1. We’re ‘Nut’ Giving Up Fundraiser
One Grand Prize
Airline tickets Montreal/Ft Lauderdale Return
3-Nights Accommodation at Marriott Fort Lauderdale
2 Tickets to Florida Panthers Alumni Box
Dinner with Florida Panthers Jesse Winchester
2 Signed Florida Panthers Jerseys
2 Tickets to Miami Dolphins game
500 tickets sold at $100 each
Should Mr. Lieff buy one?

2. Minds On!
Is there a relationship between drug costs in Canada and the US?

3. We’re ‘NUT’ Giving Up Fundraiser
Airline tickets Montreal/Ft Lauderdale Return $ 800
3-Nights’ Accommodation at Marriot Fort Lauderdale $ 450
2 Tickets to Florida Panthers Alumni Box $ 400
Dinner with Florida Panthers Jesse Winchester $ 250
2 Signed Florida Panthers Jerseys $ 400
2 Tickets to Miami Dolphins game $ 200
TOTAL $2500
E(X) = (2400)(1/500) = 5
So, you are expected to win $5 per ticket
You are better off taking your $100 to a blackjack table where
E(X) = 0.985!

4. 1.3 Trends in Data Questions? pp. 20–24 #1, 4, 9, 11, 14
Learning goals: Describe the trend and correlation in a scatter plot; Use a line of best fit to make predictions
MSIP / Home Learning: p. 37 #2, 3, (6-7 or 8)

5. Variables Variable (Mathematics) Variable (Statistics)
a symbol denoting an unknown quantity (x, y, θ, etc.)
Variable (Statistics)
A measurable attribute; these typically vary over time or between individuals
e.g., height, age, favourite hockey team
Can be discrete, continuous or categorical
Continuous: Weight (digital scale)
Discrete: Number of siblings
Categorical: Hair colour

6. Scatter Plot a graph that shows two numeric variables
each axis represents a variable
each point indicates a pair of values (x, y)
may show a trend

7. The Two Types of Variables on a Scatter Plot
Independent Variable
Horizontal axis
Time is independent (why?)
Timing is dependent (e.g., time to run 100m)
Dependent Variable
Values depend on the independent variable
Vertical axis
Syntax: dependent vs. independent or y vs. x
e.g., a graph of arm span vs. height means arm span is the dependent variable and height is the independent

8. What is a trend? the ‘direction’ of the data
a pattern of average behavior that occurs over time
e.g., costs tend to increase over time (inflation)
need two variables to exhibit a trend (time can be one)

9. An Example of a trend U.S. population from 1780 to 1960
What are some ways to describe the trend?

10. Strong, positive linear correlation
Correlations
Strength can be…
None – no clear pattern in the data
Weak – data loosely follows a pattern
Strong – data follows a clear pattern
For strong or weak correlations, the direction can be…
Positive - data rises from left to right (overall)
As x increases, y increases
Negative: data drops from left to right (overall)
As x increases, y decreases
Strong, positive linear correlation

11. Line of Best Fit A straight line that represents the trend in the data
Can be used to make predictions (using the graph or equation)
Can be drawn or calculated
Fathom has 3: movable, median-median, least squares
Gives no measurement of the strength of the trend (that’s next class!)

12. An example line of best fit
This is recycling data with a median-median line added
Describe the trend

13. Creating a Median-Median Line
Divide the points into 3 symmetric groups
If there is 1 extra point, include it in the middle group
If there are 2 extra points, include one in each end group
Calculate the median x- and y-coordinates for each group and plot the 3 median points (x, y)
If the median points are in a straight line, connect them
Otherwise, line up the two outer points, move 1/3 of the way to the other point and draw a line of best fit

14. Median-Median Line

15. Median-Median Line (10 points)

16. Lines of Best Fit – why 3? Drawing a line of best fit is arbitrary
Hit as many points as possible
Have the same number of points above and below the line
Outliers tend to be ignored
The median-median line is easy to construct and takes the spread of the data into consideration
The least-squares line takes every point into consideration but is based on a complicated formula
Good-Better-Best is a recurring theme in this course
3.3 Measures of Spread (Range, IQR, StdDev)

17. Using a regression equation – predict the independent variable
The equation of a line of best fit will be in the form y = mx + b
e.g., Toronto Maple Leafs roster on 3-Oct-13
W = 7.25H – 332
Mr. Lieff is 73.5” tall. His weight as a Maple Leaf would be:
W = 7.25(73.5) – 331.8
= or 201 lbs.

18. Using a regression equation – predict the dependent variable
e.g., Toronto Maple Leafs roster on 3-Oct-13
W = 7.25H – 332
In university Mr. Lieff weighed 186 lbs. His weight as a Maple Leaf would be:
186 = 7.25H – 331.8
= 7.25H
7.25H = 517.8
H = 71 in or 6’1”

19. Fathom Activity – How much would you weigh as an NHL player?
Part 1: Retrieve the data
Click nhl.com  Stats  Players
GAME TYPE: Regular Season
POSITION: Leave as All Skaters or choose a position
TEAM: Choose your favourite
REPORT: Bio Info
Click Go
Part 2: Copy the data to Excel
Select the data only (no headers) and Copy
Open Excel
Right-click cell A2 and click Paste | Match Destination Formatting
Type the headers into Row 1 (can’t copy )
Select and copy the data

20. Fathom NHL Activity – cont’d
Part 3: Paste the data into Fathom
Open Fathom2 (Desktop | Math or Start | enter into search)
Drag the Collection icon to the main window
Right-click the Collection | Paste cases
Double-click the name Collection1 and change it to something more descriptive (team name, position, etc.)
Double-click the cardboard box

21. Fathom NHL Activity – cont’d
Part 4: Graph the data
Create a graph in the workspace
Double click the Collection icon (cardboard box) to display the Collection Inspector
Drag Weight and Height to the respective axes
Which is dependent?
Right-click the graph and click Add Movable Line
Adjust the line so it:
Hits as many points as possible
Has the same number of points above and below
Every point has the same pull on the line
Right-click the graph and click Median-Median Line and Least Squares Line

22. Fathom NHL Activity – cont’d
Part 5: Use the equation to make predictions for the dependent and independent variables.
e.g., What would your weight be as an NHL hockey player?

23. Scatter Plots - Summary
A graph that compares two numeric variables
One is dependent on the other
May show a correlation
positive/negative
strong/weak
A line may be a good model
Median-Median and Least-Squares
If not, non-linear (can be quadratic, exponential, logarithmic, etc.)
Excel can do these

24. 1.4 Trends in Data Using Technology
Learning goal: Describe and measure the strength of trends
Questions? p. 37 #2, 3, (6-7 or 8)
MSIP / Home Learning: p. 51 #1-2, 3-5 (Fathom), 8

25. Regression The process of fitting a line or curve to a set of data
A line of best fit is a linear regression (Excel or Fathom)
A curve can be quadratic, cubic, exponential, logarithmic, etc. (Excel)
We do this to generate a mathematical model (graph or equation)
We can use the equation to make predictions
Interpolation – within the span of the data
Extrapolation – outside of the span of the data

26. Example armspan = 0.87 height + 22 y = 0.87 x + 22
What is the arm span of a student who is 175 cm tall?
y = 0.87(175) + 22
= cm
How tall is a student with a 160 cm arm span?
y = 0.87x + 22
160 = 0.87x + 22
160 – 22 = 0.87x
138 = 0.87x
x = 138 ÷ 0.87
= cm

27. Coefficient of Determination
r2 is the coefficient of determination
Takes on values from 0 to 1
r2 is the percent of the change in the y-variable that is due to the change in x
if r2 = 0.52 for the Leafs weight vs. height, 52% of the variation in weight is due to height

28. Correlation Coefficient
r is the correlation coefficient
indicates of the strength and direction of a linear relationship
r = 0 no relationship
r = 1 perfect positive correlation
r = -1 perfect negative correlation

29. Residuals
a residual is the vertical distance between a point and the line of best fit
if the model you are considering is a good fit, the residuals should be small and have no noticeable pattern
The least-squares line minimizes the sum of the squares of the residuals

30. Least Squares Line Weight vs. Height (NHL)
w = 7.23h – 325

31. Using the equation How much does a player who is 71 in tall weigh?
= lbs
How tall is a player who weighs 180 lbs?
w = 7.23h – 325  h = (w + 325) ÷ 7.23
So h = ( ) ÷ 7.23
= 69.85”
69.85 x 2.54 = 177.4cm

32. 1.5 Comparing Apples to Oranges

33. Chapter 1.5 – The Media Mathematics of Data Management (Nelson) MDM 4U
The Power of Data
Chapter 1.5 – The Media
Mathematics of Data Management (Nelson)
MDM 4U
There are 3 kinds of lies: lies, damn lies and statistics.

34. Example 1 – Changing the scale on the axis
Why is the following graph misleading?

35. Example 1 – Scale from 0
Consider that this is a bar graph – could it still be misleading?

36. Include every category!

37. Example 2 – Using a Small Sample
For the following surveys, consider:
The sample size
If there is any (mis)leading language

38. Example 2 – Using a Small Sample
“4 out of 5 dentists recommend Trident sugarless gum to their patients who chew gum.”
“In the past, we found errors in 4 out of 5 of the returns people brought in for a Second Look review.” (H&R Block)
“Did you know that 1 in 4 women can misread a traditional pregnancy test result?” (Clearblue Easy Digital Pregnancy Test)
“Using Pedigree® DentaStix® daily can reduce the build up of tartar by up to 80%.”
“Did you know that the average Canadian wastes $500 of food in a year?” (Zip-Lock Freezer bags)

39. Details on the Trident Survey
How many dentists did they ask?
Actual number: 1200
4 out of 5 is convincing but reasonable
5 out of 5 is preposterous
3 out of 5 is good but not great
Actual statistic 85%
Recommend Trident over what?
There were 2 other options:
Chewing sugared gum
Not chewing gum

40. Misleading Statements(?)
How could these statements be misleading?
“More people stay with Bell Mobility than any other provider.”
“Every minute of every hour of every business day, someone comes back to Bell.”

41. “More people stay with Bell Mobility than any other provider.”
Does not specify how many more customers stay with Bell.
e.g. Percentage of customers renewing their plan:
Bell: 30% Rogers: 29% Telus: 25% Fido: 28%
Did they compare percentages or totals?
What does it mean to “stay with Bell”? Honour entire contract? Renew contract at the end of a term?
Are early terminations factored in? If so, does Bell have a higher cost for early terminations?
Competitors’ renewal rates may have decreased due to family plans / bundling
Does the data include Private / Corporate plans?

42. “Every minute of every hour of every business day, someone comes back to Bell.”
60 mins x 7 hours x 5 days = 2 100/wk
What does it mean to “Come back to Bell”?
How many hours in a business day?

43. How does the media use (misuse) data?
To inform the public about world events in an objective manner
It sometimes gives misleading or false impressions to sway the public or to increase ratings
It is important to:
Study statistics to understand how information is represented or misrepresented
Correctly interpret tables/charts presented by the media

44. MSIP / Homework Read pp. 57 – 60 Ex. 1-2 Complete p. 60 #1-6
Final Project Example – Manipulating Data (on wiki)
Examples