Mann-Kendall Trend Test Calculator with Sen's Slope Estimator

The Mann-Kendall (MK) test is a non-parametric test for a monotonic trend in a time series: a tendency for values to consistently increase or decrease over time, not necessarily in a straight line. It was introduced by Henry Mann (1945) and its distribution was developed by Maurice Kendall (1975). It is the default trend test in hydrology, climatology, water-quality monitoring and remote sensing because it makes very few assumptions.

The idea is simple. Compare every observation with every later observation. Count how many later values are higher (+1) and how many are lower (−1). If there is no trend, pluses and minuses should roughly cancel. If there is an upward trend, pluses dominate. The total is the statistic S.

Why researchers prefer it

• No normality assumption. Only the signs of differences matter, so skewed rainfall or streamflow data are fine.
• Robust to outliers. One extreme flood year changes only a few signs, not the whole result.
• Handles missing values and non-detects. Gaps are skipped; values below a detection limit can be treated as ties.

What it assumes

• Independence of observations. This is the assumption most often violated in environmental data, and the reason the modified tests exist (see autocorrelation).
• A consistent measurement process. A gauge relocation or a sensor change can create a step that MK will report as a "trend".
• Monotonicity. A series that rises and then falls can return "no trend" even though it changed a lot.

If you need a refresher on z-scores and normal p-values used in the last step, see our z-score calculator and the bell curve guide. For drought indices that are often tested for trends with MK, read SPI and SPEI explained.

For a series \(x_1, x_2, \dots, x_n\) in time order:

Worked micro-example (n = 5)

Series: 3, 5, 4, 7, 6. Pairs with the first value 3: (5−3)+, (4−3)+, (7−3)+, (6−3)+ → +4. From 5: (4)−, (7)+, (6)+ → +1. From 4: +, + → +2. From 7: (6)− → −1. So \(S = 4+1+2-1 = 6\), \(\operatorname{Var}(S) = 5\cdot4\cdot15/18 = 16.67\), \(Z = (6-1)/4.08 = 1.22\), \(p = 0.22\). An upward tendency, but with five points it is not significant. Load the "Small example" above to see the same mechanics on 10 points, including the sign matrix.

The MK test says whether there is a trend. Sen's slope (Theil 1950; Sen 1968), also called the Theil-Sen estimator, says how fast. It is the median of the slopes between every pair of points:

The intercept of the robust line is \(b = \operatorname{median}(x) - \beta\cdot\operatorname{median}(t)\), so the line passes through the median point. Because Sen's slope is a median, about 29% of the points can be outliers before it breaks down, compared with a single point for ordinary least squares.

Environmental series are often positively autocorrelated: a wet year tends to follow a wet year, groundwater levels change slowly, and NDVI depends on last month's vegetation. Positive autocorrelation makes runs of high and low values look like trends, so the original MK test rejects H₀ far more often than the nominal 5%. The "Autocorrelated" example above is a pure AR(1) process with no trend by construction, yet the original test reports p < 0.05.

Hamed & Rao (1998) variance correction

Keep S, but inflate its variance by the effective sample size factor computed from the autocorrelation \(\rho_k\) of the ranks of the detrended series (only significant lags are used):

Trend-free pre-whitening, TFPW (Yue et al. 2002)

1. Estimate Sen's slope \(\beta\) and remove it: \(Y'_t = X_t - \beta t\).
2. Compute the lag-1 autocorrelation \(r_1\) of \(Y'\) and remove the AR(1) part: \(Y''_t = Y'_t - r_1 Y'_{t-1}\).
3. Add the trend back, \(Y_t = Y''_t + \beta t\), and run the original MK test on \(Y\) (one value shorter).

Monthly rainfall, streamflow, temperature and water-quality data have a strong annual cycle. Comparing a July value with a January value says more about seasons than about trends, and the cycle's variance hides real change. Hirsch, Slack & Smith (1982) proposed computing MK separately for each season and summing:

The "Monthly flow" example shows why this matters: the original test on 120 monthly values gives p ≈ 0.13 (no trend), while the seasonal test finds a highly significant increase. The basic seasonal test assumes seasons are independent of each other; if adjacent months are strongly correlated, its p-value is also optimistic.

A complete report states the test variant, the significance level, the statistic, the p-value and the magnitude with uncertainty. A template:

The calculator writes this sentence for you from your own numbers (tab Report & export). In the methods section, cite Mann (1945) and Kendall (1975) for the test, Sen (1968) for the slope, and the correction you used (Hamed & Rao 1998 or Yue et al. 2002). If you used Python, cite pyMannKendall (Hussain & Mahmud 2019).