question 5
Hint
What do you think? Use the recall box above as a guide.
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:
Express a distance as an absolute value.
Say the mean of a sample is given as [latex]\bar{x}=12[/latex] and the observations [latex]7[/latex] and [latex]15[/latex] are contained in the sample. Which value is closer to the mean?
For the value of [latex]7[/latex], [latex]x-\bar{x} = 7-12=-5[/latex]
For the value of [latex]15[/latex], [latex]x-\bar{x} = 15-12=3[/latex]
We might be tempted to conclude that [latex]7[/latex] is closer since [latex]-5[/latex] is a smaller number than [latex]3[/latex]. But distance is calculated using absolute value. The value of [latex]7[/latex] is [latex]5[/latex] units away from the mean (to the left) while the value of [latex]15[/latex] is only [latex]3[/latex] units away from the mean (to the right). To calculate which is closer, use absolute value.
For the value of [latex]7[/latex], [latex]|7-12|=|-5|=5[/latex]
For the value of [latex]15[/latex], [latex]|15-12|=|3|=3[/latex]
question 6
Hint
For each calculation, you subtracted the mean from the observed value. Why would some result in a negative deviation? See the interactive example above for an explanation.
question 7
Hint
Think of closer as being a distance (i.e., absolute value).
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.
