Unit 2.4 - Manual Graphing and Linear Regression - Student PDF

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MANUAL GRAPHING AND LINEAR REGRESSION Reading: Textbook, Chapter 7, p 147

Graphing Data 2

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A picture is worth a thousand words…

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Therefore, a ________ represents a set of data better than a ________

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Graphs reveal ________ in the data  Take a look at the graphs on the previous slide!

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Manual Graphing 3

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Let’s review the steps and rules for manual graphing (graphing by hand)

Manual Graphing Steps 4

1. Use _______________ graph paper (paper ruled with 10 lines per centimetre). 2. Define and the x and y variables, and label the axes with the name of the variable and its units.  _______: Dependent variable  _______: Independent variable

Absorbance

y

x Concentration of Safranin (ppm)

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Manual Graphing Steps 5

1.000

2.400

0.800

2.000

Absorbance

Which graph has better scales for x and y axes?

Absorbance

3. Set up the __________ on both axes  Set upper and lower values for each axis  Scales should be chosen such that the plot fills the sheet of graph paper as much as possible

0.600 0.400 0.200

20.0

40.0

60.0

80.0

Concenration of Safranin (ppm)

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0.800

0.000 0.0



1.200

0.400

0.000



1.600

0.0

50.0

100.0

150.0

200.0

Concenration of Safranin (ppm)

Divisions of scale must be uniform/linear (easy to interpret)  Can use divisions of 1, 2, 5, 10, 20, 50, 100, etc. but not divisions of 3.5, 4.75, etc. Use tick marks along the scale with the division values Significant figures for values on the axes should be consistent with the number of significant figures in your plotted values

Manual Graphing Steps 6

4. Plot the data points  Use symbols to define the data points. Single dots get lost on the graph.

5. Include the ____________________ of the graph. □ Goes on bottom of the graph □ General format of title: y variable versus x variable □ Readers should be able to understand what the graph represents without reference to the text  Example: Figure #1. Plot of absorbance versus concentration of Safranin (ppm).

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Manual Graphing Steps 7

6. Draw the line of best fit  If the relationship between the x and y variables is linear, draw a continuous __________ line of best fit with a ruler  Do not connect the data points together!  If line is extended past range of measured values it should be ___________, not solid  Note that not all relationships between x and y variables are ___________. If the data is non-linear, do NOT fit it to a straight line. 7. Determine the equation of the line of best fit □ Equation of the line: ______________  b is the _____________  m is the __________

Manual Graphing Steps 8

□ The y-intercept is read from the graph □ Slope is calculated using:

 

 ∆         ∆ 

□ To calculate _________, select any _____________ on the

graph: (x1,y1) and (x2,y2)  Do NOT choose experimental data points  Points must lie on the line of best fit  Points should be separated far apart from each other  Show slope calculation directly on the graph 

Write the equation of the line directly on the graph

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Example of a Bad Manual Graph Can you find all of the errors on this graph?

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Example of a Good Manual Graph

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Linear Regression 11

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Determination of the equation of the line from a manual graph using rise/run is a rough method (an estimate). This method is only as accurate as the line of best fit that was drawn.  Determination of line of best fit is subject to ________________.

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In the computer lab, you will learn how to plot data in __________ and perform linear regression to determine the equation of the line. Most of the rules for formatting manual graphs also apply to Excel graphs

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Linear Regression 12

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Mathematical calculation that is used to determine the equation of a straight line is known as the ________________________________

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Straight line is produced such that the square of the distance (residual) in the y-direction of each data point to the line is a ___________________ Each data point is given _________________ in this calculation

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y

x

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Linear Regression: Method of Least Squares 13

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Two assumptions of the least squares analysis: 1. The data falls on a ____________ line → Linear relationship exists between the x and y variables. 2. Any data point that falls off of the line of best fit is a result of ___________ (indeterminate) errors in measurement of the ________________ → Assume that NO determinate (systematic) errors occurred with regards to measurement of x variable.

Linear Regression: Method of Least Squares 14

The equations developed to calculate slope (m) and yintercept (b) using least squares analysis are:

Sxx    xi  x 

2

Sxy    xi  x   yi  y  m

S xy S xx

b  y  mx

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Linear Regression: Method of Least Squares 15

Practice Problem: Use the Method of Least Squares to determine the equation of the line for the data set given below. Concentration of Safranin (ppm) 10.0

Absorbance

20.0

0.230

40.0

0.399

60.0

0.605

80.0

0.850

0.126

Linear Regression: Method of Least Squares 16

The best way to calculate slope (m) and y-intercept (b) is to set up the data in table format: 



     

     

󰇜 󰇛 

󰇜󰇛     󰇜 󰇛 

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Linear Regression: Method of Least Squares 17

0.900 0.800

Absorbance

0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 0.0

20.0

40.0

60.0

80.0

100.0

Concentration of Safranin (ppm)

Figure #1. Plot of absorbance versus concentration of Safranin (ppm).

References 18

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Ballinger, J.; Shugar, G. Chemical Technicians' Ready Reference Handbook, 5th ed.; McGraw Hill: New York, 2011; Chapter 7, p 147. Dykes, P. CHEM10371 Lecture on Graphing Data. Presented at Sheridan College, Brampton, ON, Fall 2012. Lindberg, V. Graphing by Hand and on Computer, July 1, 2010. Department of Physics, Rochester Institute of Technology. http://www.rit.edu/~w-uphysi/graphing/graphingpart1.html (accessed Feb 8, 2014).

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