This is one of the following eight articles on 2-Independent-Sample Pooled t-Tests in Excel

2-Independent-Sample Pooled t-Test in 4 Steps in Excel 2010 and Excel 2013

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

2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

# How Sample Standard

Deviation Affects t-Test

Results

When the standard deviation in sample groups is increased, the sample groups harder to tell apart. This might be more intuitive to understand if presented visually.

Below are box plots of three sample groups each having a small sample standard deviation:

*(Click On Image To See a Larger Version)*

Each of the sample groups is visually easy to differentiate from the others. The measures of spread - standard deviation and variance - are shown for each sample group. Remember that variance equals standard deviation squared.

If each sample group’s spread is increased (widened), the sample groups become much harder to differentiate from each other. The graph shown below is of three sample groups having the same means as above but much wider spread.

*(Click On Image To See a Larger Version)*

It is easy to differentiate the sample groups in the top graph but much less easy to differentiate the sample groups in the bottom graph simply because the sample groups in the bottom graph have much wider spread.

In statistical terms, one could say that it is easy to tell that the samples in the top graph were drawn from different populations. It is much more difficult to say whether the sample groups in the bottom graph were drawn from different populations.

## Relationship Between the Two-

Independent-Sample, Pooled t-Test

and Single-Factor ANOVA

The preceding illustrates the underlying principle behind both t-tests and ANOVA tests. One of the main purposes of both t-tests and ANOVA tests is to determine whether samples are from the same populations or from different populations. The variance (or equivalently, the standard deviation) of the sample groups is what is what determines how difficult it is to tell the sample groups apart.

The two-independent-sample, pooled t-test is essentially the same test as single-factor ANOVA. The two-independent-sample, pooled t-test can only be applied to two sample groups at one time. Single-Factor ANOVA can be applied to three or more groups at one time. Both two-independent-sample, pooled t-test and single-factor ANOVA require that variances of sample groups be similar.

We will apply both the two-independent sample t-test and single-factor ANOVA to the first two samples in each of the above graphs to verify that the results are equivalent.

__Sample Groups With Small Variances (the first graph)__

*(Click On Image To See a Larger Version)*

Applying a two-independent-sample, pooled t-test to the first two of the three sample groups of this graph would produce the following result:

*(Click On Image To See a Larger Version)*

This result would have been obtained by filling in the Excel dialogue box as follows:

*(Click On Image To See a Larger Version)*

Running Single-Factor ANOVA on those same two sample groups would produce this result:

*(Click On Image To See a Larger Version)*

This result would have been obtained by filling in the Excel dialogue box as follows:

*(Click On Image To See a Larger Version)*

Both the Two-Independent-Sample, Pooled t-test and the Single-Factor ANOVA test produce the same result when applied to these two sample groups. They both produce the same p Value (1.51E-10) which is extremely small. This indicates that the result is statistically significant and that the difference in the means of the two groups is real. More correctly put, it can be stated that there is a very small chance (1.51E-10) that the samples came from the same population and that the result obtained (that their means are different) was merely a random occurrence.

__Sample Groups With Large Variances (the second graph)__

*(Click On Image To See a Larger Version)*

Applying a two-independent sample t-test to the first two of the three sample groups in this graph would produce the following result:

*(Click On Image To See a Larger Version)*

This result would have been obtained by filling in the Excel dialogue box as follows:

*(Click On Image To See a Larger Version)*

Running Single-Factor ANOVA on those same two sample groups would produce this result:

*(Click On Image To See a Larger Version)*

This result would have been obtained by filling in the Excel dialogue box as follows:

*(Click On Image To See a Larger Version)*

Both the t-test and the ANOVA test produce the same result when applied to these two sample groups. They both produce the same p Value (0.230876). This is relatively large. 95 percent is the standard level of confidence usually required in statistical hypothesis tests to conclude that the results are statistically significant (real). The p value needs to be less than 0.05 to achieve a 95 percent confidence level that a difference really exists. The sample groups with the large spread produced a p Value greater than 0.05 and we can therefore not reject the Null Hypothesis which states that the sample groups are the same. The results are not statistically significant and we cannot conclude that the two samples were not drawn from the same population.

## Showing How the Formulas For

Both the t-Test and for ANOVA

Produce the Same Result

### t-Test Formula

The Two-Independent-Sample, Pooled t-Test is used to determine with a specific degree of certainty whether there really is a difference between the mean values of two sample groups given a similar amount of variance in each of the two sample groups.

If the sample standard deviation in each of the two sample groups, s_{1} and s_{2}, is large, then the Pooled Standard Deviation will also be large, as can be seen from the following equation:

__Pooled Sample Standard Deviation__

s_{Pooled }= SQRT**[****{**(n_{1}-1)s_{1}^{2} +(n_{2}-1)s_{2}^{2}**}**/df**]**

This, in turn, increases the value of length on one Standard Error, SE_{Pooled}, as can be seen in the following equation:

__Pooled Sample Standard Error____ __

SE_{Pooled} = s_{Pooled} *SQRT(1/n_{1} + 1/n_{2})

This, in turn, decreases the t Value, as can be seen in the following equation:

t Value = (x_bar_{1}-x_bar_{2}) / SE_{Pooled}

The larger the t Value, the more likely it is that the sample groups are different, i.e., came from different populations.

The bottom line is that increased variance (or, equivalently, standard deviation) in the sample groups causes the t Value to be smaller. This makes it less likely that a t-Test will show that the sample groups are really different.

### ANOVA

The ANOVA outputs of the previous two comparisons demonstrate the following:

The smaller the p Value is, the more certainty exists that sample groups are really different, i.e., that the sample groups came from different populations.

The p Value is derived from the F value. The larger the F Value, the smaller is the p Value.

The F value can be roughly described as being the variation between groups divided by the variation within groups (the spread of the groups).

As the spread (standard deviation) of the sample groups increase, the F value become smaller. When the F Value become smaller, the p Value becomes larger. The larger the p Value becomes, the less certainty exists that the ANOVA results are statistically significant (real). If the results are not statistically significant, we cannot reject the Null Hypothesis that states that the sample different (drawn from different populations).

** Bottom line:** the larger the standard deviation of sample groups being compared with a two-independent-sample, pooled t-Test or single-factor ANOVA, that harder it is to state that the sample groups are truly different, i.e., that the sample groups come from different populations.

**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
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- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables and Pivot Charts
- SEO Functions in Excel
- Time Series Analysis in Excel
- VLOOKUP
- Simplifying Useful Excel Functions

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