Chapters
Measures of Central Tendency Part 1
In previous sections, you learned about measures of central tendency and variability, going through the basics of each concept as well as their applications in other concepts of statistics. Here, we’ll go through some more intermediate concepts involved in these descriptive statistics. That is, we’ll go through more practice problems and solutions involving these statistical measures
Changing the Units of Measures
The more you study the concepts involved in statistics, the more you’ll realize it requires a lot of data preparation. Meaning, the data that we receive from surveys and scientific studies isn’t always in the format we require to answer our questions. In higher levels of statistics, you will be able to learn about different methodologies, or processes for conducting studies, that can enable you to correct for issues in format before you even collect your data!
However, the majority of data preparation simply involves changing the units we have into another. This can be something like transforming centimetres into metres or changing a variable to follow a logarithmic scale. While changing units is a completely normal, and even routine, part of statistics, it’s important to keep in mind if and how changing units can also change certain measures of central tendency and variability.
For example, say you download the following information on a group’s weight before they join a particular exercise class.
| Observation | Observation Value |
| 1 | 61 |
| 2 | 64 |
| 3 | 67 |
| 4 | 71 |
| 5 | 73 |
| 6 | 73 |
| 7 | 85 |
| 8 | 91 |
Here, our information is displayed in kilograms. From this information, we can determine all the measures we’ve learned in previous sections, rounding to the first decimal place.
| Measure | Calculation |
| Mean | 73.1 |
| Median | 72 |
| Mode | 73 |
| Variance | 104.7 |
| Standard Deviation | 10.2 |
| Average Deviation | 7.4 |
Let’s say, however, that you also want to publish the information on these weights for an audience that uses the imperial system rather than the metric system of measurements. This means you will have to convert your units into pounds from kilograms, multiplying each observation value in your data by 2.2 to get an approximate weight. How will this affect what you’ve already measured?
Turns out, there’s a shortcut to calculating measures of central tendency and variability when the unit has been changed. Instead of changing all of the units in your data, you can simply follow the rules for changing units in the tables below.
| When adding or subtracting a constant | Effect on the Measure |
| Mean, Median, Mode | Add or subtract that constant |
| Standard Deviation, Variance, Average Deviation, IQR | No effect, they stay the same |
| When multiplying or dividing by a constant | Effect on the Measure |
| Mean, Median, Mode, Standard Deviation, Average Deviation, IQR | Multiply or divide by that constant |
| Variance | Multiply or divide by the square of that constant |
Looking back at our example, since we are multiplying our data by a constant, 2.2, we simply have to follow the rules above in order to get all the measures we desire, rather than multiplying the whole data set by 2.2 and recalculating everything.
| Measure | Calculation in Kg | Calculation in Lbs |
| Mean | 73.1 | \[ 73.1*2.2 \] |
| Median | 72 | \[ 72*2.2 \] |
| Mode | 73 | \[ 73*2.2 \] |
| Variance | 104.7 | \[ 104.7*(2.2^2) \] |
| Standard Deviation | 10.2 | \[ 10.2*2.2 \] |
| Average Deviation | 7.4 | \[ 7.4*2.2\] |
Practice Problems
In this section, you learned about the effects of transforming the units in your data on measures of central tendency and variability. These practice problems are meant to reinforce what you’ve learned while still providing you with a challenge.
Problem 1
We are studying the wages of workers in the garment industry in India. For our study, we only need information regarding the mean hourly wages for males and females. Unfortunately, the data we have found is in Indian rupees from 2013. In order to get these wages to 2019 pound sterling, we first must convert rupees to pound sterling using a 2013 conversion rate, then adjust for inflation.
Given that the conversion rate for 2013 was 1 pound sterling = 89 rupees and the inflation rate was 2.4% each year between 2013 and 2019, what was the mean hourly wage for males and females in 2019 pound sterling? Because adjusting for inflation can be complex, let’s simplify our calculation by assuming that the inflation is 2.4% of 2013 pound sterling for each year.
Use the following table:
| Male Mean Wage 2013 | Female Mean Wage 2013 |
| 217 rupees | 141 rupees |
Problem 2
We’re interested in understanding the effects of joining an exercise class on weight. After one month of courses, each participant in our study dropped 3% of their initial weight. Given information on their weights before joining the class, fill in the following information about their weight after one month of exercise course, rounding to the first decimal.
| Measures: Before Joining | Measures: After One Month |
| Minimum: 61 | Min: |
| Quartile 1: 67 | Q1: |
| Median: 72 | Med: |
| Quartile 3: 73 | Q3: |
| Maximum: 91 | Max: |
Solutions
After solving or attempting to solve the problems above on your own, check your answers below to compare what you succeeded on and what you may need to review.
Problem 1: The Effects of Changing Units on Mean Hourly Wages
In this problem, we were given the following information:
- In 2013: 1 pound sterling = 89 rupees
- The inflation rate was 2.4% each year between 2013 and 2019
Below you will find the steps involved in solving this problem.
| Step | Calculation |
| 1. Convert each 2013 wage into 2013 pounds. Following the rules of changing units, we simply divide the mean by this constant. | Males: \[\dfrac{21}{89} = 2.44 \thickspace pounds\] Females: \[\dfrac{141}{89} = 1.58 \thickspace pounds\] |
| 2. Calculate the amount of inflation for 2013 to 2014. | Males: \[2.44 \thickspace pounds * 0.024 = 0.059 \thickspace pounds\] Females: \[1.58 \thickspace pounds * 0.024 = 0.038 \thickspace pounds\] |
| 3. We assumed that the inflation rate for each year is 2.4% of 2013 pounds (not the case in reality but simplifies our calculations). Calculate the total inflation amount from 2013-2019. | Males: \[0.059 \thickspace pounds * 6 \thickspace years = 0.35\] Females: \[0.038 \thickspace pounds * 6 \thickspace years = 0.23\] |
| 4. Following the rules of changing units, we simply add the total inflation to the 2013 mean in order to get the 2019 mean. | Males: \[2.79 \thickspace pounds\] Females: \[1.81 \thickspace pounds\] |
We can conclude that the hourly wage for Indian garment workers in 2019 pounds is 2.79 for males and 1.81 for females.
Problem 2: The Effects of Changing Units on the Interquartile Range
In this problem, we were given the following information:
- After one month, each participant dropped 3% of their initial weight
We don’t need all the information about each participant’s weight before and after the study. Instead, we can simply follow the rules of changing units. Because each participant lost 3% of their initial weight, that means they all now weigh 97% of what they did before. To find the information about the IQR, we simply multiply each figure by this constant.
| Measures: Before Joining | Measures: After One Month |
| Minimum: 60 | Min: \[60*0.97 = 58.2\] |
| Quartile 1: 61 | Q1: \[61*0.97 = 59.2\] |
| Median: 67 | Med: \[67*0.97 = 65\] |
| Quartile 3: 73 | Q3: \[73*0.97 = 70.8\] |
| Maximum: 91 | Max: \[91*0.97 = 88.3\] |
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