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

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

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

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

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

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

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 http://www.seeingstatistics.com/seeing1999/gallery/CorrelationPicture.html Strong, positive linear correlation

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!)

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?

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

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

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

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.)

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

Fathom Activity – Predict your weight as an NHL player! Click http://www.nhl.com/ice/playerstats.htm 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

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

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

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 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

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

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

Using the equation How much does a player who is 71 in tall weigh? = 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

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

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.

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

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

Include every category!

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