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Introduction to Data Plots Graphing, Plotting and Modeling Data 1 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. 2 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. 3 Introduction to Data Plots Follow these 8 steps to create a graph. (1) Use a ruler to draw the x and y directions for your graph. (2) Label each axis with the independent (x) and dependent (y) variables. (3) Mark each axis with the scale values using an interval that makes it easy to plot each data point. (4) Include units of measure for each scale. 4 Introduction to Data Plots Follow these 8 steps to create a graph. (5) Use dots to plot the data as x, y ordered pairs on your graph. (6) Use a symbol, usually a circle, to enclose the data point. (7) Create descriptive title for the graph and place it at the top of the graph. (8) Place a conclusion state about your plot below your graph. 5 Introduction to Data Plots Plot the following data. day 1 2 4 5 y Number of roses 2 3 Number of roses A simple graphing activity. Dependent Variable Rose Blossom Promoter 6 5 4 3 2 1 1 2 3 4 5 day 5 6 Independent Variable x (1) Use a ruler to draw the x and y directions for your graph. The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter (2) Label each axis with the independent (x) and dependent (y) variables. (3) Mark each axis with the scale values using an interval that makes it easy to plot each data point. (4) Include units of measure for each scale. (5) Use dots to plot the data as x, y ordered pairs on your graph. (6) Use a symbol, usually a circle, to enclose the data point. (7) Create title for the graph and place it at the top of the graph. (8) Place a conclusion state about your plot below your graph. 6 Introduction to Data Plots y Number of roses When do you connect the data points together? Dependent Variable Rose Peddle Blossom Promoter 6 5 4 3 2 1 1 It depends on where the data points came from! 2 3 4 5 day Independent Variable x 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. 7 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 carpenter, it is the rise divided by the run. Slope (pitch) Rise of a roof = Run If you ask a mathematician it is the difference between two Y values divided by the corresponding difference between two X value. Slope of = a line 8 Y 2 Y 1 X 2 X 1 Introduction to Data Plots Graphic Analysis y Slope = x 2 y x 1 1 x 6 Rise= (6-2) Number of roses y Dependent Variable Fundamental calculation for plots 2 5 4 3 2 Using the two ordered pairs; Slope =? = Run = (5-1) 1 1 2 3 4 Rise = Run (6-2) = (5-1) 4 y 5,6 1,2 = 1 4 Using the two ordered pairs; 4 , 5 2,3 5 day Independent Variable x Slope = Rise Run = (5-3) (4-2) = 2 = 2 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. 9 1 Introduction to Data Plots Graphic Analysis 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. y Number of roses Fundamental calculations for plots Dependent Variable Rose Blossom Promoter 6 5 4 3 2 1 1 2 3 4 5 day Independent Variable x If you have experimental data, draw the simplest curve so that all the data points are as close to the curve as possible. Whenever possible, a straight line is the curve of choice. 10 Introduction to Data Plots Graphic Analysis Fundamental calculation for plots day Number of roses 1 2 2 3 4 5 5 6 Do these ordered pairs define a line? (5 , 6) (1 , 2) (4 , 5) (2 , 3) (2 , 3) (1 , 2) Did we do all of the possible slope calculations? (5 , 6) Almost ! No Which pairs did we miss? (4 , 5) (1 , 2) (4 , 5) 11 Slope = Slope = Slope = Slope = Slope = Rise = (6-2) Run (5-1) Rise (5-3) = Run (4-2) Rise (3-2) = Run (2-1) Rise (6-5) = Run (5-4) Rise (5-2) Run = (4-1) = 4 = 1 4 = 2 = 1 2 = 1 = 1 1 = 1 = 1 1 = 3 3 = 1 Introduction to Data Plots Graphic Analysis The following data is from a 2nd rose experiment. Plot the following data. day 1 2 3 4 5 y Number of roses 2 3 5 5 6 Number of roses One more graphing exercise. Dependent Variable Rose Blossom Promoter 6 5 4 3 2 1 1 2 3 4 5 day Independent Variable x (1) Use a ruler to draw the x and y directions for your graph. The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter (2) Label each axis with the independent (x) and dependent (y) variables. (3) Mark each axis with the scale values using an interval that makes it easy to plot each data point. (4) Include units of measure for each scale. (5) Use dots to plot the data as x, y ordered pairs on your graph. (6) Use a symbol, usually a circle, to enclose the data point. (7) Create title for the graph and place it at the top of the graph. (8) Place a conclusion state about your plot below your graph. 12 Introduction to Data Plots Graphic Analysis Good news! You have already decided that the plot is a line so you only have to do one slope calculation. y Number of roses Slope Calculation Dependent Variable Rose Peddle Blossom Promoter 6 5 4 3 2 1 1 2 3 4 5 day Bad news! You have to pick two ordered pairs from this line to use in the slope calculation. Independent Variable x The rose plant had a linear rate of rose production one day after application of Rose Blossom Promoter Good news! You can pick any two orders pair that are easy to use in the slope formula. y Slope = 13 x 2 2 y x 1 1 Introduction to Data Plots Graphic Analysis 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. 14 Introduction to Data Plots Graphic Analysis Rose Peddle Blossom Promoter Experiment # 2 Simple Area Calculation Example y Number of roses 6 5 4 3 2 1 1 2 3 4 day x Calculate the area under the curve between x = 1 and x = 5 15 5 Introduction to Data Plots Rose Peddle Blossom Promoter Experiment # 2 Area Calculation Example y Number of roses 6 5 4 Area 1 3 2 Area 2 1 1 2 3 4 day x Total Area = Area 1 + Area 2 16 5 Introduction to Data Plots Rose Peddle Blossom Promoter Experiment # 2 Area Calculation Example Area 1 is the area of a triangle A1 = 1 b 2 y x h Number of roses 6 5 4 Area 1 3 2 Area 2 1 Area 2 is the area of a rectangle A 2 = Lxw 1 2 3 4 day x Total Area = Area 1 + Area 2 Total Area = A 17 1 + A 2 5 Introduction to Data Plots Rose Peddle Blossom Promoter Experiment # 2 Area Calculation Example 1 = 1 2 b x 6 h 1 A = (5-1) (6-2) 1 2 A 1 = 16 y Number of roses A 5 4 Area 1 3 2 Area 2 1 1 Lx w = 2 A = (5-1)(2-0) 2 2 A A A 2 2 = (4) (2) Total Area = 16 + 8 = Total Area = A = 8 18 1 3 4 day x 5 24 # of roses day + A 2 Introduction to Data Plots Rose Peddle Blossom Promoter Experiment # 2 Area Calculation Example 1 = 1 2 b x 6 h 1 A = (5-1) (6-2) 1 2 A 1 = 16 If the starting numbers have units, the answer units are the product of those units Lx w = 2 A = (5-1)(2-0) 2 y Number of roses A 5 4 Area 1 3 2 Area 2 1 1 2 A A A 2 2 = (4) (2) Total Area = 16 + 8 = Total Area = A = 8 19 1 3 4 day x 5 24 # of roses day + A 2 Introduction to Data Plots Fundamental modeling activities using data plots Interpolation; 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. 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 nor is it between any two of the original x values. 20 Rose Peddle Blossom Promoter Experiment # 2 How many rose blossoms are expected on day 3? 4.1 Rose Blossoms How many rose blossoms actually showed up on day 3? 5 Rose blossoms y 6 Number of roses Interpolation Example Dependent Variable Introduction to Data Plots 5 44.1 3 2 1 1 2 3 4 5 day Independent Variable x Thus, interpolation of the plot indicates we can expect 4.1 or 4 rose blossoms on day 3. 21 Rose Peddle Blossom Promoter Experiment # 2 7.1 7 How many rose blossoms are expected on day 6? 7.1 Rose Blossoms How many rose blossoms actually showed up on day 6? We don’t know! y 6 Number of roses Extrapolation Example Dependent Variable Introduction to Data Plots 5 4 3 2 1 1 2 3 4 5 day Independent Variable x Thus, extrapolation of the plot indicates we can expect 7.1 or 7 rose blossoms on day 7. 22 Introduction to Data Plots Interpolation Extrapolation Technicians, engineers and scientists often: collect data, arrange data as ordered pairs, plot these ordered pairs. 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 ? 23 Introduction to Data Plots 24 Introduction to Data Plots 25