This is one of the following three articles on F Tests in Excel in Excel

F-Test in 6 Steps in Excel 2010 and Excel 2013

Normality Testing For F Test In Excel 2010 and Excel 2013

Levene’s and Brown- Forsythe Tests: F-Test Alternatives in Excel

# F-Test in 6 Steps in Excel

2010 and Excel 2013

The variances of two normally-distributed populations can be compared for equality using the F-Test. The F-Test is a two-sample, two-tailed population variance test. This is a hypothesis test with a Null Hypothesis stating that the variances of both populations are the same. The Null Hypothesis is shown as follows:

H_{0}: σ_{1} = σ_{2} = σ

Note that population variance = σ^{2}

The F Test is always performed as a one-tailed test in the right tail with the Alternative Hypothesis constructed as follows:

H_{0}: σ_{1} > σ_{2}

The F Test is performed as a one-tailed test in the right tail because the sample with the larger standard deviation of the two samples is designated as sample 1. The population from which that sample was taken is designated as population 1. The two parameters associated with sample 1 and population 1 are s_{1} (sample 1 standard deviation) and σ_{1} (population 1 standard deviation).

The F distribution describes the distribution of the F statistic, also called the f value. An F statistic can be calculated if two independent random samples are taken from two normally-distributed populations. The following parameters associated with the two samples and populations that must be determined are the following:

n_{1} = size of sample 1

n_{2} = size of sample 2

df_{1} = degrees of freedom 1 = n_{1} – 1

df_{2} = degrees of freedom 2 = n_{2} - 1

s_{1} = standard deviation of sample 1

σ_{1} = standard deviation of population 1

Χ^{2}_{1} = Chi-Square statistic from population 1 = df_{1} * s_{1}^{2} / σ_{1}^{2}

s_{2} = standard deviation of sample 2

σ_{2} = standard deviation of population 2

Χ^{2}_{2} = Chi-Square statistic from population 2 = df_{2} * s_{2}^{2} / σ_{2}^{2}

The F statistic can then be calculated in any of the following four equivalent ways:

*f* = [ *s*_{1}^{2}/σ_{1}^{2} ] / [ *s*_{2}^{2}/σ_{2}^{2} ]

*f* = [ *s*_{1}^{2} * σ_{2}^{2} ] / [ *s*_{2}^{2} * σ_{1}^{2} ]

*f* = [ Χ^{2}_{1} / df_{1} ] / [ Χ^{2}_{2} / df_{2} ]

*f* = [ Χ^{2}_{1} * df_{2} ] / [ Χ^{2}_{2} * df_{1} ]

The numerator of the F statistic should be the parameters associated with the larger s.

The distribution of all possible values of the *f* statistic is called the F distribution, with *v*_{1} and *v*_{2} degrees of freedom.

Since the F distribution has the chi-square distribution as a component, many of the chi-square distribution properties are also properties of the F distribution such as the following:

1) The distribution is non-symmetric.

2) The mean is approximately 1.

3) The F-values are all non-negative.

4) There are two independent degrees of freedom, one for the numerator, and one for the denominator.

5) Each different F distribution has a unique pair of degrees of freedom.

The F Test is a hypothesis test determines if the variances of two normally-distributed populations are significantly different based upon the standard deviations of samples taken from each population.

The F Test is performed by comparing the calculated F statistic to an F Critical Value, F_{α}(df_{1},df_{2}). Alpha, α, is the specified level of significance for the hypothesis test. The Null Hypothesis that the two variances are the same is rejected if the F statistic is greater than F Critical. Equivalently, the Null Hypothesis is also rejected of the p Value (the area in the right tail of the F distribution curve that is beyond the F statistic) is smaller than alpha.

It should be noted that the F Test is extremely sensitive to non-normality. It is very important to verify normality of both samples or both populations prior to performing an F Test.

## F Test Example in Excel

Determine with 95 percent certainty whether the variances of battery lifetime of Brand A and brand B are significantly different from each other.

* (Click On Image To See Larger Version)*

### F Test Step 1 – Run Descriptive Statistics on the Data Samples

Descriptive statistics run on the above data samples produces the following result:

*(Click On Image To See Larger Version)*

**Example Data**

s_{1}^{2} = 286.13 (This sample is designated sample 1 because its variance is larger)

s_{2}^{2} = 232.39

n_{1} = 16

n_{2} = 17

df_{1} = n_{1} – 1 = 15

df_{2} = n_{2} – 1 = 16

### F Test Step 2 – Verify Normality of Both Populations

The F Test is extremely sensitive to non-normality and will produce an incorrect result if either population is not normally distributed. It is therefore very important to verify the normality of both populations prior to performing the F Test.

If the normality of both populations cannot be confirmed, the normality of both samples must be confirmed. Large sample size (n > 30) does not waive the normality requirement as occurs with t Tests.

An Excel histogram is the quickest way to attain a rough assessment of the normality of a data sample. Histograms of both data samples are shown as follows. The histogram indicates that the sample data is normally distributed. The normal distribution of the sample data infers that the populations from which the sample came are also normally distributed as required by the F Test.

An in-depth analysis of the normality of the sample data will be performed at the end of this section. For brevity, this F Test’s requirement of population normality will be considered satisfied by the following bell-shaped Excel histograms of the data from each of the two samples. Excel histograms of both sample groups are as follows:

*(Click On Image To See Larger Version)*

To create this histogram in Excel, fill in the Excel Histogram dialogue box as follows:

*(Click On Image To See Larger Version)*

*(Click On Image To See Larger Version)*

To create this histogram in Excel, fill in the Excel Histogram dialogue box as follows:

*(Click On Image To See Larger Version)*

Both sample groups appear to be distributed reasonably closely to the bell-shaped normal distribution. It should be noted that bin size in an Excel histogram is manually set by the user. This arbitrary setting of the bin sizes can has a significant influence on the shape of the histogram’s output. Different bin sizes could result in an output that would not appear bell-shaped at all. What is actually set by the user in an Excel histogram is the upper boundary of each bin.

**In-Depth Analysis of Sample Normality**

The F Test is extremely sensitive to non-normality of either population from which the samples were taken. A population’s normality is confirmed when a sample taken from that population is shown to be normally distributed. The preceding F test was performed on the basis of bell-shaped histograms of each of the two samples’ data. Other methods of confirming sample normality are listed as follows:

**Evaluating the Normality of the Sample Data**

The following five normality tests will be performed on the sample data in a blog article following this one:

An Excel histogram of the sample data will be created.

A normal probability plot of the sample data will be created in Excel.

The Kolmogorov-Smirnov test for normality of the sample data will be performed in Excel.

The Anderson-Darling test for normality of the sample data will be performed in Excel.

The Shapiro-Wilk test for normality of the sample data will be performed in Excel.

**All of these normality tests listed above are performed in Excel in the next article in this blog.**

Having confirmed the F Test’s requirement of normality of both populations, the F Test can be conducted as follows:

### F Test Step 3 – Create the Null and Alternative Hypotheses

H_{0}: σ_{1} = σ_{2}

H_{1}: σ_{1} > σ_{2 }– indicates that this is a one-tailed test in the right tail

### F Test Step 4 – Calculate the F Statistic

*f* = [ s_{1}^{2}/σ_{1}^{2} ] / [ s_{2}^{2}/σ_{2}^{2} ]

s_{1} is larger than s_{2} and should therefore go in the numerator. Since the Null Hypothesis states that the population variances, σ_{1} and σ_{2}, are equal, the F statistic can be reduced to the following:

F Statistic =* f* = s_{1}^{2 }/ s_{2}^{2}

*f* = 286.13^{ }/ 232.39 = 1.226

### F Test Step 5 – Calculate F Critical

F Critical = F_{α}(df_{1}, df_{2}) = F_{α=0.05}(df_{1} = 15, df_{2} = 16)

F Critical = F_{α}(df_{1}, df_{2}) = F.INV.RT(α, df_{1}, df_{2}) = F.INV.RT(0.05,15,16) = 2.352

### F Test Step 6 – Compare the F Statistic to F Critical

F Statistic (*f* = 1.226) is smaller than F Critical (2.352) so the Null Hypothesis is not rejected. There is not sufficient evidence at α = 0.05 to state that the variances of the two populations (the battery lifetimes of brand A and brand B) are significantly different.

Equivalently, the p value can be compared to alpha as follows:

p Value = F.DIST.RT(F statistic, df_{1}, df_{2}) = F.DIST.RT(1.226,15,16) = 0.345

The p Value (0.345) is larger than alpha (0.05) so the Null Hypothesis is not rejected.

This result shown on this Excel-generated graph of the F distribution with df_{1}=15 and df_{2}=16 is as follows:

*(Click On Image To See Larger Version)*

The Null Hypothesis of an F Test states that the variances of the two groups are the same. The p Value shown in the Excel F Test output equals 0.345. This is much larger than the Alpha (0.05) that is typically used for an F Test so the Null Hypothesis cannot be rejected. A p value of 0.345 indicates that there is a 34.5 percent probability of a Type I error, i.e. a false positive. This means that there is a 34.5 percent probability that the difference in the variances shown by the test do not exist and are merely the chance result of random sampling from each population.

The p value needs to be no larger than 0.05 to be at least 95 percent certain that the test’s indication of a difference between the population variances is a true result. A p Value of 0.345 indicates that only 65.5 percent certainty exists that the a difference between the population variances really exists.

### Performing the F Test With the Data Analysis F Test Tool

The F Test can be performed in one step by using the Excel Data Analysis F Test tool. this tool can be accessed under the Data tab as follows:

**Data tab / Data Analysis / F Test Two Sample for Variances**

The F Test dialogue box then appears and should be completed as follows:

*(Click On Image To See Larger Version)*

Hitting the OK button will produce the following output. Directly below the output are the calculations that duplicate the output created by this tool.

*(Click On Image To See Larger Version)*

### F Test Alternatives That Are Less Sensitive To Data Non-Normality Than the F Test

The F Test is extremely sensitive to non-normality of data. In many cases it is better to apply variance-comparison tests that are less sensitive to non-normality than the F Test. The two most widely-used tests to compare sample group variance are Levene’s Test and the Brown-Forsythe sample variance test. **Levene’s Test and the Brown-Forsythe sample variance test will be performed on this sample data in an article that is two articles later in this blog.**

**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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