Pearson and Log-Pearson [2]. This is because the pdfs for these distributions are known, meaning that you won’t have to battle mounds of data to come up with a potential pdf. In many cases this is a very difficult or near-impossible task. Once you’ve chosen an appropriate model, you can use the pdf to plot the frequency curve.

If you cannot fit your data to a known probability distribution, another option is to sketch out the rough shape of your data — in one continuous line from left to right. While this can be a useful visualization, it won’t help with statistical analysis without the underlying pdf.

## Types of frequency curves

Frequency curves take on many different shapes, including bell-shaped, skewed, J-shaped, U-shaped, bimodal and multimodal shapes.

## Frequency polygons vs frequency curves

Frequency polygons and frequency curves both can graph a data distribution. The main difference between a frequency polygon and a frequency curve is that the frequency polygon is a discrete graph, whereas the frequency curve is a continuous graph. While a frequency curve is a smooth curve drawn through a histogram’s midpoints, a frequency polygon is a line graph created by connecting midpoints of intervals in a histogram. For this reason, they can look almost identical.

## References

- Image: USGS. https://pubs.usgs.gov/twri/twri4a2/pdf/twri_4-A2_a.pdf
- Introduction to Flood Frequency Analysis