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Understanding Linear Regression in Data Analysis

Learn about linear regression to determine the best-fit line through data points, with technology like graphing calculators and Excel. The equation of a straight line is y = mx + b, where m is the slope and b is the y-intercept. Linear regression minimizes deviations from data points. Assess goodness of fit with r2. Discover the impact of setting the intercept to 0. Use Excel for trendline analysis and calibration curves. Explore direct and indirect relationships between variables.

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Understanding Linear Regression in Data Analysis

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  1. The Best-Fit Line Linear Regression

  2. How do you determine the best-fit line through data points? Fortunately technology, such as the graphing calculator and Excel, can do a better job than your eye and a ruler! y-variable x-variable

  3. The Equation of a Straight Line y = mx + b where m is the slope or Dy/Dx and b is the y-intercept In some physical settings, b = 0 so the equation simplifies to: y = mx

  4. Linear regression minimizes the sum of the squared deviations y = mx + b y-variable deviation = residual = ydata point – yequation x-variable

  5. Linear Regression • Minimizes the sum of the square of the deviations for all the points and the best-fit line • Judge the goodness of fit with r2 • r2 x100 tells you the percent of the variation of the y-variable that is explained by the variation of the x-variable (a perfect fit has r2 = 1)

  6. r2 is low r2 is high Goodness of Fit: Using r2 y-variable How about the value of r2? x-variable

  7. Strong direct relationship 99.1% of the y-variation is due to the variation of the x-variable

  8. Noisy indirect relationship Only 82% of the y-variation is due to the variation of the x-variable - what is the other 18% caused by?

  9. When there is no trend! No relationship!

  10. In Excel • When the chart is active, go to chart, and select Add Trendline, choose the type and on option select display equation and display r2 • For calibration curves, select the set intercept = 0 option Does this make physical sense?

  11. Does the set intercept = 0 option make a difference? Using the set intercept = 0 option lowers the r2 value by a small amount and changes the slope slightly

  12. The equation becomes A = mc or A = 0.89c 99.1% of the variation of the absorbance is due to the variation of the concentration.

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