Measures of Center Math 075 Summer 2016

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.

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.

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

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)

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.

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

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

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?

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

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

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

Mean: Fair Share and Balance Point

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

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

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?

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.

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?

Mean  

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?

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.

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.

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

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

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

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

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

Where is the balance point for this data set? MEAN Sum of the distances below the mean 1+1+1+2 = 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

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

The Mean is the Balance Point We can confirm this by calculating: 2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36 36 ÷ 9 = 4 The Mean is the Balance Point 30

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

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

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

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

Which measure of central tendency should we use? .

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

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

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