Exercise 11 Page 716

Let's take a look at the given table of average monthly temperatures of two cities.

We are asked to compare the temperatures of these two cities. In order to do so, we will find the mean, median, mode, range, and interquartile range of each data set.

Mean

The mean of a data set is the sum of the values divided by the total number of values in the set. Let's start by calculating the sum of the given temperatures in Jacksonville. &52.4 + 55.2 + 61.1 + 67.0 + 73.4 + 79.1 + 81.6 &+ 81.2 + 78.1 + 69.8 + 61.9 + 55.1 = 815.9 Let's do it for Austin as well. &48.8 + 52.8 + 61.5 + 69.9 + 75.6 + 81.3 + 84.5 &+ 84.8 + 80.2 + 71.1 + 60.9 + 51.6 = 823.0There are 12 values in each set, so we have to divide each sum by 12. lJacksonville Mean: 815.9/12=67.9916 Austin Mean: 823.0/12=68.583

Median

When the data is arranged in numerical order, the median is the middle value — or the mean of the two middle values — of the set. Let's arrange the given values and find the median. &Jacksonville: 52.4, 55.1, 55.2, 61.1, 61.9 & 67.0, 69.8, 73.4, 78.1. 79.1, 81.2, 81.6 &Austin: 48.8, 51.6, 52.8, 60.9, 61.5 & 69.9, 71.1, 75.6, 80.2, 81.3, 84.5, 84.8 Since we have an even number of data values, the median is the mean of the two middle values. lJacksonville Median: 67.0+ 69.8/2=68.4 Austin Median: 69.9+ 71.1/2=70.5

Mode

The mode is the value or values that appear most often in a set of data. Since there are no repeating values in neither data set, there is no mode.

Range

The range is the difference between the greatest and least values in a data set. &Jacksonville: 52.4, 55.1, 55.2, 61.1, 61.9 & 67.0, 69.8, 73.4, 78.1. 79.1, 81.2, 81.6 &Austin: 48.8, 51.6, 52.8, 60.9, 61.5 & 69.9, 71.1, 75.6, 80.2, 81.3, 84.5, 84.8 Let's find the range of each data set. lJacksonville Range: 81.6- 52.4=29.2 Austin Range: 84.8- 48.8=36

Quartiles and Interquartile Range

Let's start by recalling the definitions of quartiles and interquartile range.

• First Quartile (Q_\bm{1}) is the median of the lower half of the data set.
• Third Quartile (Q_\bm{3}) is the median of the upper half of the data set.
• Interquartile Range is the difference between the third and first quartiles (Q3-Q1).

Let's find the first and third quartiles for each given data set. &Jacksonville: 52.4, 55.1, 55.2, 61.1, 61.9 & 67.0 | 69.8, 73.4, 78.1, 79.1, 81.2, 81.6 &Austin: 48.8, 51.6, 52.8, 60.9, 61.5 & 69.9, | 71.1, 75.6, 80.2, 81.3, 84.5, 84.8 Since each half of the data sets has an even number of data values, each quartile is the mean of the two middle values.

We can now find the interquartile range of each data set. lJacksonville Interquartile Range: 78.6-58.15=20.45 Austin Interquartile Range: 80.75-56.85=23.9

Summary

Finally, we summarize our findings in the table below so it is easier to compare the results.

The higher range and interquartile range of the temperatures at Austin show that the temperatures are more varied across the year.