Line of Best fit, slope and y-intercepts

MAP4C

Best fit lines

0A line of best fit is a line drawn through data points that represents a linear relationship between 2 variables.

0This is also called a trend line or regression line.0The line of best fit isn’t just drawn through the middle of the

data: it needs to be closest to each point.0 In a scatter plot, a point that lies off the best fit line is called

an outlier.0Outliers may be from bad data, an error or is just unusual.

(The outliers should be considered in a line of best fit!)

The effects of outliers

0Which graph is the line of best fit? Justify your choice.

A B C0Graph C is the best line of best fit.0The two outliers pull the line down slightly, not as much as

in Graph B. 0Graph A ignores outliers.

How do you know if the data is linear??

0A linear model may be unreliable if:a) There are too few data points.b) The model is based on data clustered together.c) There does not appear to be any correlation.d) There are outliers.e) The data do not appear to be linear.

Interpolation and Extrapolation

0A line of best fit can be used to predict values.0Estimating values within a set of data is called

interpolation.0Estimating values outside a set of data is called

extrapolation. (Usually your best fit line needs to be extended)

Example and Review!

0 In a grade 12 math class: pre-exam marks and exam marks were collected for some students as shown:

a) Graph the data and draw a line of best fit.b) Determine the equation of the line of best fit.c) Use the data to predict the exam mark of a student with

a pre-exam mark of 98% and 10%d) Estimate what pre-exam mark would best match an

exam mark of 65%

Term mark (%) 84 76 70 95 92 61 25 55 51 73 62

Exam mark (%) 80 72 68 96 90 58 29 60 53 77 67

Example and Review!a) Plot the points. Place a ruler on the graph so that it comes

close as possible to all the points. Draw a line along the ruler’s edge; connected to the y-axis.

b) The equation of a line is y = mx + b. Choose two points to find the slope:

b is read off the graph at ~7The equation is y = 0.9x + 7

c) Extrapolating: A pre-exam mark of 98% would get ~95% on the exam. If 10%: the exam mark would be ~16%.

d) Interpolation: An exam mark of 65% (read off y-axis) would yield a pre-exam mark of ~63%

P.S.

0 Instead of using y and x, other letters may be used to represent equations.

0We could have used P and E for the last example to give: E = 0.9P + 7

0We also could have calculated extrapolated or interpolated data by subbing in numbers into our equation: For part c) for P = 10%.......

E = 0.9P + 7= 0.9(10) + 7

= 9 + 7 = 16%

Text work

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