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How to Approximate Probability Density

Probability density sounds like a complicated concept, and in many ways it is. With a little practice, however, you will soon be able to approximate probability density as well as any social scientist. In terms of practical application, calculating probability density allows you to make a more meaningful interpretation than a regular histogram would. This is because a regular histogram simply describes the relative frequency of an event within the sample; but the relative frequency might appear much lower or much higher than the actual probability of the event it describes. Probability is more commonly used in the social sciences because it is a universal rather than a data-set-specific standard.

Things You'll Need

Data set
Paper
Pen or pencil

Instructions

- 1
  Create a relative frequency histogram. Take your paper and draw an X-Y graph in the shape of the letter "L." Place your continuous variable on the X axis. Separate your variable into subgroups, so that as the values increase you create new histogram cells. To do this, put hash marks at regular intervals across the X axis. Imagine that your continuous variable measures the distance (in hundreds of miles) that truckers drove in a given week, so that 1 = 100 miles, 2 = 200 miles, etc. Beginning with 0, create cells that distinguish every 20 miles (the cells should be in increments of 0.2) , so that there are five cells between each integer (1.0, 1.2, 1.4, etc).
- 2
  On the Y axis, place a 0 at the X/Y intersection. Draw five evenly spaced hash marks and label each hash beginning with 0.2 for the hash above the 0, followed by 0.4, 0.6, 0.8 and finally a 1 at the top of the axis. These hash marks measure the relative frequency of the truckers in each cell.
- 3
  Plot the data. If there are 45 truckers who drove between 51 and 100 miles, place them in the corresponding cell on the X axis and ensure that the top of the bar aligns with 0.45 (just above the 0.4 hash) on the Y axis.
- 4
  Rescale the relative frequency histogram that you have just created into a relative frequency density histogram. Use the formula below to calculate this rescaling:
  
  relative frequency density = relative frequency / cell width.
  
  Applying the formula to the example above, relative frequency density = relative frequency / 5 (because there are five cells in each unit of measurement).
- 5
  After rescaling, your bars should be five times as long, and the total area of the bars should be equal to 1 (multiply the length of the cells touching the X axis by the relative frequency density of the tallest bar on the Y axis). With a large "n" (sample size) and a small cell width, the histogram now approximates a probability distribution.

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