What you’ll need to know
In this support activity you’ll become familiar with the following:
You will also have an opportunity to refresh the following skills:
In the next section of the course material and in the following activity, you will need to understand and calculate the deviation from the mean. You will be extending this knowledge in the course section to understanding what the spread of a dataset is and how it is calculated. For now, let’s concentrate on refreshing necessary skills and learning about deviation from the mean.
Deviation from the Mean
Consider the following dotplot of the exam scores for two different math classes on their midterm. The class average was [latex]70[/latex] points for both Class 1 and Class 2.
question 1
In the next activity, we will be measuring the variability of a dataset. We do this by measuring the distance, known as deviation, of an observed value from the mean. Let’s look at an example from the real world.
Hurricanes cause extensive amounts of damage. In this corequisite support activity, we will consider the amount of damage in dollars of the [latex]30[/latex] most expensive hurricanes to have hit the U.S. mainland between 1990 and 2010. In order to explore this dataset, you will need to recall what you have learned about measures of center. We’ll concentrate on the mean of a quantitative distribution in this activity.
Mean vs. median
Before we move on, take a moment to recall the difference between mean and median.
recall
Do you recall the two measures of center you learned about in the previous section of the course: mean and median?
Core skill:
Let’s go to the technology to analyze the dataset “Hurricane Damage.”
Sample size and mean
Go to the Describing and Exploring Quantitative Variables tool at https://dcmathpathways.shinyapps.io/EDA_quantitative/.
Step 1) Select the Single Group tab.
Step 2) Locate the drop-down menu under Enter Data and select From Textbook.
Step 3) Locate the drop-down menu under Dataset and select Hurricane Damage.
In the descriptive statistics at the top of the applet, you will see the sample size [latex]\left(n\right)[/latex] and the mean [latex]\left(\bar{x}\right)[/latex] of the dataset.
question 2
question 3
Deviation from the mean
A deviation from the mean is the distance between an observation, [latex]x[/latex], in a dataset and the mean, [latex]\bar{x}[/latex], of the dataset. To calculate the deviation from the mean, subtract the sample mean from each observation in the dataset [latex]\left(x-\bar{x}\right)[/latex].
Practice calculating the deviation from the mean in the following interactive example. Then, for the data table that follows, calculate the deviation from the mean to answer Question 4.
Interactive Example
Seven of [latex]50[/latex] observations a student made about her commute time by bicycle from her apartment to school are listed in the following table. Her mean commute time was [latex]10[/latex] minutes, [latex]\bar{x}=10[/latex]. Use this information to calculate the deviation from the mean [latex]\left(x-\bar{x}\right)[/latex] for each of the seven observations given.
Example: See the first entry in the table: an observed 8 minute commute time. Given a mean commute time of 10 minutes, the deviation from the mean for the observation of 8 minutes is [latex]8-10=-2[/latex].
Complete the table then check your answers below.
|
Bike Ride to School (in minutes) [latex]x[/latex] |
Deviation from the Mean (in minutes) [latex]x-\bar{x}[/latex] |
| 8 | [latex]8 - 10 = -2[/latex] |
| 10 | |
| 12 | |
| 14 | |
| 5 | |
| 15 | |
| 9 |
Now it’s your turn to calculate the deviation from the mean for ten of the 30 hurricane damage observations from the dataset “Hurricane Damage.” Follow the instructions above to reopen the dataset if you have closed it.
question 4
Representations of large numbers
Before answering Question 5 below, take a moment to consider the ways in which large numbers can be represented. In the table above, we see hurricane damage in millions of dollars in the column on the left. Look at the the bottom number in the column: [latex]11,227[/latex]. Presumably, that means [latex]11,227[/latex] millions of dollars. But what does that mean in terms of a pure number? The hurricanes contributing to this data were catastrophic, causing billions of dollars of damage. Use the recall box below to see how to write a number like [latex]11,227[/latex] million dollars as $[latex]11.227[/latex] billion. You may also see the Student Resource: Number-Word Combinations.
recall
It can be helpful to communicate large numbers using a combination of numbers and words.
When reading text containing a large value, we generally comprehend a number written as a combination of numbers and words more quickly than we do the pure number form. For example, it may take a moment to make sense of $[latex]1,250,000[/latex] but we understand immediately what $[latex]1.25[/latex] million represents.
Take a moment to refresh your understanding of combining numbers and words to express large numbers.
Core Skill:
Now, try Question 5.
question 5
Signed numbers as proximities
Before answering Question 6 and 7, you may wish to refresh your understanding of distance as an absolute value.
recall
When discussing the difference between two numbers as a distance, use the concept of absolute value to help make sense of the result. For example, we would say the difference between [latex]-1[/latex] and [latex]3[/latex] is four units even though taking their difference may result in a negative or a positive depending upon which we subtract from which.
[latex]-1-3=-4\qquad\text{ and }\qquad3 - \left(-1\right)=4[/latex]
[latex]|-1-3|=4\qquad\text{ and }\qquad|3 - \left(-1\right)|=4[/latex]
See the skill below if needed for an example of how absolute value can be applied in Questions 6 and 7 and how to interpret positive and negative results when calculating deviation from the mean.
Core skill:
question 6
question 7
You’ve learned how to calculate the deviation from the mean in this activity, which you’ll be using in the upcoming section and following activity. You’ve also refreshed several mathematical skills and statistical definitions. Hopefully, you are feeling comfortable enough with these concepts to move on to the next section.
