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Determining the Best Line That Fits the Data

3.2.2 Determining the Best Line That Fits the Data

3.2.2 • Determining the Best Line That Fits the Data

Activity

Your teacher will give you a set of cards that show scatter plots.

Arrange all the cards in three different ways. Ensure that you and your partner agree on each arrangement before moving on to the next one. Sort all the cards in order from:

  • best to worst for representing with a linear model
  • least to greatest slope of a linear model that fits the data well
  • least to greatest vertical intercept of a linear model that fits the data well

Know that the line of best fit is the line that provides the best fit to the data. It should follow the trend of the data and go through the middle of the data with a similar number of points on each side of the line.

1.

What is the order of the cards that displays the best to worst representations of a linear model?

Compare your answer:

Your answer may vary, but here is a sample. B, E, A, and F, C, D, and G. There may be some debate about the exact order. It is important that you recognize that B is the best fit, and that D and G should not be fit with linear models. You should be able to reason that linear models fit more poorly the farther the data are from the line.

2.

What is the order of the cards that displays the least to greatest slope representations of a linear model that fits the data well?

Compare your answer:

E, B, A and F, C. D and G do not have linear models that fit the data well.

3.

What is the order of the cards that displays the least to greatest vertical intercept of a linear model that fits the data well?

Compare your answer:

A, C, F, B, E. D and G do not have linear models that fit the data well.

4.

For each card, write a sentence that describes how yy changes as xx increases and whether the linear model is a good fit for the data or not.

Compare your answer: Your answer may vary, but here is a sample.

  • Card A: As xx increases, yy also increases. The linear model fits the data fairly well.
  • Card B: As xx increases, yy decreases. The linear model fits the data very well.
  • Card C: As xx increases, yy increases. The linear model fits most of the data very well; however, there are two values that do not follow this trend.
  • Card D: As xx increases, yy goes up and down like a wave. The data do not follow the line very well at all.
  • Card E: As xx increases, yy decreases. Most of the data fit a linear model very well, but the one point at (7,15)(7,15) does not fit with the rest of the data very well.
  • Card F: As xx increases, yy increases. The linear model fits the data fairly well.
  • Card G: As xx increases, yy decreases and then increases. The data do not follow the line very well at all.

Building Character: Purpose

A person in an orange jacket climbs a ladder toward a glowing star in the sky, surrounded by clouds and birds, symbolizing reaching for goals or aspirations.

Having purpose is showing a commitment to making a meaningful contribution to the world. Purpose drives you to make positive contributions in line with your own interests and strengths.

Think about your current sense of purpose. Are the following statements true for you??

  • I often reflect on my life goals and the kind of person I want to be.
  • I often think about what matters most to me and why it matters.

Don’t worry if none of these statements are true for you. Developing this trait takes time. Your first step starts today!

Self Check

Tell which line is the best fit for the data and give an explanation why it is the best fit.

TWO SCATTER PLOTS WITH X-AXIS LABEL OF INDEPENDENT VARIABLE AND Y-AXIS LABEL OF RESPONSE VARIABLE. SCATTER PLOT ON THE LEFT SHOWS FIVE PLOTTED POINTS AND THE LINE Y EQUALS NEGATIVE NINE-TENTHS X PLUS FORTY-FIVE. SCATTER PLOT ON THE RIGHT SHOWS THE SAME PLOTTED POINTS AND THE LINE Y EQUALS NEGATIVE SEVEN-TENTHS X PLUS FORTY.

  1. Line 1 is the best fit because it has a steeper slope.
  2. Line 2 is the best fit because it goes through the middle of the data with about the same number of data points above and below the line.
  3. Line 2 is the best fit because it touches the last point on the right of the scatter plot.
  4. Line 1 is the best fit because it touches the first point on the left of the scatter plot.

Additional Resources

Draw a Best-Fit Line for the Plotted Data

When looking to draw a line of best fit, first make sure a linear model works for the data.

Next, place the line between the data points so that about the same number of points are above and below the line. The aim is to also have points within a close distance to the line of best fit.

Below are two examples of lines of best fit drawn through scatter plots.

1.

Scatter plot with a line of best fit drawn going through the points so that six points lie above the line and four points lie below the line.

2.

Scatter plot with a line of best fit drawn going through the points so that two points lie above the line, six points lie below the line, and two points lie on the line.

Try it

Draw a Best-fit Line for the Plotted Data

The scatter plots below are the same scatter plot, but each student drew their best fit line differently. Which line do you believe to be the best fit, and why?

Four scatter plots with body mass in pounds on the x-axis and bite force in pounds on the y-axis. Each scatter plot shows the same plotted data, but different lines of fit are drawn on the four plots. All lines increase from left to right.

Compare your answer: Your answer may vary, but here is a sample.

Patti’s line looks like it fits the data well, so it would probably produce good predictions. The line goes through the middle of the points in the scatter plot, with a similar number of points above and below the line that are not a far distance from the line of best fit.