Question Video: Finding Frequency Densities Mathematics

The frequency table below represents the times students take to walk to school. Find the frequency density for each frequency class.

03:05

Video Transcript

The frequency table below represents the times students take to walk to school. Find the frequency density for each frequency class.

In this question, we’re asked for the frequency density. The frequency density gives us the ratio of the frequency of a class to its width. We might calculate frequency density if we wish to create a histogram. We can use the formula that frequency density equals the frequency divided by the class width. First, we’ll work out the class width and then we’ll find the frequency density for each class. So in this first class interval of students walking to school, we have the time is given as zero is less than 𝑡 is less than or equal to four. This means that the time 𝑡 minutes can’t be equal to zero. But it could be one, two minutes, three minutes, or even four minutes.

This inequality here — 𝑡, time, is less than or equal to four — means that the time could also be equal to four minutes. We could say then that there are four students that took either one, two, three, or four minutes to walk to school. As there are one, two, three, four options in our class width, the class width is four. In our second column, we have the inequality four is less than 𝑡 is less than or equal to eight, which means that our time in 𝑡 minutes does not include four. But it does include five, six, seven, or eight, since the time is less than or equal to eight minutes. Therefore, the interval from four to eight gives us a class width of four.

The final two columns also have a class width of four. To find the frequency density for each interval then, we can use the formula. In our first column, we have a frequency of four divided by a class width of four. So our first frequency density will be one. In the next column, we have a frequency of 12 divided by a class width of four. And 12 divided by four is three. In the next column, a frequency of 16 divided by the class width of four gives us a frequency density of four. In the final column, the frequency of eight divided by the class width of four gives us the frequency density of two. And so we have our answer for the frequency density for each frequency class. That’s one, three, four, and two.

If we wanted to create a histogram from the given data, it would look like this. We plot the frequency density on the 𝑦-axis rather than just the frequency. And along the 𝑥-axis, we have the time in minutes. We remember that we can calculate the frequency if we’re given a histogram by finding the area of each of the bars. In this question, the class widths were all the same at a width of four. But often we find in histograms the bars will be of varying widths. In this question, however, we didn’t need to draw the histogram, just to give the frequency densities, which we found.

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