The median is the midpoint of the data set. This midpoint value is the point at which half of the observations are above the value and half of the observations are below the value. The median is determined by ranking the observations and finding the observation at the number [N + 1] / 2 in the ranked order. If your data contain an even number of observations, the median is the average value of the observations that are ranked at numbers N / 2 and [N / 2] + 1.
The sample median is an estimate of the population median of each group. The overall median is the median of all observations.
N greater than the overall median (N>)
N> (greater than the overall median). These values represent the number of observations in each group that are greater than the overall median. Minitab creates a table with the N≤ values and the N> values. Minitab uses these values to perform the chi-square test of association and to calculate the p-value for the test.
If a group has a large number of observations in this category, the median of the group is likely to be greater than the overall median.
N less than or equal to the overall median (N≤)
N≤ (less than or equal to the overall median) is the number of observations in each group that are less than or equal to the overall median. Minitab creates a table with the N≤ values and the N> values. Minitab uses these values to perform the chi-square test of association and to calculate the p-value for the test.
If a group has a large number of observations in this category, the median of the group is likely to be less than the overall median.
Interquartile range (Q3 – Q1)
The interquartile range (Q3 – Q1) measures the spread of the data in each group. The range is the distance between the 75th percentile (Q3) and the 25th percentile (Q1).
Interquartile ranges that differ substantially indicate that the groups do not have the same spread. This condition suggests that the data may not satisfy the assumption for Mood's median test that the groups have the same shape and spread.
Confidence interval (95% Median CI)
The confidence intervals are ranges of values that are likely to contain the true median of each population.
Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. But, if you repeat your sample many times, a certain percentage of the resulting confidence intervals contain the unknown population parameter. The percentage of these confidence intervals that contain the parameter is the confidence level of the interval.
The confidence interval is composed of the following two parts:
The point estimate is the estimate of the parameter that is calculated from the sample data. The confidence interval is centered around this value. For Mood's median test, the point estimate is the median estimate.
Margin of error
The margin of error defines the width of the confidence interval and is determined by the observed variability in the sample, the sample size, and the confidence level. To calculate the upper limit of the confidence interval, the margin of error is added to the point estimate. To calculate the lower limit of the confidence interval, the margin of error is subtracted from the point estimate.
Use the confidence interval to assess the estimate of the population median for each group.
For example, with a 95% confidence level, you can be 95% confident that the confidence interval contains the group median. 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.
Null hypothesis and alternative hypothesis
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.
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.
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.
The degrees of freedom (DF) equals the number of groups in your data minus 1. Under the null hypothesis, chi-square distribution approximates the distribution of the test statistic, with the specified degrees of freedom. Minitab uses the chi-square distribution to estimate the p-value for this test.
The chi-square statistic is calculated from a table comprised of cells that are based on the groups in your data and the groups' corresponding N≤ values and N> values. Minitab calculates each cell's value as the square of the difference between the observed and expected values for a cell, divided by the expected value for that cell. The chi-square statistic is the sum of these values.
A higher chi-square value indicates that the difference between the observed and expected values is higher. A sufficiently large chi-square value indicates that at least one difference between the medians is statistically significant. Minitab uses the chi-square statistic, in conjunction with the chi-square distribution, to calculate the p-value.
You can use the chi-square statistic to determine whether to reject the null hypothesis. However, using the p-value of the test to make the same determination is usually more practical and convenient.
The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.
Use the p-value to determine whether any of the differences between the medians are statistically significant.
To determine whether any of the differences between the medians are statistically significant, compare the p-value to your significance level to assess the null hypothesis. The null hypothesis states that the population medians are all equal. 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 differences between some of the medians are statistically significant
If the p-value is less than or equal to the significance level, you reject the null hypothesis and conclude that not all the population medians are equal. Use your specialized knowledge to determine whether the differences are practically significant. For more information, go to Statistical and practical significance.
P-value > α: The differences between the medians are not statistically significant
If the p-value is greater than the significance level, you do not have enough evidence to reject the null hypothesis that the population medians are all equal. Verify that your test has enough power to detect a difference that is practically significant. For more information, go to Increase the power of a hypothesis test.