1.3 trends in data

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?

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) = 2500 * 1/500 = 5 So you are expected to win $5 per $100 ticket. You are better off taking your $100 to a blackjack table where

E(X) = 98.5!

1.3 Trends in Data

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

Variables Variable (Mathematics)

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

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

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

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

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)

An Example of a trend

U.S. population from 1780 to 1960

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Correlations Strength can be…

None – no clear pattern in the data Weak – data loosely follows a pattern Strong – data follows a clear pattern

If strong or weak, 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 http://www.seeingstatistics.com/seeing1999/gallery/Corr

elationPicture.html

Strong, positive linear correlation

AGENDA for Fri-Mon 1.3 Median-Median Line

Using a regression equation Fathom Activity - Predict your weight as an NHL

player 1.4 Trends With Technology

Correlation Coefficient (R) Coefficient of Determination (R2) Residuals / Least-Squares Line

Fathom Investigation: finding the Least Squares Line

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 next class!)

An example line of best fit

this is temperature recycling data with a median-median line added

what type of trend are we looking at?

Median-Median Line

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

Median-Median Line (10 points)

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)

Using a regression equation

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

= 201.075 or 201 lbs.

Fathom Activity – How much would you weigh as an NHL player? To Generate and Import Data: Click http://www.nhl.com/ice/playerstats.htm

Pick a group of players that you want to associate with TEAM: Pick your favourite OR select Position, Country, Status, etc.

Select REPORTBIOS Click GO> Copy the URL

Open Fathom Click FileImportImport From URL Paste the URL Double-click the Collection name and shorten it Expand the Collection, right-click the first case and click Cut Case.

To create a graph of Weight vs. Height Create a scatter plot of Weight vs. Height

Double click the Collection icon (cardboard box) Click the Cases tab Create a graph in the workspace Drag Weight and Height to the respective axes

Which is dependent?

Right-click and select Median-Median Line Use the equation to:

Predict your weight based on your height Discuss with a neighbour: is the prediction

reasonable? Are there any limitations to the model? Extension: How would you predict your NHL height

based on your current weight?

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

1.4 Trends in Data Using Technology

Learning goal: Describe and measure the strength of trendsQuestions? p. 37 #2, 3, (6-7 or 8)MSIP / Home Learning: p. 51 #1-2, 3-5 (Fathom), 8

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

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 = 174.25 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 = 158.6 cm

Correlation Coefficient 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 r is 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

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

y

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Collection 1 Scatter Plot

http://www.math.csusb.edu/faculty/stanton/m262/regress/

Least Squares LineWeight vs. Height (NHL) w = 7.23h – 325

Using the equation

How much does a player who is 71 in tall weigh?

w = 7.23(71) – 325 = 188.33 lbs

How tall is a player who weighs 180 lbs? w = 7.23h – 325 h = (w + 325) ÷ 7.23 So h = (180 + 325) ÷ 7.23 = 69.85” or 177.4cm

NHL Least-Squares Line Activity See handout

1.5 Comparing Apples to Oranges http://www.smarter.org/research/apples-to-

oranges/

The Power of Data

Chapter 1.5 – The MediaMathematics of Data Management (Nelson)MDM 4U

There are 3 kinds of lies: lies, damn lies and statistics.

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

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Example 1 – Scale from 0 Consider that this is a bar graph – could it

still be misleading?

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Include every category!

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Example 2 – Using a Small Sample For the following surveys, consider:

The sample size If there is any (mis)leading language

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)

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

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.”

“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?

“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?

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

MSIP / Homework

Read pp. 57 – 60 Ex. 1-2 Complete p. 60 #1-6 Final Project Example – Manipulating Data

(on wiki)

Examples http://junkcharts.typepad.com/ http://www.coolschool.ca/lor/AMA11/unit1/U01L02.htm http://mediamatters.org/research/200503220005