Question Video: Estimating the Mean of a Frequency Distribution Mathematics

The following table represents the time taken by 35 people to travel to work. Calculate an estimation for the mean number of minutes people take to travel to work. Round your answer to two decimal places.

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Video Transcript

The following table represents the time taken by 35 people to travel to work. Calculate an estimation for the mean number of minutes people take to travel to work. Round your answer to two decimal places.

Looking at the table, we can see that this data has been presented in a frequency distribution. The top row contains the times taken, which have been organized into classes given as five dash, 10 dash, 15 dash, and 20 dash. The second row gives the frequencies for each class. So, for example, there were seven people who took greater than or equal to 15 minutes but less than 20 minutes.

We’re asked to find an estimation for the mean number of minutes people take to travel to work. It will only be an estimation because we don’t know the exact data values.

In general, the mean of a data set is calculated by finding the sum of all the data values and dividing this by the number of values in the data set. Our estimate of the mean will be found by estimating the sum of all the data values and then dividing this by the number of data values, which is the total frequency in the table. That’s 10 plus 15 plus seven plus three, which is 35.

To find an estimate of the sum of all the data values, we first need to find a single value that is representative of each class. We use the midpoint of each class, which is the average of the upper and lower class boundaries. For the first class, the lower boundary is five and the upper boundary is 10, because that is the lower boundary for the next class. The midpoint is therefore the average of five and 10, which is 7.5. For the second class, the midpoint is the average of 10 and 15, which is 12.5. And for the third class, it’s 17.5.

For the final class, we need to make an assumption about the upper boundary, because there is no class above it. We assume that the final class has the same width as the previous one, which is five. So we assume that the upper boundary of the final class is 25, and hence the midpoint is 22.5.

Next, we multiply the midpoint of each class by the frequency. This gives an estimate of the total number of minutes spent traveling for each class. This gives 75, 187.5, 122.5, and 67.5. Finally, we find the total of these four values, which gives an estimate of the total amount of time spent traveling by all 35 people.

We can now substitute these totals into the formula for the estimated mean. The estimated sum of all the data values is 452.5, and the number of data values is 35. As a decimal, this fraction is equal to 12.9285 continuing. And then we need to round the answer to two decimal places. Our estimate of the mean number of minutes people take to travel to work is 12.93 to two decimal places.

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