Deviation from the mean
A deviation from the mean is the distance between an observation, [latex]x[/latex], in a data set and the mean, [latex]\bar{x}[/latex], of the data set. To calculate the deviation from the mean, subtract the sample mean from each observation in the data set [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.
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 data set “Hurricane Damage.” Follow the instructions above to reopen the data set if you have closed it.
question 4
Representations of large numbers
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 that you have refreshed your understanding of combining numbers and words to express large numbers, try Question 5 on the next page.
