1. 1.3 Trends in Data Due now: p. 20–24 #1, 4, 9, 11, 14
Learning goal: Describe the trend and correlation in a scatter plot and construct a median-median line
MSIP / Home Learning: p. 37 #2, 3, 6, 8

2. Variables Variable (Mathematics) Variable (Statistics)
a symbol denoting a quantity or symbolic representation
an unknown quantity
Variable (Statistics)
A measurable attribute; these typically vary over time or between individuals
E.g. Height, Weight, Age, Favourite Hockey Team
Can be Discrete, Continuous or neither
Continuous: Weight (digital scale)
Discrete: Number of siblings
Neither: Hair colour

3. The Two Types of Variables
Independent Variable
horizontal axis
Time is independent (why?)
Timing is dependent
e.g., time to run a race vs. length of race
Dependent Variable
values depend on the independent variable
vertical axis
Format: “dependent vs. independent”
e.g., a graph of arm span vs. height means arm span is the dependent variable and height is the independent

4. Scatter Plots 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

5. 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

6. An Example of a trend U.S. population from 1780 to 1960
Describe the trend

7. Strong, positive linear correlation
Correlations
Strength is…
None – no clear pattern in the data
Weak – data loosely follows a pattern
Strong – data follows a clear pattern
For strong/weak, direction is…
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

8. Line of Best Fit A straight line that represents the trend in the data
Can be used to make predictions (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 tomorrow!)

9. An example of the line of best fit
this is temperature recycling data with a median-median line added
what type of trend are we looking at?

10. 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

11. Median-Median Line

12. Median-Median Line (10 points)

13. 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

14. AGENDA for Wed-Thu 1.3 Median-Median Line 1.4 Trends With Technology
Using a regression equation
Fathom Activity - Predict your weight as an NHL player
1.4 Trends With Technology
Correlation coefficient
Coefficient of Determination
Residuals
Least-Squares Line
Fathom Investigation: finding the Least Squares Line

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

16. Using a regression equation
For a line of best fit, the equation will be in the form y = mx + b
e.g., W = 7.25 H – 332
Mr. Lieff is 71.5 in tall. His weight would be:
W = 7.25(71.5) – 332
= 186

17. Fathom Activity – Predict your weight as an NHL player!
Click
Under TEAM: Pick your favourite
You can also change Position, Country, Status
Under REPORT: BIOS
Click GO>
Copy the URL
Run FathomFileImportImport From URLPaste
Create a scatter plot of Weight vs. Height
Add a median-median line
Use the equation to:
predict your weight based on your height
Is it accurate? Discuss with a neighbour.
MSIP / Home Learning: p. 51 #1-6, 7 bcd, 8

18. 1.4 Trends in Data Using Technology
Learning goal: Describe and measure the strength of trends
Due now: p. 37 #2, 3, (6-7 or 8)
MSIP / Home Learning: p. 51 #1-6, 7 bcd, 8
use Fathom and Excel

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

20. 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

21. Correlation Coefficient
The correlation coefficient, r, is an indicator of the strength and direction of a linear relationship
r = 0 no relationship
r = 1 perfect positive correlation
r = -1 perfect negative correlation
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 = 0.85, that means that 85% of the variation in y is due to x

22. 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

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

24. 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” or 177.4cm

25. 1.5 Comparing Apples to Oranges

26. 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.

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

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

29. Include every category!

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

31. 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)

32. 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

33. 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.”

34. “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?

35. “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?

36. 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

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