Explain the computation of one sample median test with the help of suitable example.

Introduction to One Sample Median Test

The one sample median test, also known as the Wilcoxon signed-rank test, is a non-parametric statistical test used to determine whether the median of a single sample differs significantly from a hypothesized value. It is particularly useful when the assumptions of parametric tests, such as normality or homogeneity of variance, are not met or when the data are measured on an ordinal or interval scale. The test compares the observed ranks of the sample data to the expected ranks under the null hypothesis of no difference.

Example Scenario

Suppose a researcher wants to assess whether the median score of a sample of students on a standardized test differs significantly from the population median score of 75. The sample consists of 10 students, and their scores are as follows: 80, 85, 70, 75, 78, 82, 79, 74, 76, and 81.

Computing the One Sample Median Test

Step 1: Formulating Hypotheses

The null hypothesis (H_0) states that the median score of the sample is equal to the population median score of 75. The alternative hypothesis (H_1) states that the median score of the sample is not equal to 75.

Step 2: Rank the Absolute Deviations

Calculate the absolute deviations of each observation from the hypothesized median (75) and rank them from smallest to largest, ignoring the sign of the deviation.

Step 3: Calculate the Test Statistic

Calculate the sum of the ranks for the positive deviations (W+) and the sum of the ranks for the negative deviations (W-). In this example, (W+) = 59.5 and (W-) = 22.5.

Step 4: Determine the Critical Value

Determine the critical value of the test statistic at a chosen significance level (e.g., α = 0.05) using the appropriate table for the Wilcoxon signed-rank test with n = 10.

Step 5: Make a Decision

Compare the absolute value of the test statistic to the critical value. If the absolute value of the test statistic exceeds the critical value, reject the null hypothesis and conclude that the median score of the sample differs significantly from 75. Otherwise, fail to reject the null hypothesis.

Step 6: Interpretation of Results

In our example, the absolute value of the test statistic (|W| = 37) exceeds the critical value at α = 0.05 (|Wcrit| = 28). Therefore, we reject the null hypothesis and conclude that the median score of the sample differs significantly from 75.

Conclusion

The one sample median test, or Wilcoxon signed-rank test, is a valuable non-parametric statistical test for comparing the median of a single sample to a hypothesized value. By ranking the absolute deviations of the sample data and comparing them to critical values, researchers can determine whether the median differs significantly from the hypothesized value, even when the data do not meet the assumptions of parametric tests.