1 Introduction to Data Plots Graphing, Plotting and Modeling Data

2 Introduction to Data Plots A plot is an important way to graphically analyze or determine relationships among data. Graphs must be constructed properly to avoid misinterpretation by others. Graphs must tell us something meaningful.

3 Introduction to Data Plots Take your time selecting the scale for each axis. The entire look and ease of understanding a graph depend on the scale of the axis. The scale must be broad enough to include all of the data that is to be plotted. The scale does not have to go beyond the values of data that are to be plotted.

4 Introduction to Data Plots Use a ruler to draw the x and y directions for your graph. Label each axis with the independent (x) and dependent (y) variables. Mark each axis with the scale values using an interval that makes it easy to plot each data point. (1) (2) (3) Follow these 8 steps to create a graph. Include units of measure for each scale.(4)

5 Introduction to Data Plots Follow these 8 steps to create a graph. Use dots to plot the data as x, y ordered pairs on your graph. Use a symbol, usually a circle, to enclose the data point. Create descriptive title for the graph and place it at the top of the graph. (6) (7) Place a conclusion state about your plot below your graph. (8) (5)

6 Introduction to Data Plots A simple graphing activity. Use a ruler to draw the x and y directions for your graph.(1) Label each axis with the independent (x) and dependent (y) variables.(2) Mark each axis with the scale values using an interval that makes it easy to plot each data point.(3) Dependent Variable y Include units of measure for each scale.(4) Plot the following data dayNumber of roses Independent Variable x day Use dots to plot the data as x, y ordered pairs on your graph.(5) Use a symbol, usually a circle, to enclose the data point. Create title for the graph and place it at the top of the graph. (6) (7) Place a conclusion state about your plot below your graph.(8) Rose Blossom Promoter The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter

7 Introduction to Data Plots Independent Variable Dependent Variable x y Number of roses day It depends on where the data points came from! If the numbers came from an equation, you can connect the data points. If the numbers came from an experiment, try to draw the simplest curve that has the smallest distance possible away from the data points. When do you connect the data points together? Rose Peddle Blossom Promoter

8 Introduction to Data Plots Fundamental calculation for plotting data Slope – a value that indicates the vertical to horizontal change in the data between two data points. Area – a value that indicates the area under the curve between two selected x values. Slope –like the slope of a hill, a graph’s slope indicates how much the data is going up or down between these two data points. If you ask a mathematician it is the difference between two Y values divided by the corresponding difference between two X value. Rise Run If you ask a carpenter, it is the rise divided by the run. X 1 X 2 Y 1 Y 2 Slope (pitch) of a roof = Slope of a line =

9 Introduction to Data Plots Fundamental calculation for plots Independent Variable Dependent Variable x y Number of roses day Run = (5-1) Rise= (6-2) = 1 Using the two ordered pairs; 5, 6 1, 2 Slope Using the two ordered pairs; 2, 3 4, 5 = Run Rise = (4-2) (5-3) = 2 2 = 1 Slope If the slope value is positive, the data points are going up hill. If the slope value is negative, the data points are going down hill. If the slope value is zero, the data points are horizontal. = xy x 1 y 1 y 2 x 2 Slope Graphic Analysis =? = Run Rise = (5-1) (6-2) = 4 4 Run Rise = (4-2) (5-3) (5-3) = 2 2 = 1

10 Introduction to Data Plots A line is a curve that has the same value for the slope calculation when any data point ordered pair is used in the calculation. Independent Variable Dependent Variable x y Number of roses day Rose Blossom Promoter Whenever possible, a straight line is the curve of choice. If you have experimental data, draw the simplest curve so that all the data points are as close to the curve as possible. Fundamental calculations for plots Graphic Analysis

11 Introduction to Data Plots Slope Fundamental calculation for plots Slope= Run Rise = (5-4) (6-5) = 1 1 = 1 (5, 6) (4, 5) (2, 3) (4, 5) = Run Rise = (4-2) (5-3) = 2 2 = dayNumber of roses Slope= Run Rise = (2-1) (3-2) = 1 1 = 1 (2, 3) (1, 2) (4, 5) Slope= Run Rise = (4-1) (5-2) = 3 3 = 1 = Run Rise = (5-1) (6-2) = 4 4 = 1 (5, 6) (1, 2) Do these ordered pairs define a line? Did we do all of the possible slope calculations? No Almost ! Which pairs did we miss? Graphic Analysis

12 Introduction to Data Plots One more graphing exercise. The following data is from a 2 nd rose experiment. Plot the following data dayNumber of roses 56 Use a ruler to draw the x and y directions for your graph.(1) Label each axis with the independent (x) and dependent (y) variables.(2) Mark each axis with the scale values using an interval that makes it easy to plot each data point.(3) Include units of measure for each scale.(4) Use dots to plot the data as x, y ordered pairs on your graph.(5) Use a symbol, usually a circle, to enclose the data point. Create title for the graph and place it at the top of the graph. (6) (7) Place a conclusion state about your plot below your graph.(8) 35 Independent Variable Dependent Variable x y Number of roses day Rose Blossom Promoter The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter Graphic Analysis

13 Introduction to Data Plots Slope Calculation Independent Variable Dependent Variable x y Number of roses day Rose Peddle Blossom Promoter The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter Good news! You have already decided that the plot is a line so you only have to do one slope calculation. Bad news! You have to pick two ordered pairs from this line to use in the slope calculation. Good news! You can pick any two orders pair that are easy to use in the slope formula. = x 1 y 1 y 2 x 2 Slope Graphic Analysis

14 Introduction to Data Plots Fundamental calculation for plots Slope – a value that indicates the vertical to horizontal change in the data between two data points. Area – a value that indicates the area under the curve between two selected x values. Area – The area under a plot is a number that is directly related to the sum of all of the (x,y) pair products of points on the curve between the first x and last x value. Since there are an infinite number of x,y pairs on a curve between two points on the curve, it is usually easier to find the area under the curve than to add the x times y product for every possible (x,y) pair on the plot. Graphic Analysis

15 Introduction to Data Plots y Number of roses day Rose Peddle Blossom Promoter Experiment # Simple Area Calculation Example Calculate the area under the curve between x = 1 x and x = 5 Graphic Analysis

16 Introduction to Data Plots y Number of roses day Rose Peddle Blossom Promoter Experiment # Area Calculation Example Total Area = x Area 1 +Area 2 Area 1 Area 2

17 Introduction to Data Plots y Number of roses day Rose Peddle Blossom Promoter Experiment # Area Calculation Example Total Area = x Area 1 +Area 2 Area 1 Area 2 Area 1 is the area of a triangle Area 2 is the area of a rectangle A 1 = 1 2 h x b A 2 = x L w Total Area = A 1 + A 2

18 Introduction to Data Plots y Number of roses day Rose Peddle Blossom Promoter Experiment # Area Calculation Example Total Area = x Area 1 Area 2 A 1 = 1 2 h x b A 2 = x L w Total Area = A 1 + A (5-1) (6-2) A 1 A 1 = A 2 (5-1)(2-0) A 2 = (4)(2) A 2 = 8 =24# of roses day 16 = =

19 Introduction to Data Plots y Number of roses day Rose Peddle Blossom Promoter Experiment # Area Calculation Example Total Area = x Area 1 Area 2 A 1 = 1 2 h x b A 2 = x L w Total Area = A 1 + A (5-1) (6-2) A 1 A 1 = A 2 (5-1)(2-0) A 2 = (4)(2) A 2 = 8 =24# of roses day 16 = = If the starting numbers have units, the answer units are the product of those units

20 Introduction to Data Plots Fundamental modeling activities using data plots Interpolation; Extrapolation; Predicting what the dependent variable (y) value will be for an independent variable (x) value that is not one of the original x values but is between any two of the original x values. Predicting what the dependent variable (y) value will be for an independent variable (x) value that is not one of the original x values nor is it between any two of the original x values.

21 Introduction to Data Plots Independent Variable Dependent Variable y Number of roses day Rose Peddle Blossom Promoter Experiment # x Interpolation Example How many rose blossoms are expected on day 3? 4.1 Rose Blossoms Thus, interpolation of the plot indicates we can expect 4.1 or 4 rose blossoms on day 3. How many rose blossoms actually showed up on day 3? 5 Rose blossoms

22 Introduction to Data Plots Independent Variable Dependent Variable y Number of roses day Rose Peddle Blossom Promoter Experiment # x Extrapolation Example How many rose blossoms are expected on day 6? 7.1 Rose Blossoms Thus, extrapolation of the plot indicates we can expect 7.1 or 7 rose blossoms on day 7. 7 How many rose blossoms actually showed up on day 6? We don’t know!

23 Introduction to Data Plots Technicians, engineers and scientists often: arrange data as ordered pairs, collect data, Then, they interpolate or extrapolate as a modeling tool to determine y values they did not do experiments for. Is it a better idea to interpolate a data plot or to extrapolate a data plot? Why ? InterpolationExtrapolation plot these ordered pairs.

24 Introduction to Data Plots

25 Introduction to Data Plots