The original analysis showed also the effects of facial expressions and eye contact on photo effectiveness. Profile pictures of women making eye contact are more effective than those without eye contact, for all of the facial expressions. Flirty-face pictures with eye contact are the most effective, while flirty-face pictures without eye contact are least effective.

Bar charts show the data reasonably well. When the bars overlap, they may lead to conclusion, because the front bars partially obscure the back bars, and the back bars may appear smaller than they actually are because of this obstruction. It’s also not as easy to see the relative effects.

Main Effects Plots

When performing a statistical analysis, one of the simplest graphical tools at our disposal is a Main Effects Plot. This plot shows the average outcome for each value of each variable, combining the effects of the other variables as iff all variables were independent. In the chart below, we see that the averages for smiling (with and without eye contact) is highest, the average for not smiling is lowest, and flirty-face lies in between. The average for eye contact is greater than for no eye contact (for all facial expressions combined). We could have estimated these effects from the bar chart above, but it’s helpful to take the time to plot these effects.

The data is shown below, with ranges shaded to match the color of the plotted points. The two columns to the left of the data are used to produce the two-level category axes in the charts. There is no need to label the series, since the series identification is simplified by these dual axis labels.

Note: I’ve used simple averages from the data in the original article’s charts, but in a real analysis you would have to weight the averages by the proportion of individuals using each level of each factor.

The main effects plot is simple and does not provide a great deal of information. Showing just the main effects of each factor level without accounting for the levels of other factors is simplistic and misleading.

For example, we might assume that the best success would come from smiling and making eye contact, and the worst from not smiling and not making eye contact. In fact, the flirty-face picture accounts for the best success (with eye contact) and the worst (without eye contact).

Interaction Plots 1

Another graphic statistical tools at our disposal is called an Interaction Plot. This type of chart illustrates the effects between variables which are not independent. Such a plot looks like the charts here. There are two versions, to illustrate better the effects of eye contact and of facial expression.

In fact, the original bar chart above shows interactions between factors, but it is more effective to use markers with connecting lines to display the data.

The two charts show the effect of expression for the two eye contact categories (left) and the effect of eye contact for the three expressions (right). The charts can be made easily using data with the appropriate arrangement. The chart on the right shows that the flirty-face expression is most effective, with eye contact. Without eye contact, this expression is a loser.

- – -

The two charts need independent data ranges. The two columns to the left of the data are used to produce the two-level category axes in the charts. There is no need to label the series, since the series identification is simplified by these dual axis labels.

Interaction Plots 2

An alternative and perhaps more common layout for interaction charts is shown below. As above, the two versions emphasize the effects of facial expression and of eye contact.

The series use the same single tier of category labels, and the lower tier of labels has been replaced by data labels on the series themselves. This alternative has the possibility of becoming more cluttered, but it also shows interaction effects more clearly.

For example, the nearly parallel smiling and not smiling lines in the right chart above indicate only a very weak interaction between eye contact and smiling, but the much steeper flirty-face line shows a stronger effect (or an interaction) of eye contact when the facial expression is a flirty face. The nearly parallel segments in the left hand chart show the same weak interaction between smiling and eye contact, while the huge difference between the two flirty-face data points show a strong interaction.

- – -

These two charts can use the same data range, using either columns or rows for the series data. Using series names as data labels adjacent to data points effectively identify the data.

Extension

The same approach above provides insights into the photo effectivemess for male subjects.

Here is the original bar chart.

Main effects plot.

Interactions plots, both variations.

- -

Peltier Technical Services, Inc., Copyright © 2010.
 
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
 

This is what the top section of the data looks like when normalized in this way. I’ve converted the range to a list for all the usual reasons.

Marginal Means Plots

Instead of constructing a table of formulas to analyze the data, I can instead create a simple pivot table. The data area contains Average of Result, while Factor and Level are stacked in the rows area, with Level inside Factor. The empty rows come from checking the Empty Rows Between Items for the Factor field.

I could make a pivot chart, but I dislike pivot charts. The plot area cannot be resized, and the pivot fields and buttons clutter them up.

You can hide the fields and buttons, but you still have limited formatting capability. What’s worse, the blank spaces in the data range are ignored by the pivot chart, so point for one factor are connected to points from the next factor.

Instead I’ll make a regular chart.

This is almost like the first chart from the previous article, except that the factor names aren’t centered under the factor values. This is fixed by changing the layout a bit, by showing data in the Factor field in outline form, not table form, and retaining the space after each item.

Now the labels are centered. The chart based on a grid of formulas has less space between groups of labels, but we’ll accept this in exchange for how easy pivot tables make our work.

The second set of formulas in the previous example allowed for separate series for each factor. To get this with a pivot table, we need a duplicate Factor field. When added to the list containing the data, Excel renames it Factor2.

Now we place Factor2 in the columns area, and this splits the table into six columns, one for each factor.

These six columns become six separately formatted series in the chart.

Interaction Plots

In order to work interactions into the pivot table, you need to reintroduce the original columns for the factors and their values. Since we’ve already changed the table so that one observation requires six rows, we need to add six times the original data. If someone knows more about pivots than I do. please explain to me how to do this without using scads of data.

Anyway, here is a pivot table, one of the six needed to display all of the interactions. You can only neatly show one separate factor in the columns area. It’ easy enough to change the pivot table, of course: drag one factor off the table and drag another on.

Here’s pivot table two of six. Pivot tables three through six are not shown here.

An interaction plot can be built from each of these pivot tables. These are essentially the same as those in the earlier analysis, and the same conclusions can be drawn from them.

Even though some data preparation was required, and for some of the analysis a lot of data had to be duplicated, but creating an analysis using pivot tables can be easier, faster, and much more interactive than one using formulas.

The Data

If anyone would like to use this data and try to add to this preliminary analysis, the data is available in the following two CSV files: Table1.csv and Table2.csv.

Peltier Technical Services, Inc., Copyright © 2010.
 
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
 

The data consists of 71 mixtures; here is a subset of the data. There are six factors, each with multiple levels, and one result from each. For this process, it is desirable to have as small a result value as possible. Like most industrial experiments, this one has a large matrix, and it is only partially filled in. Perhaps the missing data represents such poor performance that the mixtures were discarded without being analyzed. Perhaps the experimental project ran out of time, or materials, or budget. The missing data makes it difficult to perform your typical statistical analysis, but we can borrow statistical methods and still gain some valuable insight.

In this example, we’ll look at Marginal Means plots and Interaction Plots.

Marginal Means Plots

Here is a simple looking table used to calculate marginal means, which are the means for each level of a factor, averaged over all levels of all other factors. For example, of all the mixtures, the average for all mixtures that involve Mixer A is 1.603. As described, this is not a statistically rigorous marginal means analysis because of the unfilled nodes in the experimental design matrix. Regardless, it can provide useful insights into the manufacturing process.

The shaded cells will be used as the source data of a marginal means plot. Green shaded cells must be empty. Tan cells should contain a space character, which doesn’t show but makes the category axis labels line up. Yellow cells are the rest of the source data.

The formula in cell M44, shown below, is copied into the non-blank cells in M44:M66.

=SUMIF(OFFSET($A$2:$A$72,0,$J44),$L44,$H$2:$H$72)/
COUNTIF(OFFSET($A$2:$A$72,0,$J44),$L44)

A2:A72 is the first column of data, showing the mixture number. Here’s how it works:

• Offset finds the column for the given factor, which is offset from A2:A72 (the mixture number column) by the value in column J. The Mixer data is one column right of the Mixture Numbers.
• Sumif compares this factor column to the appropriate factor level in column L, and if it matches, includes in the total the value in the results column, H2:H72.
• Countif makes the same comparison, but only counts one for each match, without adding the results value.
• Sumif divided by Countif produces the average.

A line chart is produced from the shaded cells. The empty cells (green) provide the gaps between the connected points of each factor.

.

A marginal means plot resembles a Main Effects Plot

An alternative arrangement of worksheet formulas produces one column of calculations per factor, and therefore one series per factor, allowing for easy formatting of different factor averages. The color scheme is the same as above: shaded cells are the chart’s source data range, and green cells must be empty. I’ve used conditional formatting to de-emphasize but not completely hide the #N/A values.

The formula in cell m5, which is copied into the entire range M5:R22, is

=IF($J5=M$3,SUMIF(OFFSET($A$2:$A$72,0,$J5),$L5,$H$2:$H$72)/
COUNTIF(OFFSET($A$2:$A$72,0,$J5),$L5),NA())

The Sumif and Countif are the same as above, but the calculation is only performed if the index in column J matches the number in row 3. Otherwise the formula returns NA(), or #N/A, which will not be plotted in our line chart. The resulting table has values along the diagonal.

The shaded area is used to create an alternate marginal means plot. No breaks in the line is needed, because the breaks already occur between different series.

What does the marginal means plot tell us? Recalling that smaller result values are better, we see that Mixer B produces the best results, higher values of Amount are bad. Carp beats Slug, results improve steadily as Pctg values increase, as are larger Add values. The Gap form is a bit better than Ant, and both are far superior to None.

This is a good first cut of the data. We can eliminate None in the Form factor and lower Pctg values. But the plant manager needs to evaluate the data more closely, because he has two nonnegotiable constraints:

• The result must be less than 1, and preferably less than 0.5.
• An acceptable result must be achieved using any of the Mixers.

Interaction Plots

We now need to look at interactions between the different factors. As with the marginal means analysis, missing values in the experimental matrix make a rigorous statistical analysis impossible. But we can still gain insights from such an analysis.

The interactions table is pretty complicated. The color scheme is the same: shaded indicates chart data, green are cells that have to remain empty, and tan should contain a space character.

Whereas our extended marginal means table had data on the diagonal, the interactions table has no plottable data along the diagonals, because the interaction of a factor with itself in undefined.

The formula in W6, which is copied into all of the non-blank cells of the table, is

=IF($T6=W$3,NA(),
IF(SUMPRODUCT(--(OFFSET($A$2:$A$72,0,$T6)=$V6),--(OFFSET($A$2:$A$72,0,W$3)=W$5))=0,NA(),
SUMPRODUCT(--(OFFSET($A$2:$A$72,0,$T6)=$V6),--(OFFSET($A$2:$A$72,0,W$3)=W$5),$H$2:$H$72)/
SUMPRODUCT(--(OFFSET($A$2:$A$72,0,$T6)=$V6),--(OFFSET($A$2:$A$72,0,W$3)=W$5))))

Here’s how the formula works:

• If the index in column T matches that in row 3, it returns a non-plotting #N/A error.
• The first Sumproduct counts how many mixtures match the indicated level of one factor with that of another factor. If this value is zero, we return the #N/A error. Otherwise we’d get a #DIV/0! error, which plots as if it were a zero value.
• The second Sumproduct adds up the results values for each mixture that is counted in the first Sumproduct.
• The third Sumproduct is the same as the first, and it counts the matching mixtures.
• The results total divided by the number of results is the average.

We could plot this entire table, but the chart would be crowded and impossible to decipher.

It makes more sense to break this into six different plots, one per factor.

Looking at the first one gives insight into what factor levels would produce acceptable results for all mixers. 0% Amount, Carp, 4.0% Pctg, Gap or Ant (but probably Ant), and Add 2 or 4 (probably 4). Follow-up mixtures should be evaluated to test these conditions and to determine how tight the limits should be. For example , can we get away with Add of 1, especially if this is an expensive Add. Or, can we really use either Gap or Ant Form, since having a choice provides flexibility in the process.

The other interaction plots support the observations made above.

The next two plots give rise to another observation. When Amount is 0%, Mult is Carp. When Amount is 35% or 50%, Mult is Slug. The plant manager confirms that Slug is added to pure Carp to bring the amount from 0% to 35% and 50%. This means we can remove the factor Mult from the analysis to clean things up a bit.

If we pore over the other interaction plots we may come up with additional simplifications, or perhaps other higher-order interactions to investigate.

Summary

There are a large number of standardized statistical methods which can help us gain insight into the important processes around us. Even if we lack the required data for a rigorous statistical treatment, we can still make useful observations using these methods. And we often find that, instead of answering questions, such analyses merely show us what other questions we should ask.

Follow Up

In Preliminary Data Exploration with Excel Pivot Tables I show how to carry out this analysis using pivot tables and a somewhat tweaked data source, rather than cumbersome formulas.

Peltier Technical Services, Inc., Copyright © 2010.
 
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
 

Chandoo: June was Pointy Haired Dilbert Blog’s Best month ever
J-Walk’s Spreadsheet Page Blog: June Visitor Stats
Daily Dose of Excel’s June Stats
PTS Blog’s Web Stats – June 2009
Debra’s Contextures 200906 Site Stats

I thought it would be interesting to compare some of my favorite blogs and web sites. Unless you have access to the data for each site, it’s not so easy. One service that lets you make comparisons is Alexa. Using Alexa you can compare up to five sites in a number of categories. I usually look at Reach, Pageviews, and Traffic Rank, which are defined by Alexa as follows:

Reach measures the number of users. Reach is typically expressed as the percentage of all Internet users who visit a given site.

Pageviews measure the number of pages viewed by site visitors. Multiple page views of the same page made by the same user on the same day are counted only once.

Traffic rank is based on three months of aggregated historical traffic data from millions of Alexa Toolbar users and data obtained from other, diverse traffic data sources, and is a combined measure of page views and users (reach).

I plotted a few groups of data, since Alexa only allows five curves on a chart. Since this is my blog, I plotted the Peltier Tech data on all charts, as a benchmark. In the first group I plotted Daily Dose and Spreadsheet Page. Then I wondered whether John Walkenbach’s old site, J-Walk.com, is still getting traffic, taking away from Spreadsheet Page. Finally I added old pal Tushar Mehta’s site, since he’s been around a long time (after j-walk.com but before peltiertech.com).

I’m pulling a slightly larger audience than the other sites, according to the Reach figures. I don’t know what the spike is in early May: it doesn’t show up in any other website analytics service I follow.

Daily Dose and Spreadsheet Page are close to each other, and it looks like j-walk.com still has a significant presence. John could do some .htaccess magic to transfer the link juice to his newer site.

It would be nice to be able to rescale the Y axis, maybe 0 to 0.007%, to provide a little resolution at the low end. Or maybe use a log scale.

Pageviews is a hard chart to analyze. Again, stretching the Y axis to scale from 0 to 0.0002% or using a log scale would spread out the low-end values.

Traffic Rank is shown on a logarithmic scale. I don’t know what kind of algorithm Alexa uses to get Rank from Pageviews and Reach, but it must be somehow multiplicative. A slight advantage in both measures of Peltier Tech over the others means Peltier Tech is visible near the bottom (actually, alternating above and below the X axis), while the others are apparently below the axis.

In my next group, I included The Spreadsheet Page (again), and I added Chandoo’s Pointy Hairde Dilbert, since he started this whole discussion. I’ve also included Debra Dalgleish’s popular Contextures web site, and also Chip Pearson’s encyclopedic site.

Chip pretty much owns the rest of us in terms of Reach, except for my early May spike, and Chandoo’s Lifehacker spike in mid-June. Debra’s site is consistently close to Chip’s and a bit higher than mine. PHD and Spreadsheet Page track each other fairly closely, except for Chandoo’s spike in June.

As before, the inflexible Y axis scale makes Pageviews simply a reminder to mow the lawn.

Traffic Rank shows all of us venturing into visible territory, with the relative placements similar to those shown in the Reach chart.

To put this all into perspective, I decided to compare my site to a few larger ones, namely Amazon, Microsoft, and Google. These sites are a little more popular, so Alexa’s Y axes expand upwards to accommodate them (and push my data ever lower in the charts).

If you look closely at the Reach chart you can see Peltier Tech: it’s the blue line obscuring the X axis. Compared to Google, Microsoft and Amazon are near the bottom. On a log scale, they’d be much closer to Google than to Peltier Tech.

Again, Peltier Tech is coloring the X axis blue in the Pageviews chart. Again, Google’s numbers dwarf Amazon’s and Microsoft’s.

At least the Peltier Tech site shows a little texture in Traffic Rank. Good thing that’s a logarithmic scale! Google’s rank serves as the gridline for Y=1. The log scale also helps MS and AMZ stay near the top of the chart, with ranks around 15 and 30.

So what does all of this mean? I don’t know, but it’s a fun way to kill a little time. I guess I can rest assured that the Peltier Tech web site is in the middle of the pack of the popular Excel sites. But if all of you readers tell your friends, and twitter all about my site, and your friends all post on FaceBook about Peltier Tech, and I don’t know, we throw in a little MySpace and Digg and Technorati and other funny sounding words, and the momentum grows, I’ll probably still be firmly lodged in the middle of the pack.

Peltier Technical Services, Inc., Copyright © 2010.
 
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
 

Pointy Haired Dilbert statistics for June 2009

John Walkenbach responded with his own visitor stats.

The Spreadsheet Page statistics for June 2009

John asked, so here are my June stats, first for the entire Peltier Tech website (which includes the blog), then for the PTS Blog.

PeltierTech website statistics for June 2009

PTS Blog statistics for June 2009

This was almost the best month for the web site, but March (below) was slightly better (February and April were in between March and June).

PeltierTech website statistics for March 2009

June was the highest month for visits and visitors on the blog, but April had more pageviews.

PTS Blog statistics for April 2009

My stats show “typical” daily variation, with high weekday and low weekend, for visits, visitors, and pages, while the other measures are flat. John has the same daily variation in visits, visitors, and pages, while hispages per visit and time per visit are higher on weekends. I think this means that people spend more time on his site on weekends, when they have more time to poke around.

Chandoo’s stats in the second week of June are crazy, higher and less regular than the rest of the month, which may show a regular variation except for the distraction of the second week. The peak in Chandoo’s numbers probably corresponds to the June 12 announcement of the new Chart Busters feature he and I are working together. This announcement led to some controversy, as it at first used the name Chart Doctor, which is also the name of a  feature on Kelly O’Day’s Process Trends web site.

Peltier Technical Services, Inc., Copyright © 2010.
 
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
 

I plotted the data and gave it the hairy eyeball. Not a linear trend, maybe something quadratic.

Attempted Regression

The blog reader had fitted a 6th order polynomial trendline, and was having trouble using it to predict values. My fit is shown below, and I had no such problems with predictions matching the trendline. I suspect the user had insufficient precision in his coefficients, which is covered in Trendline Fitting Errors.

The 6th order fit isn’t really all that great. I decided it really isn’t much better than the quadratic fit I had initially suspected.

Then I thought the data almost fit two line segments over different ranges of data. I’ve plotted these below.

I replied to the user with this suggestion, and he said that wouldn’t work, because the data would have to be fitted with many line segments, because the data he gave me was only part of a much larger sequence of values.

Run Charts

I thought a moment and realized that with many weeks of repeated data, what the user needed was an approach based on Statistical Process Control. I wrote about Control charts in Introducing Control Charts (Run Charts). This is an opportunity to illustrate another set of run charts. In this example, I relied on techniques from a small, 136-page book called Understanding Variation.

Understanding Variation: The Key to Managing Chaos
Donald J. Wheeler

I added a column to my table to calculate the Moving Range, which is simply the absolute value of the difference between the current value and the previous value. This is an easier measure of variation to compute than the standard deviation, though with modern computer hardware and software that’s not an important consideration.

In any case, I plotted the weekly values data and the moving range data.

I computed the averages of the values data and of the moving ranges. I added horizontal lines to indicate the averages (see Run Chart with Mean and Standard Deviation Lines for detailed instructions).

Then I used simple factors to determine upper and lower control limits for these quantities, and I added the limits to the charts.  For the values, the control limits are given by:

Limit = Average Value ± 2.66 * Average Moving Range
 

For the moving range, the lower control limit is zero and the upper control limit is given by:

Limit = 3.27 * Average Moving Range
 

What this tells me is that the values and the moving ranges fall within limits, so the variability is given not by anything we can fit a curve to, but simply by normal variation within the process. Closer examination of some of the data would probably point to an out-of-control process (for example, the last five values show continuing decline). Let’s just worry about violations of the control limits.

I calculated 70 more values with the same mean and standard deviation as the original 10 values, to simulate an ongoing process (because the blog reader did not provide more data). I plotted these values on the same chart with the original ten values, using the limits calculated based on the original ten values.

The values look pretty good, all within the limit except for a single point, which should be examined for any special causes of variation. All of the moving range points fall within the upper control limit. I recalculated the averages and limits using the entire data set and replotted the data.

There was little difference; the limits were slightly more generous. The value that exceeded the control limit in the first chart of all the data still is out of control, and still deserves a closer look.

One final note: The polynomial regression breaks down completely in a process like this which is successfully modeled using SPC. A linear fit may be useful to detect a possible trend of the average over time.

Further Reading about Statistical Process Control

ISO 9001 – Introduction to SPC

Control Charts on Wikipedia

Interpreting Control Charts

Selecting the Right Control Chart

I thought I’d see just how PC my readers are. I checked my FeedBurner stats for the last 30 days, ending 9/21/2008, and here is the breakdown of operating systems.

This fits pretty closely with my overall web site stats, too, which totals around ten times the traffic, but in FeedBurner it’s easier to get a quick list of numbers, and FeedBurner lists the versions of each OS.

So what’s all the buzz about Macs lately? Macs make up around 4% of my visitors. For every Mac visitor, I get more than 26 Windows visitors.

What’s all the buzz about Linux? Linux counts for 1% of my traffic if I round up.

Windows includes 95% of my traffic if I round down, and over 80% of that is XP. I guess the significant buzz is that under 20% of my Windows hits are from Vista.

I decided to compare the past month with the same month last year, which meant Google Analytics, since FeedBurner only goes back to March when I launched the blog. When I plotted the two months, the raw visits showed only that Windows and Mac visitors both increased, but there seemed to be a gain in percentage of visitors using Mac at the expense of percentage of visitors using Windows. (I neglected all the minor operating systems, but I will note that during each of these months I got two hits on Nintendo Wii.)

This needed further investigation, so I collected data for the months in between. Raw visits showed a general up trend in Windows users, while Mac and Linux were lost in the weeds.

On a percentage basis, Windows was up in the mid 90s, with a hint of a downward trend, and a gain, Mac and Linux were too low to resolve.

Putting the data on a log scale flattened any trend in the Windows line, while it made Mac and Linus easier to see. There is possibly an upward trend in Mac and a downward trend in Linux. It’s sough to call, given the fluctuation in both lines.

To evaluate the trends, I added trendlines. These do in fact show slight slopes in the directions indicated above. The slopes are very small, and correspond to changes on the order of a fraction of a percent per year. The R² values are also pretty low, indicating rather poor correlations.

My conclusion is that Linux and Mac make up a very small fraction of my readership, averaging 1.3% and 3.5% respectively over the past year. Sorry, guys. No offense, but this supports my decision to concentrate on Windows versions of my utilities. My utilities are built in Excel 2000 and VBA 6, and so are compatible with Excel 2002, 2003, and even 2007. However, they are incompatible with the older VBA 5 found in Excel 97 and Mac Excel. In fact, Mac Excel 2008 doesn’t even support VBA at all.

The first step is to plot the data using XY charts. The charts below show time to drain vs height for various hole sizes and time to drain vs hole size for various diameters. We will investigate various trendline formulas using these plots.

In Trendline Fitting Errors I warned about picking an appropriate type of trendline relationship. In Graphical Approach to a Simple Physics Problem I automatically looked for a power law relationship. What if I didn’t have a scientific background? How might I go about fitting trendlines to the data. Let’s just look at the highest curve in each of the charts above, and apply a few trendlines.

Obviously the lines are not linear, but let’s start with linear trendlines. The original data are shown with points connected by dotted lines, and the trendlines by solid lines.

Since there is curvature in the data, we need a higher power of the X variable. These two charts show polynomial fits of order 2. The fits come close to fitting all of the points, but the curve for height seems to be approaching a maximum at a height of 30, implying that if we double the height, the time will decrease. The curve for diameter is even stranger, with a minimum just above 4; this implies that if we widen the hole further, the water will leak out more slowly. Both of these scenarios are rather unlikely.

To carry the polynomial fitting to an extreme, let’s look at third order fits. Fortunately with only four points, we will not be able to go beyond this. In the height relationship, the poly fit predicts a slowing of the increase of time with height, then after about 20 it accelerates. The diameter relationship actually changes direction twice. To smooth this out, we’d have to make another measurement at a height of 20 or a diameter of 4, then apply a higher order fit.

This turns into an exercise of reductio ad absurdum, and fortunately Excel doesn’t get more absurd than a polynomial fit of order 6. In my previous life as a scientist/engineer, there were very few physical relationships that theoretically followed polynomial relationships. Sometimes a poly fit was acceptable when interpolated values were required, but it’s generally better to find a simpler relationship. A simpler relationship is one with fewer fitting parameters, and which follows the form of any theoretical description of the phenomenon.

A great many physical phenomena follow power law relationships. The following show power law fits to the data.

Time to drain the bucket follows the reciprocal of diameter to the second power. This fits nicely since the volume of water passing through the hole is proportional to the area of the hole, which is proportional to the diameter squared. It’s an inverse relationship, because the larger the hole, the more water can flow, and the shorter time it will take.

I’m less certain of the height relationship. The time is obviously additive as more water is added to the initial height. Also, the added water should push harder on the water beneath, forcing it out faster. An equal increment of water height will produce a decreasing increment of time. If I weren’t too lazy to dig my old calculus book out of the basement, I’d find the similar example and determine the exact form of the drainage curve.

Below is another view of the power law fits on a double logarithmic scale (on which a power law fit traces a straight line). The visual goodness of fit lets me assert that the power law fits are appropriate.

The person had a motor, and had measured horsepower (HP) at particular rotational speeds, in RPM (revolutions per minute).  The data is charted and tabulated below:

Note: Some versions of Excel have problems performing statistics on some data sets. For the most part, Excel is adequate for this purpose, particularly if the data aren’t “extreme” (a statistical term) and if the data is not overfitted.

Applying a Trendline

Adding a trendline is straightforward. The easiest way is to right click on a series, and choose Add Trendline from the context menu.

This pops up a dialog from which you can select a type of trendline to fit to the series, as well as choose options for the trendline. This dialog also appears when you right-click on an existing trendline and choose Format from the context menu, with an additional tab for patterns (to format the line). The specifics of this dialog are a topic for a different discussion.

The added trendline is dumped onto the chart, obliterating details in its vicinity.

You can right click the trendline, choose Format, and make it much more presentable. Below I’ve formatted the trendline and trendline formula to match the line series, and changed the line series to display markers only.

Trendlines are really valid only for charts with a numerical category axis, such as an XY chart, or a line chart with a date-scale axis.

You can add trendlines to a line chart; Excel makes no value judgments here. Note that the categories are not numerical, and a trend between discrete categories may be meaningless (e.g., Cat, Dog, Ferret, Goldfish). I discuss use of trendlines on the wrong chart types in the next section.

You can also add trendlines to a clustered column chart. Note that the trendlines don’t necessarily match up with their corresponding columns, but with the center of the cluster (centered over the category labels). This adds to the confusion of non-numeric categories.

You can even add trendlines to a horizontal bar chart, but their usefulness is even less than that of trendlines in a line or column chart.

You cannot add trendlines to a stacked series: the command is disabled.

For qualitative purposes of illustration, I’ll show how to add trendlines to this type of chart. You must promise never to use them for evil purposes. Note that each trendline captures the variability of not only its respective series, but all other series stacked beneath it.

In the sheet, calculate the sums of the first series, then the first and second, then the first, second, and third series, etc., and add them to the chart as line chart series, formatted below as markers without lines.

Apply trendlines to the unstacked line series, format the trendlines, format the line series to display no markers, and remove all the unneeded entries from the legend.

You cannot add trendlines to a 3D chart: the command is disabled. This is Yet Another Reason not to use 3D charts, when will you ever learn?

Error 1: Wrong Chart Type

The first problem that many people encounter when fitting a trendline is caused by using the wrong chart type. The person who asked about fitting this data didn’t make this mistake, but this is a fine place to illustrate it.

In Line Charts vs. XY Charts I described differences between XY and Line charts. For our purposes here, suffice to say that XY charts treat both X and Y data as continuously variable numerical data, while line charts treat the X values as non-numerical text labels, and if necessary, treats them using the counting numbers 1, 2, 3, etc.

The first mistake people make while fitting trendlines to charts is when they start with a line chart. Notice the X values: The axis doesn’t start at zero, and although the differences between adjacent numbers are not all the same (some differ by 1000, others by 500), the spacing between labels is constant.

In this chart I have applied a fourth order trendline to the data, removed the lines between the points, and formatted the curved trendline to match the series. The fit doesn’t look too bad, but as I pointed out, the X values are not appropriate for the fit.

When the coefficients and actual X values are plugged into the trendline formula, we get the following actual HP values and fitted values (“Line”).

The calculated values are way too high: 5.1E+10 is 51 billion. The RPM values (1000 through 5000) were plugged into the formula, but it was calculated by Excel using the counting numbers 1 through 5. The fitted data matches the actual pretty well.

I discussed trendlines on other improper chart types (clustered and stacked column and bar charts) in the preceding section.

Error 2: Wrong Precision

The following shows the trendline for the same data in an XY chart. It fits the points pretty well, with a little curvature even over the lower few points, which seem like they should fit a straight line.

When we plug the RPM data and the fitting coefficients into the trendline equation, we get the following horrendous match.

This illustrates the second mistake people make. They accept the trendline formula from the chart without thinking about the coefficients. Notice the second through fourth order coefficients above (-1E-5, 3E-9, and -3E-13): these are shown with only one significant digit. That’s a miscalculation waiting to happen. Select the trendline formula, and apply a scientific number format with lots of digits, and you’ll plainly see the source of the error.

Here are the actual and fitted data points: much better.

Error 3: Manually Transcribing Coefficients

If you want to use the trendline coefficients in the worksheet, there’s a better approach than manually transcribing data from the trendline formula to cells. Excel has a function called LINEST which performs linear regression calculations. Select a range 5 rows tall and N+1 columns wide, where N is the order of the regression, type this formula in the top left cell, and array-enter it by holding Ctrl+Shift while pressing Enter:

=LINEST(B2:B7,A2:A7^{1,2,3,4},,TRUE)
 

If you’ve entered the formula correctly, Excel signifies that it is an array formula by enclosing it in curly braces (typing them yourself will only produce an error):

{=LINEST(B2:B7,A2:A7^{1,2,3,4},,TRUE)}
 

B2:B7 is the range of Y values, A2:A7 is the range of X values, A2:A7^{1,2,3,4} signifies that the X values are to be raised to the first through fourth powers for the regression. For more details on the formula and the results (below), refer to the help files.

The first row of the resulting range contains the coefficients, from fourth power to constant. The first cell in the third row contains the fitted R². These are identical to the coefficients in the XY chart’s trendline formula.

Error 4: Overfitting

I noted earlier how the first several points look like a straight line fit. Below I’ve computed the straight line fits for the first 5 points (blue dottedline) and the first 4 points (dashed red line). These lines fit all but the last point nearly perfectly.

This illustrates another error people make when fitting trendlines: overfitting. A 4th order polynomial fit has no physical significance in any model I’ve ever heard of.  It makes a “nice” curve, but is valid for interpolation only, for purposes of looking up intermediate values along smooth curves.  First and even second order fitting may have some reasonable theoretical basis, but if the data curves systematically, you should consider applying logarithmic, exponential, or trigonometric transformations to your data prior to calculating a trendline.

Error 5: Ignoring the Physics of the Problem

This leads to my third point. One should not blindly apply statistics without first using one’s own trained eyeball. Yes, your eye may see patterns that are not there, but your eye can be better than statistical techniques at analyzing results.  The high correlation of the 4th order fit (0.9998) might lead one to believe that the HP is truly proportional to RPM to the 4th power. A quick glance led me to look at the first part of the data, up to 4000 or 4500 RPM, which looked linear.  It felt good in my gut that horsepower increases linearly with RPM.  Then as RPM exceeds some threshold value, something breaks down. I don’t know what it may be, but I know where I would look. I would take multiple measurements in the 4500-5500 rpm range, to see whether I get a smooth curve, perhaps approaching some maximum HP asymptotically.  Perhaps there is some slipping in the linkage, or some thermal effect from friction, or some deformation in the mechanism.  The point is, don’t let a seemingly nice statistical fit prevent you from noticing what your eye is seeing, and always consider some real physical mechanism underlying the data.

Summary

To avoid errors in the use of trendlines, one should follow this approach:

1. Consider the system under analysis. Does it lend itself to a curve fit? Do you know anything about the system that suggests a particular physical relationship?
2. Consider the type of fit and the purpose for the fit. Is the fit used to make the chart “look nice”? A poly fit is probably fine. Are you using the relationship to interpolate between known values? A poly fit is probably fine, unless you know the relationship is logarithmic, exponential, or power law. Will you try to interpolate beyond the existing  data? Not generally wise, especially using a polynomial fit.
3. Use LINEST to get coefficients into the worksheet. This is easier and generally more reliable (subject to certain caveats) than manually transcribing coefficients from the trendline formula label in the chart.
4. Use an appropriate chart type. This almost always means use an XY chart. You can get away with using a line chart (particularly with a date-scale axis), so long as you understand what the categories mean.
5. Apply a suitable number format to the trendline formula. Generally a scientific format with lots of digits (14 or 15 digits) will provide sufficient resolution.

What’s wrong with this chart (which I’ve reproduced below)?

• Axis label is faulty. “Japan and Suicide Rate” should be “Japan Long Term Unemployment and Suicide Rates”.
• Rates are not defined. Suicide Rate is number of suicides per 100,000 population. Long Term Unemployment Rate is percentage of total unemployed who have been unemployed for over twelve months.
• Frequency of data is different. Using markers as in my version, it is clear that suicide rates are reported with much less frequency than long term unemployment. At least they didn’t use smoothed lines*.
• Such unrelated rates should not be plotted on the same axis. This chart makes it look like the rates were about the same in 1980, then unemployment dropped below suicides, then it rose above suicides, then both remained the same from about 1992 through 2000.

*Formatting the chart with smoothed lines and no markers gives the reader no indication that the data is reported on a different time scale, and may make it look as though the correlation is even closer.

The first step is to address the overlapping scales. The chart above using one scale is misleading. I can add a secondary axis, and adjust the relative scales of the two series to mislead the reader in any way I want.

The only way to make the scales free from confusion is to plot he series on completely different, non-overlapping scales. I’ve split the chart into two panels to show the series separately on their own scales. It actually looks like the rise in suicides is leading the rise in long term unemployment. If I thought higher unemployment led to increased suicides, I’d expect unemployment to lead.

If we’re looking for a relationship between the two variables, we should plot them on an XY chart. I’ve labeled the points with year, so that one can trace the evolution of the relationship. If I do the math, I get a correlation (R²) of 0.60, not a very strong relationship. By eye, I can see two regions in the chart.

• Up to 1995, when both suicides and unemployment was low, there is a very good negative relationship between suicides and unemployment. Are the number of suicides great enough to reduce the number of long term unemployed?
• From 2000 onward, there seems to be a constant suicide rate despite the large increase in long-term unemployment.

Six data points isn’t much to base a hypothesis on, so I made a (probably invalid) attempt to rectify this, by interpolating suicide rates for the in-between years with no data. The chart looks very much like the one above; I’ve used filled markers to denote actual data, and unfilled for interpolated points.

The correlation plot looks pretty much the same, a bit more convoluted. I did not bother calculating correlation coefficients, as that would stretch the validity of this exercise beyond my comfort zone.