Methods and formulas for 1-Sample Wilcoxon

Select the method or formula of your choice.

In This Topic

Pairwise averages
Estimated median
Wilcoxon statistic
P-value
Confidence interval

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.

Formula

= all pairwise averages for i ≤ j.

= the total number of pairwise averages

Notation

Term	Description
Y_i	i^th value in the data set
Y_j	j^th value in the data set
n	sample size

Estimated median

Let W_{( 1 )}< W_{( 2 )}< ... < W_{( M )} denote the ordered values of pairwise averages (also called Walsh averages), where M = n(n+1)/2. If M is odd, the estimated median is the middle value. If M is even, the estimated median is the average of the two middle values. Minitab obtains the point estimate of the population median using an algorithm based on Johnson and Mizoguchi (1978)¹.

D.B. Johnson and T. Mizoguchi (1978). "Selecting the Kth Element in X + Y and X1 + X2 + ... + Xm," SIAM Journal of Computing 7, pp.147-153.

Wilcoxon statistic

The Wilcoxon statistic is the number of pairwise averages (also called Walsh averages) that are larger than the hypothesized median, plus one half the number of pairwise averages that are equal to the hypothesized median. The Wilcoxon statistic is denoted as W. Minitab obtains the test statistic using an algorithm based on Johnson and Miizoguchi (1978)¹.

D.B. Johnson and T. Mizoguchi (1978). "Selecting the Kth Element in X + Y and X1 + X2 + ... + Xm," SIAM Journal of Computing 7, 147-153.

P-value

The Wilcoxon test statistic, W, is the sum of the ranks associated with the observations that exceed the hypothesized median. Minitab calculates the test statistic by using pairwise (Walsh) averages as described in Johnson and Mizoguchi¹:

The number of observations, N, is reduced by one for each observation that is equal to the hypothesized median. The sample size that results is n.
Exclude observations that are equal to the hypothesized median. Calculate n(n + 1) / 2 pairwise Walsh averages (Y_i + Y_j) / 2 for i ≤ j of the observations.

For large sample sizes, the distribution of W is approximately normal. Specifically:

is approximately distributed as a normal distribution with a mean of 0 and a standard deviation of 1, N(0,1).

The normal approximation p-value for the three alternative hypotheses uses a continuity correction of 0.5.

Alternative hypothesis	P-value
H₁: Median > Hypothesized median
H₁: Median < Hypothesized median
H₁: Median ≠ Hypothesized median

Notation

Term	Description
n	the observed number of data points after the observations that are equal to the hypothesized median value are omitted
W	the Wilcoxon test statistic
w	the number of Walsh averages that exceed the hypothesized median, plus half of the number of Walsh averages that equal the hypothesized median.
k

D.B. Johnson and T. Mizoguchi (1978). "Selecting the Kth Element in X + Y and X1 + X2 + ... + Xm," SIAM Journal of Computing 7, pp.147-153.

Confidence interval

The confidence interval is the set of values (d) for which the test of H₀: median = d is not rejected in favor of H₁: median ≠ d, using the confidence level (α = 1 - (percent confidence) / 100). The 1-sample Wilcoxon test does not always achieve the confidence level that you specify because the Wilcoxon statistic is discrete. Because of this, Minitab uses a normal approximation with a continuity correction to calculate the closest achievable confidence level.