Nonparametric Trend Test Calculator

Calculator Input

Enter values separated by commas, spaces, semicolons, or new lines. If time values are left blank, the calculator uses 1, 2, 3, ... as the ordered index.

Observed Values

Time or Ordered Index

Significance Level (Alpha)

Confidence Level (%)

Alternative Hypothesis

Displayed Decimal Places

Example Data Table

Time	Value
1	10
2	12
3	11
4	14
5	15
6	16
7	18
8	19
9	21
10	22

This sample produces an increasing pattern, so it is useful for checking whether the trend test identifies a significant positive monotonic trend.

Formula Used

This calculator applies the Mann-Kendall nonparametric trend test and Sen slope estimator. It is appropriate when you want to detect monotonic trends without assuming normality.

1) Mann-Kendall S statistic

S = Σ sgn(y_j − y_i) for all i < j

2) Variance with tie adjustment

Var(S) = [n(n−1)(2n+5) − Σ t_p(t_p−1)(2t_p+5)] / 18

3) Standardized Z statistic

Z = (S−1)/√Var(S) if S > 0, Z = 0 if S = 0, and Z = (S+1)/√Var(S) if S < 0

4) Kendall tau-b

Tau-b standardizes S to measure the direction and strength of monotonic association.

5) Sen slope

Sen slope = median[(y_j − y_i) / (x_j − x_i)] for all i < j

How to Use This Calculator

Enter the observed data values in their natural order.
Enter matching time values, or leave them blank to use sequential indices.
Set alpha for the hypothesis test, such as 0.05.
Choose a two-sided, upward, or downward alternative.
Set the confidence level for the Sen slope interval.
Click Run Trend Test to show results above the form.
Review S, Z, p value, tau-b, Sen slope, and the chart.
Use the CSV or PDF buttons to export the results.

FAQs

1. What does this calculator test?

It tests whether the data show a monotonic upward or downward trend over time. The method focuses on ranked order, not strict linearity, so it works well with skewed or non-normal observations.

2. When should I use a nonparametric trend test?

Use it when your data may violate normality assumptions, contain outliers, or follow an unknown distribution. It is common in environmental, hydrology, finance, and observational trend analysis.

3. What is the difference between S and tau-b?

S counts the net direction of pairwise comparisons. Tau-b rescales that information into a standardized coefficient, making interpretation easier when comparing trend strength across data sets.

4. What does the p value mean here?

The p value measures how surprising the observed trend is under the null hypothesis of no monotonic trend. Smaller values give stronger evidence against that null hypothesis.

5. Why is Sen slope included?

Sen slope estimates the median rate of change per time unit. It adds practical meaning because it tells you how quickly values are rising or falling, not only whether change exists.

6. Can I leave the time field empty?

Yes. When time values are omitted, the calculator assigns 1, 2, 3, and so on. That is useful when observations are equally spaced and already entered in chronological order.

7. How are ties handled?

Repeated observed values are allowed. The variance formula adjusts for ties so the significance test remains appropriate. Repeated time values are not allowed because Sen slope requires distinct denominators.

8. What does a non-significant result imply?

It means the calculator did not find enough evidence for a monotonic trend at your chosen alpha. It does not prove the series is stable, only that evidence was insufficient.