This is one of the following sixteen articles on Single-Factor ANOVA in Excel

Overview of Single-Factor ANOVA

Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013

Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group

Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013

Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison

Single-Factor ANOVA - All Excel Calculations

Overview of Post-Hoc Testing For Single-Factor ANOVA

Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA

Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA

Overview of Effect Size For Single-Factor ANOVA

ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013

ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013

ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013

Power of Single-Factor ANOVA Test Using Free Utility G*Power

# Power of Single-Factor

ANOVA Test Using Free

Utility G*Power

The accuracy of a statistical test is very dependent upon the sample size. The larger the sample size, the more reliable will be the test’s results. The accuracy of a statistical test is specified as the Power of the test. A statistical test’s Power is the probability that the test will detect an effect of a given size at a given level of significance (alpha). The relationships are as follows:

α (“alpha”) = Level of Significance = 1 – Level of Confidence

α = probability of a type 1 error (a false positive)

α = probability of detecting an effect where there is none

Β (“beta”) = probability of a type 2 error (a false negative)

Β = probability of not detecting a real effect

1 - Β = probability of detecting a real effect

Power = 1 - Β

Power needs to be clarified further. Power is the probability of detecting a real effect *of a given size at a given Level of Significance (alpha)* at a given total sample size and number of groups.

The term Power can be described as the accuracy of a statistical test. The Power of a statistical test is related with alpha, sample size, and effect size in the following ways:

1) The larger the sample size, the larger is a test’s Power because a larger sample size increases a statistical test’s accuracy.

2) The larger alpha is, the larger is a test’s Power because a larger alpha reduces the amount of confidence needed to validate a statistical test’s result. Alpha = 1 – Level of Confidence. The lower the Level of Confidence needed, the more likely a statistical test will detect an effect.

3) The larger the specified effect size, the larger is a test’s Power because a larger effect size is more likely to be detected by a statistical test.

If any three of the four related factors (Power, alpha, sample size, and effect size) are known, the fourth factor can be calculated. These calculations can be very tedious. Fortunately there are a number of free utilities available online that can calculate a test’s Power or the sample size needed to achieve a specified Power. One very convenient and easy-to-use downloadable Power calculator called G-Power is available at the following link at the time of this writing:

http://www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3/

## Power calculations are generally

used in two ways:

### 1) *A priori*

*A priori*

- Calculation of the minimum sample size needed to achieve a specified Power to detect an effect of a given size at a given alpha. This is the most common use of Power analysis and is normally conducted *a priori* (before the test is conducted) when designing the test. A Power level of 80 percent for a given alpha and effect size is a common target. Sample size is increased until the desired Power level can be achieved. Since Power equals 1 – Β, the resulting Β of the targeted Power level represents the highest acceptable level of a type 2 error (a false negative – failing to detect a real effect). Calculation of the sample size necessary to achieve a specified Power requires three input variables:

a) **Power level **– This is often set at .8 meaning that the test has an 80 percent to detect an effect of a given size.

b) **Effect size** - Effect sizes are specified by the variable f. Effect size f is calculated from a different measure of effect size called η** ^{2}** (eta square). η

**= SS**

^{2}**/ SS**

_{Between_Groups}

_{Total}_{ }These two terms are part of the ANOVA calculations found in the Single-factor ANOVA output.

The relationship between effect size f and effect size η** ^{2}** is as follows:

*(Click Image To See a Larger Version)*

Jacob Cohen in his landmark 1998 book *Statistical Analysis for the Behavior Sciences* proposed that effect sizes could be generalized as follows:

η** ^{2}** = 0.01 for a small effect. A small effect is one that not easily observable.

η** ^{2}** = 0.05 for a medium effect. A medium effect is more easily detected than a small effect but less easily detected than a large effect.

η** ^{2}** = 0.14 for a small effect. A large effect is one that is readily detected with the current measuring equipment.

The above values of η** ^{2}** produce the following values of effect size f:

f = 0.1 for a small effect.

f = 0.25 for a medium effect.

f = 0.4 for a large effect.

c)** Alpha** – This is commonly set at 0.05.

## Calculating Power With Online

Tool G Power

### 1) *A Priori*

- An example of *a priori* Power calculation would be the following. Power calculations are normally used a priori to determine the total ANOVA sample size necessary to achieve a specific Power level for detecting an effect of a specified size at a given alpha.

The single-factor ANOVA example used in this chapter has three groups. The G-Power utility could be used *a priori* in this way:

Calculate the total sample needed achieve the following parameters:

Power level = 0.8 (80 percent chance of detecting the effect)

Effect size f = 0.4 (a large effect)

Number of Groups = 3

Alpha = 0.05

The G-Power dialogue box would be filled in as follows and calculates that a total sample size of 66 would be needed have attain a Power of 0.818 (81.8 percent) to detect a large effect of effect size f = 0.4. The example used in this chapter has a total of 63 data observations. That would be nearly a large enough total size to have an 80 percent chance of detecting a large effect (f = 0.4) at an alpha = 0.05.

*(Click Image To See a Larger Version)*

### 2) *Post hoc*

- Calculation of a test’s Power to detect an effect of a given size at a given alpha for a given sample size. This is usually conducted *post hoc* (after a test has been performed). If a test’s Power is deemed unacceptably low, the test’s results are usually considered invalid.

An example of a *post hoc* Power calculation would be the following. Power calculations are normally used *post hoc* to determine the current Power level of an ANOVA test for detecting an effect of a specified size at a given alpha given the total sample size.

The single-factor ANOVA example used in this chapter has three groups. The G-Power utility could be used *post hoc* in this way:

Calculate the total sample needed achieve the following parameters:

Effect size f = 0.25 (a medium effect)

Number of Groups = 3

Total sample size = 63

Alpha = 0.05

The G-Power dialogue box would be filled in as follows and calculates that this single-factor ANOVA test achieves a Power level of 0.391 (39.1 percent chance) to detect a medium effect (effect size f = 0.25) with three groups of 63 total data observations.

*(Click Image To See a Larger Version)*

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

- Histograms in Excel
- Bar Chart in Excel
- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013
- One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013
- Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013

- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- 1-Sample t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013

- 2-Independent-Sample Pooled t-Tests in Excel
- 2-Independent-Sample Pooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2- Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in 12 Steps in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
- Chi-Square Goodness-Of-Fit Tests in Excel
- F Tests in Excel
- Correlation in Excel
- Pearson Correlation in Excel
- Spearman Correlation in Excel
- Confidence Intervals in Excel
- z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression in 6 Steps in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer- Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

- Two-Factor ANOVA Without Replication in Excel
- Randomized Block Design ANOVA in Excel
- Repeated-Measures ANOVA in Excel
- Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel 2010 and Excel 2013
- Sphericity Testing in 9 Steps For Repeated Measures ANOVA in Excel 2010 and Excel 2013
- Effect Size For Repeated-Measures ANOVA in Excel 2010 and Excel 2013
- Friedman Test in 3 Steps For Repeated-Measures ANOVA in Excel 2010 and Excel 2013

- ANCOVA in Excel
- Normality Testing in Excel
- Creating a Box Plot in 8 Steps in Excel
- Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel

- Nonparametric Testing in Excel
- Mann-Whitney U Test in 12 Steps in Excel
- Wilcoxon Signed-Rank Test in 8 Steps in Excel
- Sign Test in Excel
- Friedman Test in 3 Steps in Excel
- Scheirer-Ray-Hope Test in Excel
- Welch's ANOVA Test in 8 Steps Test in Excel
- Brown-Forsythe F Test in 4 Steps Test in Excel
- Levene's Test and Brown-Forsythe Variance Tests in Excel
- Chi-Square Independence Test in 7 Steps in Excel
- Chi-Square Goodness-of-Fit Tests in Excel
- Chi-Square Population Variance Test in Excel

- Post Hoc Testing in Excel
- Creating Interactive Graphs of Statistical Distributions in Excel
- Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
- Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013
- Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
- Solving Poisson Distribution Problems in Excel 2010 and Excel 2013

- Optimization With Excel Solver
- Maximizing Lead Generation With Excel Solver
- Minimizing Cutting Stock Waste With Excel Solver
- Optimal Investment Selection With Excel Solver
- Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
- Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
- Optimizing a Bond Portfolio With Excel Solver
- Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers

- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables
- SEO Functions in Excel
- Time Series Analysis in Excel
- VLOOKUP

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