1. Measures of Center
Math 075 Summer 2016

2. Tell me how I am doing?
I am constantly striving to be a better teacher.
I need to hear from you
It’s time for you to grade me
Please take a moment to take my anonymous survey
Go to by website and click the link.

3. How Many???
It’s early but I bet you have been on social media. 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.

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

5. 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)

6. 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.

7. 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

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

9. 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?

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

11. 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

12. Exit Ticket Go to my website and click on the link
After you are done you can take a ten minute break

13. Mean: Fair Share and Balance Point

14. Understanding the Mean
Each person at the table should:
Grab a handful of candy.
Count them and write the number down.

15. Understanding the Mean
Work together at your table to answer
the following question:
If you redistributed all of the candy from your handfuls so that everyone had the same amount (so that they were “shared fairly”), how many cubes would each person receive?
Mean….also known as fair share

16. Understanding the Mean
What was your answer?
- How did you handle “leftovers”?
- Add up all of the numbers from the
original handfuls and divide the sum
by the number of people at the table.
- Did you get the same result?
Obviously, the answer represents the mean – but make sure they understand it represents a “fair share” as the underlying concept of the mean.
- What does your answer represent?

17. Understanding the Mean
Place it on a magnet on your number, so they are ordered…
Horizontally: Low to high, left to right; leave one space if there is a missing number.
Vertically: If your number is already on the wall, place your sticky note in the next open
space above that number.
Building a line plot for the activity at the end of Slide 79.

18. Understanding the Mean
Looking at our dot plot, how can we
describe our data set?
- Find the range?
- Find the mode?
- Find the median?
- Find the mean?

19. Mean

20. Simulation Time
Each group will need one ruler, a pencil, and 6 pennies.
Place the six pennies at 6 and balance the ruler using the pencil.
Where did you place the pencil?
After each move ask what is the mean?

21. Simulation Time Move one of the pennies to 8.
Now move another penny to balance the ruler.
Where did you place your other penny?
What is the mean? Write answer on board or have students share out.

22. Simulation Time
Move another penny off the stack to the 2 in mark. (Don’t move the pencil).
Now move the last two pennies off the 6 in mark and place them on the ruler so that the ruler is balanced.
What is the mean? Write answer on board or have students share out. The final step have students place on board in dot plot where they placed the last two pennies. Talk about the sum of the distance above and below the mean.

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?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. 26

27. 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. 27

28. 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. 28

29. 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 29

30. The Mean is the Balance Point
We can confirm this by calculating:
= 36
36 ÷ 9 = 4
The Mean is the Balance Point 30

31. 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. 31

32. 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 32

33. Now, Bill Gates walks into the diner...
Find the mean and the median using Stat Crunch
Are the mean and the median similar? Would both or either represent a ‘typical’ or ‘average’ customer’s salary?
Should we use the mean or the median in this case?
Graph the data (histogram; box plot) using Stat Crunch. What shape is the distribution?
$45,000
$48,000
$52,000
$40,000
$35,000
$58,000
$46,000
$3,710,000,000

34. 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

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

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

38. 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

39. Homework Read OLI 37-39 Exam Reflection #1 Check point Module 7
Brain ology Survery