Transcript A 2

Introduction to Data Plots
Graphing, Plotting and Modeling Data
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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)
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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.
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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
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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 ?
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Introduction to Data Plots
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Introduction to Data Plots
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