Measures of Center Math 075 Fall 2016

But First We Review

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….

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?

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.

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.

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

Mean: Fair Share and Balance Point

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?

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?

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

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.

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.

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

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

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? 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. 26

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

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 28

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

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

Kobe Bryant walks into a dinner

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