Section 7.1 Scatter Plots & Best-Fitting Lines

Drawing a scatterplot Identify what your “x-values” (horizontal axis) and “y-values” (vertical axis) will be. Identify what your “x-values” (horizontal axis) and “y-values” (vertical axis) will be. Use the range of values to determine an appropriate scale for the axes. Use the range of values to determine an appropriate scale for the axes. Plot points. Plot points.

Best-fitting Lines A “line of best fit” is all about finding a line that seems to fit the general trend of points on a scatter plot. A “line of best fit” is all about finding a line that seems to fit the general trend of points on a scatter plot. Sometimes, a line fits very well, other times it doesn’t. This is where the CORRELATION, and CORRELATION COEFFICIENT comes in – they describe the trend of the data and whether it seems close to a line. Sometimes, a line fits very well, other times it doesn’t. This is where the CORRELATION, and CORRELATION COEFFICIENT comes in – they describe the trend of the data and whether it seems close to a line.

Two ways to describe correlation General terms: General terms: “positive correlation” “positive correlation” “negative correlation” “negative correlation” “no apparent correlation” “no apparent correlation” Correlation coefficient (an actual number) Correlation coefficient (an actual number)

Using the ordered pairs (x, y) below, draw a scatterplot of the data ( 0, 5.1 ) ( 0, 5.1 ) ( 1, 6.4 ) ( 1, 6.4 ) ( 2, 7.7 ) ( 2, 7.7 ) ( 3, 9 ) ( 3, 9 ) ( 4, 10.3 ) ( 4, 10.3 ) ( 5, 11.6 ) ( 5, 11.6 ) ( 6, 12.9 ) ( 6, 12.9 )

Finding the line of best-fit There are several methods for finding “best-fit lines.” We’re going to look at 2 of those methods: visually estimating visually estimating Median-median line Median-median line

Visually estimating Here, we look at the data and guesstimate where the line should go. Then, we use two points on that line to find an equation…they can be an original data pt, but do NOT have to be We use point-slope form to write the equation.

Median-median line 1 st – we’ll review what we learned yesterday. Then we’ll finish the problem w/ step 5 Finally, you’ll try a few examples on your own.

Let’s practice median-median lines Use the following data: (1, 15) (2, 15) (4, 18) (5, 21) (1, 15) (2, 15) (4, 18) (5, 21) (8, 23) (8, 26) (9, 27) (11, 32) (8, 23) (8, 26) (9, 27) (11, 32)

Step #1 Arrange the data from smallest to largest and then divide all the data into three groups. Make the groups the same size, if possible. If not possible, make the first and third groups the same size: Arrange the data from smallest to largest and then divide all the data into three groups. Make the groups the same size, if possible. If not possible, make the first and third groups the same size: (1, 15) (2, 15) (4, 18) (5, 21) (8, 23) (8, 26) (9, 27) (11, 32)

Step #2 Find the median x-value and the median Find the median x-value and the median y-value for each group, separately: y-value for each group, separately: Group 1 Group 2 Group 3 Medianx-value Mediany-value

Step #3 Rewrite the median x-value and y-value as one ordered pair and call it the “summary point” for each group: Rewrite the median x-value and y-value as one ordered pair and call it the “summary point” for each group: Group 1 Group 2Group 3 Group 1 Group 2Group 3 (2, 15) (6.5, 22) (9, 27) (2, 15) (6.5, 22) (9, 27)

Step #4 Find an equation for the line between the two outer points: Find an equation for the line between the two outer points:

Step #5 Move the equation 1/3 of the way toward the middle summary point, as follows: Move the equation 1/3 of the way toward the middle summary point, as follows: Plug in the x-value of the middle summary point and see what your equation says y “should be”—let’s call this the “expected y-value” Plug in the x-value of the middle summary point and see what your equation says y “should be”—let’s call this the “expected y-value” Take 1/3 (actual y-value – expected y-value) Take 1/3 (actual y-value – expected y-value) Add this number to the y-intercept of your equation. Add this number to the y-intercept of your equation. The result is the equation of the median- median line! The result is the equation of the median- median line!

Try this one on your own… (12, 42), (15, 72), (17, 81), (11, 95), (8, 98), (14, 78), (9, 83), (13, 87), (13, 92)

In Order…and in 3 Groups GROUP 1: (8, 98), (9, 83), (11, 95) GROUP 2: (12, 42), (13, 87), (13, 92) GROUP 3: (14, 78), (15, 72), (17, 81)

The Median of Each Group… GROUP 1: (9, 83) GROUP 2: (13, 87) GROUP 3: (15, 72)

Writing the 1 st Equation… Find an equation for the line between the two outer points: Find an equation for the line between the two outer points:

Step #5 Plug in the x-value of the middle summary point and see what your equation says y “should be”—let’s call this the “expected y-value” Plug in the x-value of the middle summary point and see what your equation says y “should be”—let’s call this the “expected y-value” Take 1/3 (actual y-value – expected y-value) Take 1/3 (actual y-value – expected y-value) Add this number to the y-intercept of your equation. Add this number to the y-intercept of your equation.

The answer is…