Issue: Volume 4, Number 5
Date: May 2004
From: Mark J. Anderson, Stat-Ease, Inc. (

Dear Experimenter,

Here's another set of frequently asked questions (FAQs) about doing design of experiments (DOE), plus alerts to timely information and free software updates. If you missed previous DOE FAQ Alerts, please click on the links at the bottom of this page. If you have a question that needs answering, click the Search tab and enter the key words. This finds not only answers from previous Alerts, but also other documents posted to the Stat-Ease web site.

Feel free to forward this newsletter to your colleagues. They can subscribe by going to If this newsletter prompts you ask to your own questions about DOE, please address them to

Here's an appetizer to get this Alert off to a good start: To see experimental evidence that dogs know calculus, click There you find data gathered by math professor Tim Pennings on how his dog Elvis showed an uncanny talent for taking the optimal path when retrieving objects thrown into the water.  For simplified versions of the story, and more pictures of Elvis and his master, fetch or

Here's what I cover in the body text of this DOE FAQ Alert (topics that delve into statistical detail are designated "Expert"):

 1. FAQ: Explanation for "degrees of freedom"
 2. FAQ: How Stat-Ease does pair-wise testing
 3. Info alert: An opportunity to publicize your success with DOE!
 4. Events alert: Talk in Toronto at the Annual Quality Congress
 5. Workshop alert: See when and where to learn about DOE—A class is coming to San Jose

PS. Quote for the month—Why does physicist Freeman Dyson say that "The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often"? The answer is provide by Littlewood's Law of Miracles at the end of this Alert.


1. FAQ: Explanation for "degrees of freedom"

-----Original Question-----

"No matter how many books I read about statistics, I still don't understand 'degrees of freedom.'  I need an explanation and many examples showing how to calculate and use them.  Also, how can I be sure that all degrees of freedom in a stat problem have been accounted for?"

Answer (from Stat-Ease Consultant Shari Kraber and Pat Whitcomb):

"You are right about degrees of freedom, they can get a bit complex and it's tough to find any really good explanation. Degrees of freedom (df) are the number of independent pieces of information that can be obtained from a data set. The corrected total number of df available is N-1, or one less than the total number of data points.  Correcting the data for the overall mean uses 1 df.  For example with 10 pieces of data, there are 9 pieces of information that are independent of the overall mean. I always check the df for the corrected total sum of squares (SS) at the bottom of the analysis of variance (ANOVA)—this should be 1 less than the number of data points you have.

Degrees of freedom account for independent pieces of information. For example, let's say that you perform an experiment in four blocks of runs.  Block coefficients are corrections to the overall mean used to obtain the mean of each block.  Therefore n-1, where n is the number of blocks, df are used by blocks.  In our example 3 df (=4-1) must be used to estimate the block coefficients for the predictive model since only 3 of the 4 block corrections are independent of the overall mean.  Individual model terms then use up differing amounts of df, depending on the nature of your experimental factors.  Main effect terms for modeling factorials correct the overall mean to obtain the mean of each factor level. Therefore n-1, where n is the number of factor levels, df are used by each main effect.  Thus in two-level factorials each main effect has 2-1, or 1 df.  If a factor has three levels the main effect has 3-1 or 2 df.

Factorial interaction df are calculated by multiplying the df from the main effects involved in the interactions.  For example, in a two-level design on two factors, the df for AB interaction requires 1 df because both A and B take up 1 df and 1x1=1.  If factor A has 3 levels and factor B has 4 levels then the AB interaction has (3-1)(4-1)=2x3=6 df.

Now let’s put it all together—What if you studied three suppliers of four material types?  In this case the df for A is 2 (=3-1) and B requires 3 df (=4-1), so the AB takes up 6 df(=2x3). In this example there are 12 different combinations of A(supplier) with B(material).  Therefore our model has to support prediction of a different mean for each of those 12 combinations. Start with the overall mean (allowing estimation of 1 mean). Then the main effect for A has 2 df (allowing estimation of 2 more means. That’s 3 df total at this stage. Continuing on, the main effect for B has 3 df (allowing estimation of 3 more means. We are now at 6 total. Finally the interaction AB has 6 df, allowing estimation of 6 more means. That brings us to 12 df in total.

In ANOVA, after placing the correct df's into the block and model terms, the residual gets the 'leftovers.'  If you do replicates, extra df's become available for estimation of pure error.  For instance, if a center point was replicated four times, then the pure error df would be 3 (=4-1)."

Shari and Pat have tackled a very tough topic and wrestled it down to the ground.  Here's a link to another explanation of degrees of freedom by Gerard E. Dallal of Tufts University: from his "The Little Handbook of Statistical Practice" (table of contents at - Mark

(Learn more about degrees of freedom and ANOVA by attending the three-day computer-intensive workshop "Experiment Design Made Easy."  See For a course description.  Link from this page to the course outline and schedule.  Then, if you like, enroll online.)


2. Expert-FAQ: How Stat-Ease does pair-wise testing

-----Original Question-----
An Indiana statistician

"I'm trying to help a student with multiple comparisons done with Stat-Ease software.  I found the table of pairwise t-tests in the ANOVA, but I'm not sure how the calculations are done."

For pairwise testing, our software makes use of "Fisher's LSD" (Least Significant Difference), an example of a "two-step, step-down" procedure.*  The first step is use of 'Fisher's protected test' (that is: analysis of variance (ANOVA)) to produce an F-test on overall significance on the spread of means.  The second step, taken only if the null hypothesis is rejected (statistically significant F, say at p<0.05) is application of the t-test pair-wise at the same p (say <0.05).  In this procedure the p-values are not adjusted for multiple comparisons.  The degrees of freedom (df) on the pair-wise tests are based on the standard deviation from the ANOVA which pools variation within each of the treatments.

*(For details, see "Multiple Comparison Methods for Means" by Rafter, Abell and Braselton in Society for Industrial and Applied Mathematics (SIAM) Review, Vol. 44, No. 2, 2002, pp 259-278, posted at for an abstract.)


3. Info alert: An opportunity to publicize your success with DOE!

Gain technical recognition in your company and scientific field. Share a DOE with our writer at:

Rename proprietary factors if needed to maintain confidentiality. A thirty-minute interview, some e-mails, a review, and it's done!


4. Events alert: Talk in Toronto at the Annual Quality Congress

I will present a talk titled "Screening Process Factors in the Presence of Interactions" at session W202 of the American Society of Quality (ASQ) Annual Quality Congress (AQC) in Toronto on Wednesday, May 26.  For information on this event, see

My talk, co-authored by Pat Whitcomb, introduces a new, more efficient type of fractional two-level factorial design of experiments (DOE) tailored for screening of process factors. These designs are referred to as "Min Res IV" because they require a minimal number of factor combinations (runs) to resolve main effects from two-factor interactions (resolution IV).  We set the stage for these state-of-the-art minimum-run medium-resolution designs by first reviewing traditional screening approaches—standard two-level fractional factorials (2^k-p) and Plackett-Burman designs.  To provide an element of realism, we will show how well each screening design identifies main effects in the presence of two-factor interactions (2FI’s) affecting a hypothetical machine operated by computer numerical control (CNC).

Our ultimate objective is to equip technical professionals with cost-effective statistical tools for making breakthroughs in process efficiency and product quality.

I hope you can make it to AQC in Toronto for my talk.  If not, do not fret—the proceedings will be posted to our web site and I will provide a link in a future issue of the DOE FAQ Alert.

Click for a list of appearances by Stat-Ease professionals.  We hope to see you sometime in the near future!


5. Workshop alert: See when and where to learn about DOE—A class is coming to San Jose

I will teach the Experiment Design Made Easy (EDME) workshop in San Jose, California on June 8-10.*  You will see this and other scheduled classes at  To enroll, click the "register online" link on our web site or call Stat-Ease at 1.612.378.9449.  If spots remain available, bring along several colleagues and take advantage of quantity discounts in tuition, or consider bringing in an expert from Stat-Ease to teach a private class at your site.  Call us to get a quote.

*PS. I was just told that the San Jose EDME may be sold out.  If so, please consider going to the next one scheduled July 13-15 at our home training center in Minneapolis.  -- Mark


I hope you learned something from this issue. Address your general questions and comments to me at:



Mark J. Anderson, PE, CQE
Principal, Stat-Ease, Inc. (
Minneapolis, Minnesota USA

PS. Quote for the month—Why does physicist Freeman Dyson say that "The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often"?

"Littlewood was a famous mathematician... at Cambridge University.... He defined a miracle as an event that has special significance when it occurs...with a probability of one in a million...Littlewood's Law of Miracles states that in the course of any normal person's life, miracles happen at a rate of roughly one per month.  The proof of the law is simple.  During the time that we are awake and actively engaged in living our lives, roughly for eight hours each day, we see and hear things happening at a rate of about one per second.  So the total number of events that happen to us is about thirty thousand per day, or about a million per month...The chance of a miracle is about one per million events.  Therefore we should expect about one miracle to happen, on the average, every month."

-- From a review by Freeman Dyson of the book "Debunked! ESP, Telekinesis, Other Pseudoscience" by Georges Charpak and Henri Broch at  I became aware of this paradox from a fascinating word-game you can subscribe to at   Mark

Trademarks: Design-Ease, Design-Expert and Stat-Ease are registered trademarks of Stat-Ease, Inc.

Acknowledgements to contributors:

—Students of Stat-Ease training and users of Stat-Ease software
—Fellow Stat-Ease consultants Pat Whitcomb and Shari Kraber (see for resumes)
—Statistical advisor to Stat-Ease: Dr. Gary Oehlert (
—Stat-Ease programmers, especially Tryg Helseth (
—Heidi Hansel, Stat-Ease marketing director, and all the remaining staff


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#1 Mar 01
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