Find definitions and interpretation guidance for every statistic that is provided with the 1-sample Wilcoxon analysis.

The sample size (N) is the total number of observations in the sample.

The sample size affects the confidence interval and the power of the test.

Usually, a larger sample size results in a narrower confidence interval. A larger sample size also gives the test more power to detect a difference. For more information, go to What is power?.

The median is the midpoint of the pairwise averages. Pairwise averages (also called Walsh averages) are the means of each possible pair of values in your data set, including the pair of each value with itself. The median is determined by ranking the pairwise averages and finding the value that is at the number [N + 1] / 2 in the ranked order. If the number of observations are even, then the median is the average value of the pairwise averages that are ranked at numbers N / 2 and [N / 2] + 1.

The median of the pairwise averages is an estimate of the population median.

Because the median is based on sample data and not on the entire population, it is unlikely that the sample median equals the population median. To better estimate the population median, use the confidence interval.

To get both the confidence interval and the test results you must perform the analysis twice because Minitab only calculates one item at a time.

The confidence interval provides a range of likely values for the population median. Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. But, if you repeated your sample many times, a certain percentage of the resulting confidence intervals or bounds would contain the unknown population median. The percentage of these confidence intervals or bounds that contain the median is the confidence level of the interval. For example, a 95% confidence level indicates that if you take 100 random samples from the population, you could expect approximately 95 of the samples to produce intervals that contain the population median.

An upper bound defines a value that the population median is likely to be less than. A lower bound defines a value that the population median is likely to be greater than.

The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size.

Because of the discreteness of the Wilcoxon statistic, it is not always possible to achieve a confidence interval at the requested confidence level. Minitab calculates the closest achievable value using a normal approximation with a continuity correction.

To get both the confidence interval and the test results you must perform the analysis twice because Minitab only calculates one item at a time.

Sample | N | Median | CI for η | Achieved Confidence |
---|---|---|---|---|

Time | 16 | 11.55 | (9.2, 12.6) | 94.75% |

In these results, the estimate of the population median for reaction time is 11.55. You can be 94.75% confident that the population median is between 9.2 and 12.6.

Because of the discreteness of the Wilcoxon statistic, it is not always possible to achieve a confidence interval at the requested confidence level. Minitab calculates the closest achievable value using a normal approximation with a continuity correction.

The achieved confidence indicates how likely it is that the population median is contained in the confidence interval. For example, a 95% confidence level indicates that if you take 100 random samples from the population, you could expect approximately 95 of the samples to produce intervals that contain the population median.

The null and alternative hypotheses are two mutually exclusive statements about a population. A hypothesis test uses sample data to determine whether to reject the null hypothesis.

- Null hypothesis
- The null hypothesis states that a population parameter (such as the mean, the standard deviation, and so on) is equal to a hypothesized value. The null hypothesis is often an initial claim that is based on previous analyses or specialized knowledge.
- Alternative hypothesis
- The alternative hypothesis states that a population parameter is smaller, larger, or different from the hypothesized value in the null hypothesis. The alternative hypothesis is what you might believe to be true or hope to prove true.

In the output, the null and alternative hypotheses help you to verify that you entered the correct value for the test median.

To calculate N for a 1-sample Wilcoxon test, Minitab eliminates the observations that are equal to the hypothesized median. N for a 1-sample Wilcoxon test equals the number of remaining observations.

N for a 1-sample Wilcoxon test affects the power of the test. A larger value gives the test more power to detect a difference. For more information, go to What is power?.

The Wilcoxon statistic equals the number of pairwise averages (also called Walsh averages) that are greater than the hypothesized median, plus one half of the number of pairwise averages that are equal to the hypothesized median.

Minitab calculates the Wilcoxon statistic as follows:

- Minitab eliminates any observations that are equal to the hypothesized median.
- Minitab forms pairwise (Walsh) averages, (Y
_{i}+ Y_{j}) / 2 for i < j. - Minitab calculates the statistic as stated above.

Minitab uses the Wilcoxon statistic to calculate the p-value, which is a probability that measures the evidence against the null hypothesis.

Because the interpretation of the Wilcoxon statistic depends on the sample size, you should use the p-value to make a test decision. The p-value has the same meaning for any sample size.

The p-value is a probability that measures the evidence against the null hypothesis. A smaller p-value provides stronger evidence against the null hypothesis.

Use the p-value to determine whether the population median is statistically different from the hypothesized median.

To determine whether the difference between the population median and the hypothesized median is statistically significant, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

- P-value ≤ α: The difference between the medians is significantly different (Reject H
_{0}) - If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis. You can conclude that the difference between the population median and the hypothesized median is statistically significant. Use your specialized knowledge to determine whether the difference is practically significant. For more information, go to Statistical and practical significance.
- P-value > α: The difference between the medians is not significantly different (Fail to reject H
_{0}) - If the p-value is greater than the significance level, the decision is to fail to reject the null hypothesis. You do not have enough evidence to conclude that the population median is significantly different from the hypothesized median. You should make sure that your test has enough power to detect a difference that is practically significant.