1. Measures of Center
Math 075 Fall 2016

2. But First We Review

3. Rubric anyone? Read the rubric and grade your paragraph.
Think like a teacher.
How did you do?
What could you change?
What is one strength?
What is one weakness?
Go to my website and click on sugar model paper. Compare your paragraph to the model paper….

4. One More Time Go to my website and download the Protein paper help
Let’s grade the paragraph? Did you improve from the last one? What can you change?

5. OK One more time…Promise
This time you are creating your histogram.
Go to my website and download the cereal data
Upload or cut and paste this into stat crunch
Create two histograms for carbohydrates: One for adults and one for child.
Cut and paste these into a word document
You will now analyze these two and upload to canvas or print and give to me.

6. How Many???
Come up to the board an write the number of different types of social media you have been on this morning.
No need to organize
Female write your answer in black
males write your answer in blue
You have one minute to talk to your partner about one observation.

7. Graphical representations...
Bar Graph and pie graphs are the odd man out. They represent categorical (qualitative) data while the other graphs represent quantitative data.

8. No matter what...
We always want to create a graphical representation; visuals help us process information, indentify trends more easily We always label & scale our graphical representations We always use technology when available (no need to create graphical representations by hand)

9. Let’s create…. Working in groups answer the following questions:
Do you think males or females consume more caffeine on a daily basis? Why?
What age do you think consumes more caffeine on a daily basis? Why?
Type up: We will be using this throughout class.

10. Let’s create…. Stat crunch time
Go to my website and open up the caffeine consumption data
Cut and paste it into stat crunch
Let’s create graphs (you will cut and paste these into your document)
Dot plot
Histogram
Box plot
Scatter plot

11. Let’s create
Why did we not use pie charts or bar graphs?

12. Let’s describe Group 1: Box plot Group 2: Histogram Group 3: Dot Plots
You will have ten minutes to describe the data using SOCS. Be ready to share out.
Type up your observation on your paper
Answer the questions we started with (were your assumptions wrong/right? Support with the data:
Do you think males or females consume more caffeine on a daily basis? Why?
What age do you think consumes more caffeine on a daily basis? Why?

13. Histograms Let’s create a histogram of caffeine consumption
What happens when you change the bin width?
Does it change the description?

14. Caution... Bar graphs vs. histograms...
On left is bar graph; on right is histogram
Be sure you understand the difference between the two graphical representations

15. Mean: Fair Share and Balance Point

16. Here are two sets of exam scores, one for a class that has 4 students and one for a class that has 15 students.
Class A: 80, 90, 90, 100
Class B: 60, 65, 65, 70, 70, 70, 75, 75, 80, 80, 80, 80, 80, 85, 100 
Without doing any calculations, which class do you think will have a larger mean? Why?
Now calculate the mean for each class. Which is larger? Why does this make sense?

17. The dot plot gives quiz scores for a small class.
What is the mean? Show your work or explain how you got your answer.
What is the median? Show your work or explain how you got your
answer.
Which measure (the mean or the median) is the better way to represent typical performance on this quiz? Why?

18. This table gives quiz scores for a different class.
What is the mean? Show your work or
explain how you got your answer.
b) What is the median? Show your work
or explain how you got your answer.
c) Which measure (the mean or the median) is
the better way to represent typical
performance on this quiz? Why?
Scores
Number of students 5 1 6 3 7 8 9

19. Challenge!!!
For this problem, use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
List five digits that have a median of 7 and a mean of 7 (repeats allowed). Find a different set of 5 digits that work. 
List five digits that have a median of 7 and a mean that is less than 7 (repeats allowed.) Give the mean of your 5 digits. Find a different set of 5 digits that work.
List five digits that have a median of 7 and a mean that is more than 7 (repeats allowed.) Give the mean of your 5 digits. Find a different set of 5 digits that work.
Construct a data set where neither the mean nor the median is a reasonable “typical” value.

20. What if?
Imagine that you have a bag filled with 9 numbers. The mean and the median of the numbers in the bag are both 6.
You draw a number out of the bag. It is a 4. You replace it with a 1. Does the mean of the numbers in the bag get bigger, smaller, or stay the same? What about the median? Jot down some notes to explain how you figured this out.
 You draw a number out of the bag. It is a 4. You replace it with a 1. Does the mean of the numbers in the bag get bigger, smaller, or stay the same? What about the median? Jot down some notes to explain how you figured this out.

21. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 21

22. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 22

23. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 23

24. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 24

25. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 25

26. Where is the balance point for this data set?
MEAN
Sum of the distances below the mean
= 5
Sum of the distances above the mean
2 + 3 = 5 X X X X X X
Now refer back to the original line plot. We know that 3 is the balance point or mean. Talk about the sum of the distances above the mean being the same as the sum of distances from the mean below the mean. 26

27. Where is the balance point for this data set?
Move 2 Steps
Move 2 Steps
Move 2 Steps
Move 2 Steps
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean.
4 is the Balance Point 27

28. The Mean is the Balance Point
We can confirm this by calculating:
= 36
36 ÷ 9 = 4
The Mean is the Balance Point 28

29. Where is the balance point for this data set?
If we could “zoom in” on the space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point.
Move 1 Step
The Balance Point is between 10 and 11 (closer to 10).
Move 2 Steps
Move 1 Step
Move 2 Steps
Sticky Note Activity:
Work with the whole group to use this strategy to find the mean number of cubes in one handful based on our data set.
If it doesn’t work out to be whole number, discuss how we could find the exact decimal value of the mean if we could “zoom in” and how we could estimate the mean based on the modified line plot.
*You may want to have a calculator handy to find the actual mean of your whole group data set. 29

30. Median
The middle
Line up the data in numerical order and find the middle of the data
If there are two numbers in the middle find the average….add them up and divide by 2.
Is resistance to outliers 30

31. Kobe Bryant walks into a dinner

32. What’s the moral of this story?
Means are excellent measures of central tendency if the data is (fairly) symmetric
However, means are highly influenced by outlier(s)
So, if the data has an outlier(s), then a better measure of central tendency is the median, which is not influenced by outliers; this is called ‘resistant’
So, consider the shape of data/distribution, then wisely choose an appropriate measure of central tendency

33. Which measure of central tendency should we use?.

34. Which is larger: mean or median
Which is larger: mean or median? Which should we use to describe the ‘typical’ or middle value?

35. Let’s graph the data
Go to stat crunch and plug in the values on the board
Create a box plot/histogram
Cut and paste into a document
Describe the data distribution using SOCS
Print and turn it in…this is your exit ticket for this section