ANOVA Analysis of Variation

Photo by NASA

Photo by NASA

Looking at my notes from prepping for the exam, I see that I took it to mean that I had to cover One-way,Two-way (without replicates), and Two-way (with replicates). I made sure I understood ANOVA’s use in a variety of Factorial Experiments as well as Random Block design.

An ANOVA usually is used to compare the means of three or more factors by using the F Distribution.

Sir Robin Fischer developed ANOVA to determine if the within treatment variation is significant in comparison to the treatment means.

All variation is accounted for in the analysis.

Equality of sample means can be tested by comparing sample variances. Within treatment variation and between treatments variation can be totaled.

When discussing ANOVA, the error variance is the repeatability. Technical variance is the reproducibility.

The “Between Groups” row represents what is often called “explained variance” or “systematic variance”. We can think of this as variance that is due to the independent variable, the difference among the three groups. For example the difference between a person’s score in group one and a person’s score in group two would represent explained variance. The “Within Groups” variance represents what is often called “error variance”. This is the variance within your groups, variance that is not due to the independent variable. For example, the difference between one person in group one and another person in group one would represent error variance. – From

  • Understand the concept of ANOVA
  • Learn the types of ANOVA
  • Review the procedure to complete ANOVA
  • Discuss how to apply ANOVA


Fundamental equation of analysis of variance – Total SS = SST+SSE ;Total SS = Total SUm of Squares; SST = SUM of Squares among treatments; SSE = SUm of squares within treatments; In other words, the total sum of squares of deviations from the grand mean is equal to the sum of squares of deviations among treatment means and the grand mean + sum of squares of deviation within treatments

Reason for using ANOVA for significance testing is that the “EQUALITY OF SAMPLE MEANS CAN BE TESTED BY COMPARING SAMPLE MEANS”

Types of ANOVA

  • One-way
    • Measures single factor from multiple sources
    • Uses only one technician / measurer
  • Two-way (without  replicates)
    • Measures 2 factors
    • Uses only one technician (unless technicians are one of the factors)
  • Two-way (with replicates)
    • Measures 2 factors, but has multiple repetitions of each combination.
    • Uses only one technician (unless the technicians are one of the factors)


One-way ANOVA


Underlying assumption of ANOVA single factor is that the variation within each factor treatment or factor treatment combination is the same. This is called homogeneity of variance

Ratio for no treatment effect is MS Treatments / MS Residual

Three different brands of chlorine are used at a local pool over the course of the summer – as one brand’s four week supply runs out, the staff begins using the next brand of chlorine. Management is interested in finding out if the different brands have a significant effect on the ability to maintain safe pH levels. Once the previous brand is no longer present in any measurable amount, the pH levels for each brand are collected through random sampling. Is there a significant difference in the three brands of chlorine? Test at the 5% significance level. Complete the ANOVA table (except the p value) manually.


One way ANOVA

One way ANOVA


Two-way ANOVA

The study of the variation caused by different machines and different material thickness conducted by one technician who conducts multiple repetitions of different sets of conditions is called a

Two-way ANOVA with Replicates



Three-way ANOVA


Procedure for ANOVA

  1. Randomly select parts / processes
  2. Identify parts, processes, technicians
  3. Collect data
  4. State H0 and H1
  5. Choose α
  6. Calculate the F Statistic
  7. Find Fα in the table.
  8. Test the F cal vs the F α.

To Be Added

Degrees of freedom, treatment DoF, Error DoF, MANOVA (and why / how it’s different from ANOVA including the benefits.


ANOVA Videos

Six Sigma Black Belt Certification ANOVA Questions:

Question: To assess the significance of factors in either a fractional or a full-factorial experiment structure, a black belt can use: (Taken from ASQ sample Black Belt exam.)

(A) analysis of variance (ANOVA)
(B) fault tree analysis (FTA)
(C) failure mode and effects analysis (FMEA)
(D) evolutionary operation (EVOP)

Answer: (A) analysis of variance (ANOVA). The other options just don’t make any sense. An ANOVA is designed to compare three or more factors against each other- which is what happens in a designed experiment.

Also see fault tree analysis (FTA) and failure mode and effects analysis (FMEA).

2 comments… add one
  • Praful bhusari

    What is the purpose of ANOVA?

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