This is one of the following seven articles on 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

# The Sign Test in Excel

As an Alternative to

the Paired t-Test

The Sign Test along with the Wilcoxon Signed-Rank Test are nonparametric alternatives to the paired t-Test when the normality of the difference sample cannot be verified and the sample size is small.

The Wilcoxon Signed-Rank Test is significantly more powerful than the Sign Test but has a requirement of symmetrical distribution about a median for the difference sample data (the data set of the sample points minus the Constant of the Null Hypothesis). The Wilcoxon Signed-Rank Test is based upon a normal approximation of its Test Statistic’s distribution. This requires that the difference sample be reasonably symmetrically distributed about a median.

The Sign Test has no requirements regarding the distribution of data but, as mentioned, is significantly less powerful than the Wilcoxon Signed-Rank Test.

The Sign Test counts the number of positive and negative non-zero differences between difference sample data and the Constant from the Null Hypothesis in the paired t-Test. In this case that Constant = 0 because the Null Hypothesis of the one-tailed t-Test is as follows:

H_{0}: x_bar_{diff} = Constant = 0

The after-minus-before difference sample for this problem is calculated as follows:

* (Click On Image To See Larger Version)*

The final difference sample is created by subtracting the Constant from the Null Hypothesis, 0, from the after-minus-before difference as follows:

* (Click On Image To See Larger Version)*

A count of positive and negative differences in this sample is taken as follows:

* (Click On Image To See Larger Version)*

The minimum count of positive or negative non-zero differences is designated as the Test Statistic W for the Sign Test. Test Statistic W is named after Frank Wilcoxon who developed the test.

The objective of the one-tailed, paired t-Test was to determine whether to reject or fail to reject the Null Hypothesis that states that the mean difference between the number of clerical errors before and after the training for all employees is equal to 0.

If the mean difference is equal to 0, then the probability of a difference being positive (greater than zero) is the same as the probability of being negative (less than zero). This probability is 50 percent.

The Null Hypothesis for this one-tailed, Sign Test states that the probability of a difference being positive (p) is 50 percent. This can be expressed as follows:

H_{0}: p=0.5

The Alternative Hypothesis would state the following:

H_{1}: p≠0.5

Each non-zero difference is classified as either positive or negative. This is a binary event because the classification of each difference has only two possible outcomes: the non-zero difference is either positive or negative.

The distribution of the outcomes of this binary event can be described by the binomial distribution as long as the following two conditions exist:

1) Each binary trial is independent.

2) The data from which the differences are derived are at least ordinal. The data can be ratio, interval, ordinal, but not nominal. The differences of “less than” and “greater than” must be meaningful even if the amount of difference is not, as would be the case with ordinal data but not with nominal data.

3) Each binary trial has the same probability of a positive outcome.

All of these conditions are met because of the following:

1) Each sample taken is independent of any other sample.

2) The differences are derived from continuous (either ratio or interval) data.

3) The proportion of positive differences versus negative differences is assumed to be constant in the population from which the sample of differences was derived.

The counts of the positive and negative differences both follow the binomial distribution. The lowest count, whether it is the count of positive differences or the count of negative differences, is designated as W, the Test Statistic. This Test Statistic follows the binomial distribution because W represents the count of positive or negative outcomes of independent binary events that all have the same probability of a positive outcome.

As stated, the Null Hypothesis of this one-tailed, paired Sign Test is the following:

H_{0}: p=0.5

If the training program was successful, there would be a reduction in the number of clerical errors. If the number of clerical errors were reduced, there would be a larger number of negative after-minus-before differences than positive differences.

The difference count indicates that there are 11 negative differences and 6 positive differences. These counts are distributed according to the binomial distribution that has a probability of positive outcome, p, equal to 0.5 and the total number of trials, N, equal to 17. As Excel-generated graph of this binomial distribution is shown as follows:

* (Click On Image To See Larger Version)*

This test evaluates whether a count of 11 negative differences is significant at an alpha level of 0.05. The area under the PDF curve beyond 11 differences is equal to the probability that this outcome did not occur by chance. This is the p value for this test.

The Null Hypothesis would be rejected if the p Value calculated from this test is less than alpha, which is customarily set at 0.05.

The binomial distribution is symmetric and the curve area in the right tail beyond 11 differences is the same as the curve area in the left tail beyond 6 differences.

Test Statistic W in the Sign Test is always set to the lower count. The area outside the lower count is equal to the area outside the upper count. That area is the p Value for the one-tailed Sign test.

That p Value is equal to the probability that the number of positive outcomes is less than W = 6 if the total number of nonzero counts = N = 17 and every binary trial has the same probability of a positive outcome = p = 0.5.

Given that variable x is binomially distributed, the CDF (Cumulative Distribution Function) of the x ≤ X is calculated in Excel as follows:

p Value = F(X;n,p) = BINOM.DIST(X, n, p, 1)

This calculates the probability that up to X number of positive outcomes will occur in n total binary trials if the probability of a positive outcome is p for every trial. “1” specifies that the Excel formula will calculate the CDF and not the PDF.

Therefore the following can be calculated:

p Value = Pr (p = No. of Positive Differences ≤ W |p=0.5, N = Total No. of Non-Zero Differences) =

p Value = BINOM.DIST(W, N, p,1)

p Value = BINOM.DIST(7,20,0.5,1) = 0.1316

This is shown in the following Excel-generated graph of the PDF of the binomial distribution for this sign test. The parameters of this binomial distribution are Total Trials = N = 17 and the Probability of a Positive Outcome of Each Trial, p, equal 0.5.

* (Click On Image To See Larger Version)*

The p Value (0.1661) is larger than alpha (set at 0.05). The Null Hypothesis is therefore not rejected at this alpha level. This would be equivalent to stating that there is not enough evidence to reject the Null Hypothesis which ultimately states that there has been no reduction in clerical errors as a result of the training program.

This example demonstrates how much less powerful the one-sample Sign Test is than the paired t-Test or the Wilcoxon Signed-Rank Test. The Sign Test did not come close to detecting a difference at the same alpha level that the other two tests did.

**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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