Graphing Standards

7/25/2019 Graphing Standards

1/20

1 General

1.1 Graphs must be plotted on proper graph paper or on plain white paper if done on the computer.

Special graph paper for grading curves and liquid limit flow curve are supplied on the instructor's

website. It is recommended that these be used to plot the data manually. For manual plots on linear,

scaled axes proper graph paper, available in any stationery or drafting supply store, MUST be used.

1.2 Each graph must have a unique title that states the variables plotted or type of graph and the material(s)

tested.

1.3 When the graph is to be included as part of a technical report, it is necessary and desirable toannotate the graph as a Figure (eg., Figure 1, Figure 2, etc.). This makes it easier to discuss the

graph in the text of the report (eg., "... the effect of increasing the water-cement ratio is to decrease

the compressive strength of the concrete as illustrated in Figure 1 ...").

2 Reasons for Graphing

2.1 Some apparati require calibration graphically (eg. Pycnometers, hydrometers). In these instances it is

desirable to use Excel's plotting capabilities to determine a calibration equationby fitting a

trendline to the calibration data.

2.2 Some test procedures require graphical solutions to determine or illustrate material properties.

For example, the Liquid Limit of clay is determined by plotting a Flow Curve (water content versus

the logarithm of the number of drops) or the grain size distribution of coarse aggregate which is

obtained by plotting percent passing versus the logarithm of particle diameter.

2.3 Many of the graphs plotted are to illustrate relationships between the dependent and independent

variable. In these cases is is highly desirable to use Excel's plotting capabilities to determine an

equation for the relationship by fitting a trendline to the plotted data.

3 Creating Excel Graphs

3.1 The vast majority of graphs plotted for marterials and geotechnical labs are scatter plots. These are

launched in Excel by clicking on the chart icon ( ) or choosing "Chart" from the "Insert"

pulldown menu:

W. Kirk Smeaton 1 of 20

Building and Construction Sciences Department

Graphing Standards for Laboratory Reports


2/20

3.1 Continued

Next, the chart type is selected:

Select only the chart sub-type shown.

3.2 When "Next" is selected, the next window is used to specify the ranges of columns within the

spreadsheet containing the independent variable (x) and the dependent variable (y).





3/20

3.3 To illustrate the remaining steps, the following sample data will be used:A B C D E F G H I

12 Diameter Series # %Passing

13 0.074 0%

14 CA FA MF 0.074 #200 100%

15 " 19.1 100.0% 100.0% 100.0% 0.149 0%

16 " 12.7 99.9% 100.0% 100.0% 0.149 #100 100%

17 " 9.52 91.8% 100.0% 100.0% 0.297 0%

18 #4 4.76 12.8% 97.2% 100.0% 0.297 #50 100%

19 #8 2.38 6.4% 91.0% 100.0% 0.59 0%

20 #16 1.19 5.0% 82.5% 100.0% 0.59 #30 100%

21 #30 0.59 4.2% 65.4% 100.0% 1.19 0%22 #50 0.297 3.4% 33.4% 90.5% 1.19 #16 100%

23 #100 0.149 2.7% 11.6% 69.0% 2.38 0%

24 #200 0.074 2.4% 5.8% 43.0% 2.38 #8 100%

25 Series #: CA FA MF 4.76 0%

26 4.76 #4 100%

27 9.52 0%

28 9.52 " 100%

29 12.7 0%

30 12.7 " 100%

31 19.1 0%

32 19.1 " 100%

3.4 Select "Add". Specify "Name:" as the series

name or label. Place cursor into "Name:" window,

click on the specify icon ( ) and then place

cursor in cell D15 on spreadsheet (where series

label resides) and then click on select icon ( ).

Repeat process for "X Values:" and "Y Values:",

specifying the ranges explicitly.


Sieve

Designation



Diameter

(mm)

%Passing


4/20

3.5 Select "Add" again and repeat for FA and MF. Add series for the vertical lines that mark thesieves in the series.

You think this looks crazy ...

wait 'til you see it with titles and labels!

3.6 Choose "Next" and the preliminary formatting is done: Titles, Axes, Gridlines, Legend and Data Labels.

First add the Graph Title and Axis

Titles.

Most of the time you can skip Axes.





5/20

3.7 Then specify major AND minorgridlines for both axes. This is notcarved in stone!

Since there are three different aggregates

each with its own curve, the reader

(and you) needs to be able to distinguish

among them so specify to "Show legend".

To begin with, place it along the

bottom (again, this can be changed).

For data labels, we need to label the

sieve designation and actual size of

opening, so select "Series name" and

"X Value". The default location for labels

is to appear to the right of the

plotted point.

3.8 Finally, select "Next" and the last chart building window appears.





6/20

3.8 Continued

It is recommended that you

always select that the

new chart be embedded

as an object in the

spreadsheet.

3.9 VIOLA! Your chart! Yes, I know it looks ridiculous. So read on!

4 Editing Excel Graphs

4.1 As you can see, Excel doesn't understand what we want aesthetically in a graph. In fact, Excel'sgraphing capabilities seem to have been developed for Business and Marketing applications as

oppposed to engineering and science. To make the graph recognizable to a technoperson like yourself,

you need to first right-click just above the legend, on the blank area (if you hit an element of the graph,

like an axis title, a box will appear around it and an element-specific menu will pop up). The menu

you want to see is called the "Format Chart Area" popup menu shown on the next page.





7/20

4.2 The "Format Chart Area" menu allows you to change any of the settings

you chose in creating the chart that you now don't like.

4.3 The size of the lettering that Excel uses as default may be a tad overstated,

as if the words are the most important feature on the plot! If you right-click

on the title, you can change the fonts, size and styles.

If you left click on the title, you can edit the text, and if you select it you can

change style and font over specific ranges. I have reduced the size to 12 point,

(was 16) changed the style to regular (was Bold) and underlined it.

4.4 The % Passing scale is shown from 0 to 150% but the data ends at 100%. The

numbers also have one decimal place and the size of the numbers is too big...

all these things take up space! By double left-clicking on any number in the vertical

scale, you can make some serious changes from this popup menu:

4.5 The axis lines and ticks are not at issue so we start with

"Scale":

If we change the "maximum" value to 1, then the "Major Unit" (posted numerical interval) changes to 0.2

and the "Minor Unit" (horizontal gridline interval) changes to 0.04. This means the numbers displayed

will be the 0%, 20%, 40%, 60%, 80% and 100% and the interval between gridlines will be 4%. This is too

small so let's change it to 10%.

Selecting "Font" we can change the size from 12 point to 9 point and the style from Bold to Regular.

Selecting "Number" we can change the precision to 0 decimal places.

4.6 The "Alignment" is fine as it is (Excel got somethingright!)





8/20


9/20

4.8 Well, its starting to look a bit better but does that grey fill in the plot area enhance our

understanding of the graph? Does it make it any clearer? Obviously not and this is why graphs

produced for engineering and technical purposes should NOT use fill. To remove the fill and change

the border to black, right click in a cell on the graph, taking care not to touch a gridline or plot element.

Choose "Format Plot Area"

Change "Area" to "None".

Change "Color" to "Automatic" (which is black).

4.9 All those data labels are now whats giving us grief. Let's reposition them so that they appear above

the plotted points. To do this, right click on one of the MF labels and a Data Labels menu pops up:





10/20

4.9 Continued

All the data labels for the MF data series are highlighted, which allows you to make the change to

all of them at once. Choosing "Format Data Labels", another menu appears which allows you to

modify the label's "Patterns" (border and

background), the "Font", the "Number" format and

the "Alignment". The default settings are fine for

patterns (no border, no background) but choosing

"Font", we can set the size to 8 point and choosing

"Number", change to "General". Choosing

"Alignment" as shown we select "Above".

This is repeated for each data series (yes, it

takes a while).

4.10 Next we can edit the individual data series by right-clicking on any plot symbol in a series.

For example, the right-clicking on MF series produces this popup:

Choosing "Format Data Series", the Marker Background colour is

first set to "No Color" as shown below:





11/20

4.10 Continued

Next, the highly ambiguous marker ( ) is

converted to a more precise marker, by

changing the "Foreground" colour to black

and then selecting , or . The size

of the markers is also augmented to 7 for

better visibility.

The line is changed from "None" to a

continuous line with a slightly heavier

weight, colour is set to black.

This is repeated for the CA and FA series

only using a line style of short dashes for

the FA and long dashes for the CA.

The result is the graph shown below, now its starting to actually look like a graph. However, the labels

that remain have too much information in them. The intention was to have the sieve designation above

the series and the sieve opening (mm) below. Hence, by right-clicking on each label, the unwanted

information can be eliminated and the label positioned explicitly.





12/20

4.11 Now the remaining data series are edited,

again by right-clicking on one of the two

markers in each series and first setting

the "Marker" to "None", then changing the

style of the "Line" to short dashes,

the colour to dark blue and the weight

increased slightly. In this way a vertical line

is plotted for each sieve size.

There should only be 3 series in the legend:

CA, FA and MF. By left-clicking on the

legend label (not the line or marker) the

entire legend entry is framed. Then by

right-clicking on the framed entry, the

following menu pops up:

Choosing "Clear" as shown will eliminate the entry from the legend. If you framed only the marked and

line of the entry you may find that the labels for that line also disappears from the plot so it's a good

idea to check this with every entry cleared. The text size for the remaining entries is then reduced

by left-clicking on the legend box and using the format legend popup menu. The legend now is small

enough to reposition as shown below:





13/20

4.12 As an alternative to using a legend, the drawing toolbar can be used to label the curves explicitly:

First, select the entire graph, click on the textbox icon ( ) and then draw a text box somewhere

on the graph. If there is a border and/or background in the textbox you can remove these by using

the format textbox menu that is activated by left-clicking in the textbox and then right-clicking

on the highlighted border. The text can be entered as soon as the textbox is created.

Leader lines could then be drawn using the arrow icon ( ). These are illustrated on the final

plot of this example graph on the next page.

4.13 The graph created, embedded in the spreadsheet, can be presented as part of the spreadsheet

printout or it can be printed separately. The embedded graph is in portrait orientation unless

changed in the print preview to landscape. All the example plots shown so far are in portrait.

The elements of the graph can be moved around explicitly as can the plot dimensions.

Some plots should be plotted in portrait using as much of the vertical space on the page as possible,

especially when the plotted data is then used manually for a graphical solution.

When a chart is plotted in isolation from the spreadsheet, the chart is selected first by left-clicking

near a corner of the chart and then selecting print preview ( ). The chart is then shown in

landscape orientation as shown on the next page. Choosing print will then send the chart to the printer

and it will print in landscape orientation.

5 Fitting Curves to Data

5.1 There are many times when mathematical models are required of the data plotted. Excel has the

capability of performing "best fit" analysis of the data plotted to produce an equation that can be

used to estimate the y-axis variable for given values of the x-axis variable. Excel calls these

"Trendlines". Several types of mathematical models are available (linear, logarithmic, polynomials,

power functions, exponentials and moving average) and the equations and R2statistic* can be

displayed.

The problem is that Excel doesn't distinguish between "good" data and "outliers". Quite often,

especially when using the data from and entire class, there are one or two points that are so wrongthat they totally distort the mathematical model produced by the "Trendline" feature.

(*) An R2value of 100% or 1.0 means that the data matches the mathematical model perfectly. As the scatter of

the data around the model increases, the R2value decreases.





14/20





15/20

5.2 To illustrate the points discussed in section 5.1, another example is used.

A B C D E F G H

7 % AC Gmm Gmb Stability Pa VMA Flow

8 4.5 2.484 2.339 3.733 5.86 14.30 16.2

9 5.0 2.469 2.349 4.029 4.86 14.40 20.2

10 5.5 2.448 2.377 3.284 2.90 13.80 30.8

11 6.0 2.427 2.375 3.714 2.15 14.30 30.2

12 6.5 2.412 2.363 3.663 2.04 15.20 35.7

13 7.0 2.395 2.359 3.088 1.50 15.80 48.0

Using %AC (Percent Asphalt Cement) as the independent variable (x-axis), each of the other

variables is plotted as the dependent variables (y-axes) using the steps and conventions illustrated

in the previous example. These graphs are used to determine an optimum % AC and so it is more

useful to display them all on one page as shown on the following page.

Each graph can be positioned accucately within the page by holding down the ALT key while moving it

or resizing; this technique makes graph borders coincide with cell boundaries within the spreadsheet.

5.3 To illustrate the fitting of mathematical models to the plotted data, we will focus on the middle

two plots (Stability versus %AC and Percent Air Voids versus %AC).

Right-clicking on the 6.5% AC The "Add Trendline - Type" menu then appears. One

point generates the popup menu of the six options is selected. This defines the form

shown. of the mathematical model to be used to fit to the data.

Choose "Add Trendline". A parabola (2ndOrder Polynomial) is selected. Before

selecting "OK" however, the "Options" menu should be

selected to specify display of the equation and R2.





16/20

5.2 Continued





17/20

5.3 Continued

The "Options" menu allows you to specify that

the equation of the trendline and its R2value

should be printed on the graph.

Another very useful option is to set the intercept

to 0 for functions that should logically pass

through the origin.

The graph will then look like this.

The title and equation can be explicitly repositionedto improve the readability.

5.4 The R2value is only 42% and the reason should be obvious: the 5.5% AC point is way too low.

This could be due to the testing conditions, if for example the specimens spent too much time

in the water bath, thus making them too weak. The Flow plot on the previous page reinforces the

"too soft" theory as the 5.5% point appears to be too high. This is an instance where an outlier in the

data is distorting the results. The solution is to separated the outlier from the rest of the data but

still plot it on the graph by creating a series of its own. J K

7 % AC Stability

Some adjustments are required to the data. 8 4.5 3.733

The data is copied into two new columns (J and K) but the 5.5% 9 5.0 4.029

data is extracted and moved to the bottom as shown. 10 6.0 3.714

The old graph is copied (right-click in a blank portion of the graph) 11 6.5 3.663

and the new graph is edited by redefining the source data and 12 7.0 3.088

trendline. 13 5.5 3.284





18/20

5.4 Continued

Redefined Source data for Stability Series Source data for Outlier Series

5.5 The resulting graph shows the new trendline with

an R2of 92%. That sure beats 42%! Note that

although the important thing may seem to be the

trendline, we also need to see the data points plotted

to have an idea of how much scatter there was in it.

The outlier should appear to be markedly different

than the rest of the data so to show that it was

excluded from the data used to develop the trendline.

A linear model was used for the % Air Voids.

Even though a better fitting model, such as a parabola

(R2= 97%), logarithmic (R2= 93%), power function

(R2= 97%) or exponential (R2= 96%), could have been

selected, the linear model is simpler and would be

easier to use and an R2of 90% is quite acceptable.

If, for example, it was desired to know what value of

%AC would produce 4% air voids, the manipulation of a

line equation is trivial compared to the solution of a

quadratic equation (the answer is 5.3% by the way).





19/20

5.6 One advantage of knowing the equations of the trendlines is when it is required to find the values

of x and y when the function maximizes or minimizes. The first derivative of a parabolic function

is the function of the slope of the curve as x varies. Parabolic curves maximize or minimize at points

where a tangent to the curve (its slope) is a horizontal line. In such cases the slope has a value of 0.

The function of the Stability in kN versus %AC was: y = -0.3091x2+ 3.2933x - 4.7908

The first derivative of this function is: y' = -0.6182x + 3.2933

Setting y' (the slope) equal to 0 and solving for x, Stability maximizes at a %AC = 5.33%

Therefore, the value of the maximum Stability (for the curve) = -0.3091(5.33)2+ 3.2933(5.33) - 4.7908

or 3.981 kN.

5.7 The equation coefficients that were obtained in the example had 4 and 5 significant digits which is

desireable. However, sometimes the scale of the data or the units in which they're expressed can

conspire to rob us of this required precision.

For example, the units of measurement for Stability was Newtons and the %AC was actually the fraction

of AC multiplied by 100. If the %AC had been entered as the fraction of AC with a percentage format

and the Stability in Newtons were used, then the graph looks pretty much the same, but the equation

coefficients are in a different time zone (and precision):

The coefficient of the quadratic (x2) term is too big

to fit in the equation display so it is reported as

-3E+06 (Engineering Notation) which translates to

-3,000,000 which has but ONE significant digit!

As you can see, the farther away from 1.000 thedata gets, the harder it is to maintain the required

precision. It is sometimes necessary to artificially

scale the data to get reasonably precise coefficients.

To convert to an equation that produces stability in

Newtons given fraction of AC, the quadratic

coefficient of the original equation should be

multiplied by 107, the linear coefficient should be multiplied by 105and the constant term should be

multiplied by 103: Stability (N) = -3091000x2+ 329334x - 4790.8 where x is the fraction of AC.

Notice that the linear coefficient was actually more precise (6 significant digits) in the rescaled

example above. That's because there was no need to take up space with a decimal point so one moredigit was printable.

6 Graphing Equations

6.1 Sometimes it is required to graph a given equation or function. In these instances Excel is used to

generate data for plotting, however there are some minor formatting differences.

Let's say it is required to plot the equation: Nq= etan()tan2(45 + /2)




Note: When you copy a chart, any drawing elements such as text boxes and arrows are not included


20/20

6.2 First, set up a column of "x" values (i.e., ). Since the tan(90) =, we can't get too carried away.Since (angle of internal friction for a soil) is rarely above 45, 0 - 50 should suffice for a range.

A B C

Then input the equation: Nq

=EXP(PI()*TAN(B6*PI()/180))*(TAN((45+B6/2)*PI()/180))^2 6 0 1in the second column and copy over the row range. 7 5 1.567698

8 10 2.471436

9 15 3.941147

10 20 6.399394

11 25 10.66214

12 30 18.40112

13 35 33.29609

14 40 64.19521

15 45 134.8738

16 50 319.0573

6.3 Then generate the graph using the scatter style as before. The y-axis is logarithmic and there is no

need to fit a trendline since it really isn't data, after all! The point markers are also suppressed

and only straight lines are used to join the points. The more points included in the graph, the

smoother the curve will appear. As a rule of thumb, if you can visually tell that the curve is a series

of straight lines, then you need to generate more points to plot.

Remember, when you're plotting a known mathematical relation and plotting "data" that has been

calculatedrather than measured, no individual points should be visible, only the curve.

Incidently, if anyone knows why my y-axis labels keep getting truncated or how I can fix this, I

would be most grateful for the information (I have been known to be generous with bonus marks!).



Graphing Standards

Documents

Transcript of Graphing Standards