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Question Video: Finding a Frequency Density Mathematics

The frequency table provided shows the ages of teenagers in a neighborhood. Find the missing density from the table.

03:40

Video Transcript

The frequency table provided shows the ages of teenagers in a neighborhood. Find the missing density from the table.

In this question, we’re looking at frequency density rather than just frequency. We would find frequency density if we wanted to go on and create a histogram from the data. Frequency density tells us the ratio of a frequency of a class to its width. We can calculate frequency density by using the formula that frequency density is equal to frequency over the class width. It can often be helpful to add in a row to our table to work out the class width for each interval.

Let’s take a look at our first interval and work out what exactly we mean by this. 13 is less than 𝑎 is less than or equal to 14. The age in years means that the age must be greater than 13 but not equal to it. The second part of this inequality 𝑎 is less than or equal to 14 means that the age can be 14 or less than. So if 𝑎 must be bigger than 13 and not equal to it and less than 14, it leaves us really only with one possible value. And that’s the age of 14 years old. 14 is larger than 13, and although it isn’t less than 14, it can be equal to it. The class width here would simply be one.

In the second inequality, we’re told that the age must be greater than 14. So teenagers in this category could not be 14. And the ages go up to and including 17. So the teenagers in this category would either be 15, 16, or 17 years old. So the class width here would be three. In our final column, we have ages greater than 17 but not including it and less than or equal to 19 years old. The teenagers in this category would either be 18 years old or 19 years old. So the class width here would be two.

Let’s have a look at the frequency densities that have already been calculated. In the first column, we can see that the frequency density given is 21. The formula tells us that we take the frequency of 21 and we divide by the class width, in this case, one. And 21 over one does indeed give us 21. In the second column, to find the frequency density, we will take our frequency of 24 and divide by the class width of three. And 24 divided by three does indeed give us a frequency density of eight. To find the missing frequency density in the final column, we take the frequency of 10 and divide by the class width of two. 10 divided by two gives us five, the answer for the missing frequency density.

In this question, we weren’t asked to create a histogram from the data. But if we did, what would it look like? When we create a histogram on our 𝑦-axis, we don’t have frequency like we would in a bar chart. But instead, we plot frequency density. The age would be plotted on the 𝑥-axis. And we can see that each bar in this histogram goes up to each individual frequency density. The bars are all next to each other. And in a histogram, we would calculate the frequency by finding the area of each bar. Histograms can be a very good way for visualizing the distribution of values in data sets. In this question, we didn’t need to draw the histogram just to give the missing frequency density of five.

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