This is one of the following six articles on 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

# Overview of 1-Sample t-

Test Nonparametric

Alternatives in Excel

There are two nonparametric tests that can be substituted for the one-sample t-Test when normality of the sample or population cannot be verified and sample size is small. These two tests are the Wilcoxon One-Sample, Sign-Rank Test and the Sign Test. The one-sample t-test is used to evaluate whether a population from which samples are drawn has the same *mean* as a known value. The nonparametric tests evaluate whether the sample have the same *median* as a known value.

The Sign Test is significantly less powerful alternative to the Wilcoxon One-Sample Signed-Rank test, but does not assume that the differences between the samples and the known value is symmetrical about a median, as does the Wilcoxon One-Sample Signed-Rank test when used as a nonparametric alternative to the one-sample t-test. The Sign Test is non-directional and can be substituted only for a two-tailed test but not for a one-tailed test.

The Wilcoxon test is based upon the sum of rankings of values while the Sign Test is based upon the sum of positive versus negative values.

The Wilcoxon One-Sample Signed Rank is much more powerful (able to detect a difference) than the Sign Test but has a required assumption that sample data are distributed about a median is a relatively symmetric fashion. The Sign Test does not have this assumption.

## Wilcoxon One-Sample, Signed-Rank

Test in Excel

The Wilcoxon One-Sample, Signed-Rank Test is an alternative to the one-sample t-Test when sample size is small (n < 30) and normality cannot be verified for the sample data or the population from which the sample was taken.

The Wilcoxon One-Sample, Signed-Rank Test calculates the difference between each data point in the sample and the Constant from the t-Test’s Null Hypothesis (186,000 in this case). The absolute values of each difference and ranked and then assigned the sign (positive or negative) that the difference originally had. These signed ranks are summed up to create the Test Statistic W.

Test Statistic W will be approximately normally distributed if the required assumptions are met for this test. The Test Statistic’s z Score can then be calculated and compared with the Critical z value. The decision whether or not to reject the test’s Null Hypothesis is made based on the results of this comparison.

The Null Hypothesis for this test states that the median of the difference population equals a Constant. This is somewhat similar to the Null Hypothesis of the one-sample t-Test which states that the mean of a population equals a Constant.

The Wilcoxon One-Sample, Signed-Rank Test is performed on this data by implementing the following steps:

### Step 1) Calculate the Difference Between Each Sample Data Point and the Constant to Which the Sample Is Being Compared.

The original Null Hypothesis from the one-sample t-Test stated that the **mean** monthly retails sales for the stores in a single region is equal to the nation average which is 186,000. The Null Hypothesis for this t-Test was as follows:

H_{0}: x_bar = Constant =186,000

A difference sample consisting of the differences between each sample data point and the Constant (186,000) is created as follows:

(Click On Image and See a larger Version)

### Step 2) Create the Null and Alternative Hypotheses.

The one-sample t-Test attempts to determine whether the **mean** monthly retails sales for the stores in a single region is equal to the nation average which is 186,000.

The Wilcoxon One-Sample, Signed-Rank Test attempts to determine whether the **median** monthly retails sales for the stores in a single region is equal to 186,000.

If the median monthly retail sales for the region’s stores equals 186,000, then the median of the difference will equal zero. The Null Hypothesis is based on this and is stated as follows:

H_{0}: Median_Difference = Constant = 0

The Alternative Hypothesis is non-directional because the test’s overall purpose is to determine only whether or not the regional mean monthly retail sales equals the national average of 186,000. The Alternative Hypothesis for this Wilcoxon One-Sample, Signed-Rank Test will therefore be stated as follows:

H_{1}: Median_Difference ≠ Constant = 0

H_{1}: Median_Difference ≠ 0

### Step 3) Evaluate Whether the Test’s Required Conditions Have Been Met

The Wilcoxon One-Sample, Signed-Rank Test has the following requirements:

a) Data are ratio or interval but not categorical (nominal or ordinal). This is the case here.

b) Sample size is at least 10.

c) Data of the Difference sample are distributed about a median with reasonable symmetry. Test Statistic W will not be normally distributed unless this assumption is met.

The following Excel-generated histogram shows that the difference data are distributed symmetrically about their median of 14,000:

(Click On Image and See a larger Version)

This histogram and the sample’s median were generated in Excel as follows:

(Click On Image and See a larger Version)

### Step 4 – Record the Sign of Each Difference

Place a “+1” and “-1” next to each non-zero difference. This can be automatically generated with an If-Then-Else statement as follows:

(Click On Image and See a larger Version)

Placing a plus sign (+) next to a number automatically requires a custom number format available from the Format Cell dialogue box. One custom format that will work is the following: “+”#:”-“# . This is demonstrated in following Excel screen shot:

(Click On Image and See a larger Version)

### Step 5 – Sort the Absolute Values of the Differences While Retaining the Sign Associated With Each Difference

Sort both columns based upon column of difference absolute values.

(Click On Image and See a larger Version)

### Step 6 –Rank the Absolute Values, Attach the Signs, and Sum up the Signed Ranks to Create Test Statistic W.

The absolute values are ranked in ascending order starting with a rank of 1. Absolute values that are tied area assigned the average rank of the tied values. For example, the first four absolute values are 6000. Each of these four absolute values would be assigned a rank of 2.5, which is equal to the average rank of all four, i.e., (1 + 2 + 3 + 4) / 4 = 2.5.

Test Statistic W is equal to the sum of all signed ranks.

(Click On Image and See a larger Version)

### Step 7 – Calculate the z Score of W

The distribution of Test Statistic W can be approximated by the normal distribution if all of the required assumptions for this test are met. The difference data consists of more than 10 points of ratio data that are reasonably symmetrically distributed about their median. The assumptions are therefore met for this Wilcoxon One-Sample, Signed-Rank Test.

The standard deviation of W, σ_{W}, is calculated as follows:

σ_{W} = SQRT[ n(n + 1)(2n + 1)/6 ] = 53.57

z Score = ( W – Constant – 0.5) / σ_{W}

z Score = ( 110 – 0 – 0.5) / 53.57 = 2.04

The constant is the Constant from the Null Hypothesis for this test, which is the following:

H_{0}: Median_Difference = Constant = 0

The z Score must include a 0.5 correction for continuity because W assumes whole integer values (except in the event of a tie of ranks).

### Step 8 – Reject or Fail to Reject the Null Hypothesis Based Upon a Comparison Between the z Score and the Critical z Value

Given that α = 0.05 and this is a two-tailed test, the Critical z Value is calculated as follows:

Z Critical_{α=0.05,Two-Tailed} = ±NORM.S.INV(1 – α/2) = ±NORM.S.INV(0.975)

Z Critical_{α=0.05,Two-Tailed} = ±1.9599

The Null Hypothesis is rejected if the z Score is further from the standardized mean of zero than the Critical z Values. This is the case here since the z Score (2.04) is further from the standardized mean of zero than the Critical z Values (±1.9599). These results from the Wilcoxon Signed-Rank Test are shown in the following Excel-generated graph:

(Click On Image and See a larger Version)

Rejection of the Null Hypothesis for this test can be interpreted to state that there is at least 95 percent certainty that the median of the difference sample does not equal zero. This would mean that there is 95 percent certainty that the **median** monthly sales of the retail stores in the region does not equal the national average of 186,000.

The results of this Wilcoxon One-Sample, Signed-Rank Test were very similar to the results of the original one-sample t-Test in which the Null Hypothesis was rejected because the t value (2.105) was further from the standardized mean of zero than the Critical t Value (2.093). The results of this t-Test indicate 95 percent certainty that the **mean** monthly sales of the retail stores in the region does not equal the national average of 186,000.

The results of the t-Test are shown in the following Excel-generated graph of this non-standardized t Distribution:

(Click On Image and See a larger Version)

The Wilcoxon One-Sample Signed-Rank Test detects that the ** median** difference between the region’s retail store monthly sales and the national average is significant at an alpha level of 0.05.

The one-sample t- Test detects that the ** mean** difference between the region’s retail store monthly sales and the national average is significant at an alpha level of 0.05.

**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
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- 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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