March 26, 2020
Chapters
In previous sections of this guide, we showed you how to calculate the mode, median, mean and variance. In addition, we provided you with some tips on how to interpret each measure and when one is more appropriate to use than another. Apply what you’ve learned here with these practice problems. Afterwards, be sure to check your answers and compare your responses with the solutions provided.
Mean, Median, and Mode
As a brief recap, the mean, median and mode are measures of central tendency. This simply means that they strive to capture the centre of the data. These three measures make up the three most common ways of describing data. Below, you’ll find their definitions as well as the best times to use each.
Measure  Description  Formula  Uses 
Mean  The average of the variable 
 When you want to find the average value When there are not any, or many, extreme values 
Median  The midpoint of the variable  No standard formula  When you want to know the middle value When there are extreme values 
Mode  The most occurring value of the variable  No standard formula  When you want to know the most frequent value 
Variance
The variance is part of a group of measurements that make up the other main group of calculations in statistics: measures of variability. Measures of variability try to capture how far spread or how close together the data are.
Measure  Description  Formula  Uses 
Variance  Measures the spread of the data 
 When you want to understand the spread When you want to detect outliers 
Grouped Variance  Measures the spread of grouped data 
 When you want to understand the spread of grouped data 
Practice Problems
Problem 1
Data on temperatures are taken every day. You are interested in analysing the data for the month of august but notice there are some extreme values at the beginning and end of the month. Which measure of central tendency would be most appropriate to use given the following information?
Day of the Month  Temperature (C) 
12  32 
34  26 
56  10 
78  17 
910  16.5 
1112  16.5 
1314  18 
1516  23 
1718  15.5 
1920  18.5 
2122  18 
2324  19 
2526  17.5 
2728  18.5 
2931  15.3 
Problem 2
You’re interested in investigating the efficiency of a brand of batteries. You run a study, recording the number of hours each round of batteries last. Find the average number of hours this brand of batteries lasted from the following data set.
Hours Lasted  Number of Batteries 
0  5  15 
6  10  25 
11  15  38 
16  20  9 
21  25  4 
Problem 3
You’re interested in interpreting the distribution of ages across preferred modes of communication. Given the following information, give at least three statements on the data provided using measures of central tendency.
Text Message  WackyBook  InstantBam  ClickClock  Phone Call  
0  5  15  0  3  0  0  32 
6  10  29  1  10  2  0  8 
11  15  30  1  12  5  0  2 
16  20  21  2  20  3  1  3 
21  25  8  20  15  0  2  5 
26  30  6  23  8  0  3  10 
Problem 4
You want to illustrate the example that changing units effect different measures differently. You have data on the average heights of males in the UK from 1900 to 1980. Find the average and variance of these heights, then calculate these measures for heights in feet and inches (1 in = 2.54cm). Choose an appropriate chart or plot to illustrate these changes.
Year  Heights in Cm 
1900  169.4 
1910  170.9 
1920  171 
1930  173.9 
1940  174.9 
1950  176 
1960  176.9 
1970  177.1 
1980  176.8 
Problem 5
You are analysing data on test grades for a high school biology class. You want to give an interpretation of the distribution of your variable. Using the information below, describe the distribution of test grades using the measures of central tendency and variability.
Measure  Value 
Group Variance  296 
Group Mean  7779 
Group SD  17 
Solution to Practice Problems
Solution Problem 1
In this problem, you were asked to:
 Determine the most appropriate measure of central tendency for the data
Given the graph, we can see that most days of the month have the same average temperature except for the first week of August. Given that these are almost 10 degrees higher than the rest of the month, it would be most beneficial to use either the median or the mode.
Solution Problem 2
In this problem, you needed to:
 Find the average number of hours this brand of batteries lasted
Because the data is grouped, we need to find the group mean. We do this using the following formula.
Hours Lasted  Number of Batteries  
0  5  15 


6  10  25 


11  15  38 


16  20  9 


21  25  4 


Total  91  64.5  985.5 
Plugging this into the formula, we get,
Meaning the group mean is between the interval 6 to 15. In other words, the batteries last, on average, between 6 and 15 hours.
Solution Problem 3
In this problem, you were asked to
 Make at least three statements on the data provided using measures of central tendency
Because we have grouped data, the easiest measure of central tendency to approximate by simply looking at the graph and table is mode. Take a look at the table below for sample answers for each age group.
Age Group  Preferred Mode of Communication 
0  5  Most children use the telephone as their primary mode of communication 
6  10  Children between 6 and 10 use mostly text message to communicate 
11  15  Teenagers between 11 and 15 use text message the most for communicating 
16  20  Those aged 16 to 20 use text message the most and InstantBam almost as much 
21  25  Those aged 21 to 25 use WackyBook as their primary mode of communication 
26  30  Adults aged 26 to 30 mostly use WackyBook to communicate 
Solution Problem 4
In this problem you were asked to:
 Find the mean and variance for the data in cm and in
 Graph the data
The mean for the data is found by calculating the following,
To find the variance, we follow the formula,
To find the mean and variance in inches, we simply need to follow the rules for changing units.
Solution Problem 5
In this problem, you were tasked with:
 Interpreting the chart using the information given
Find sample answers in the table below
Measure  Value  Interpretation 
Group Variance  296  The variable has a large variability, which is reflected in the fact that the data is spread unevenly about the mean. 
Group Mean  7779  The group mean here is slightly to the right of what we would expect it to be from looking at the graph. This is because there are some extreme values on the right. 
Group SD  17  The standard deviation makes up quite a significant portion of the mean, which indicates that the spread is quite large. 