Name: runs test for randomness in ordered data
Uploaded: 2025-01-09T15:14:08.000Z
Duration: 1 min 17 s
Channel: JoVE

A scientist collected data on the body lengths of 30 baboons approaching a water source.
So, the Wald-Wolfowitz test can determine if the sequence in which baboons approached the water source is random or relates to their body length.
Here, the null hypothesis states that the data are in a random sequence, whereas the alternative hypothesis states the data are not in a random sequence.
For these numerical data, the runs&#8212;G&#8212;are computed by assigning binary signs to the values greater and lesser than the median 74.5.
Here, the value of G is 17.
Notice that the number of values smaller than the median&#8212;n1 &#8212; and greater than the median&#8212;n2 &#8212; are less than 20.
So, the critical values at &#945; = 0.05 can be obtained from the standard table.
The Wald-Wolfowitz test is two-tailed. So, to reject the randomness, the test statistic needs to be beyond the range of critical values.
Here, G falls within this range, providing evidence in favor of the null hypothesis.

wald-wolfowitz runs test ii

Wald-Wolfowitz Runs Test II

The Wald-Wolfowitz runs test, commonly referred to as the runs test, is a nonparametric test used to assess the randomness of ordered data. The test evaluates the number of runs, which are consecutive sequences of similar elements within the data. If the number of runs is significantly higher or lower than expected, the data is considered non-random, indicating a detectable pattern or structure.
For binary data, runs are identified using symbols such as + and &#8722;, or equivalently, 1s and 0s. In the case of categorical data with two categories, these categories are converted into binary choices. However, in cases like DNA sequences, where categories such as A, T, G, and C are inherent, no conversion is necessary, and the test can be applied directly to the sequences. For numerical data, values are often converted by assigning a + (or 1) to those above a certain threshold (like the mean or median) and a &#8722; (or 0) to those below it, enabling the identification of runs.
The test's null hypothesis (H0) states that the data follows a random sequence, while the alternative hypothesis (H1) suggests that the data has an underlying pattern or order. The test statistic, denoted as G, represents the number of observed runs in the data. This value is then compared to critical values in a two-tailed test. If G falls outside the critical range (i.e., it is too high or too low), we reject the null hypothesis, concluding that the data is non-random and exhibits a particular sequence. Conversely, if G falls within the critical range, we fail to reject the null hypothesis, suggesting that the data is likely random without any specific order.
The critical values are obtained from a standard table of runs when the number of elements in the data having a particular characteristic (for instance, values smaller than the median), denoted by n1, and the number of elements in the data having a different characteristic (for instance values greater than the median) denoted by n2 is less than or equal to 20 and at the significance level &#945; = 0.05 (Note: do not confuse these numbers with the sample size n).
When these conditions are not fulfilled, i.e., when n1 and n2 and greater than 20 or when the significance level &#945; is anything else but 0.05, the test statistic z is used, and it is calculated using the following equation:
<img alt="z-score formula equation, statistical analysis, data normalization, mathematical expression." src="/files/ftp_upload/14683/PT_14683_Equation_1.jpg" style="text-align: center; height: 36px;" data-altupdatedat="1761830439000" data-alt="Equation 1" />
Where &#181;G and &#963;G are calculated by the following equations:
<img alt="Gravitational microlensing formula, distance calculation equation, physics diagram." src="/files/ftp_upload/14683/PT_14683_Equation_2.jpg" style="text-align: center; height: 38px;" data-altupdatedat="1761830439000" data-alt="Equation 2" />
<img alt="Geometrical standard deviation, formula, statistical analysis, data distribution calculations." src="/files/ftp_upload/14683/PT_14683_Equation_3.jpg" style="text-align: center; height: 46px;" data-altupdatedat="1761830439000" data-alt="Equation 3" />
The critical z values, i.e., negative (left-tailed) and positive (right-tailed) critical z values, are obtained from the standard z distribution table. When the test statistic z (calculated from the above equations) is beyond the range of -z and +z, the randomness is rejected, concluding that there is evidence of a specific sequence in the data. If the test statistic is within the range, the randomness in the data is not rejected.
The runs test is unaffected by sample size or the underlying distribution of the population and sample, making it versatile for various types of sequential data to detect randomness. However, while it can identify whether a sequence is random, it does not measure the degree or magnitude of randomness within the data.

A scientist collected data on the body lengths of 30 baboons approaching a water source.
So, the Wald-Wolfowitz test can determine if the sequence in ...

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runs test for randomness in ordered data

The Wald-Wolfowitz runs test checks whether ordered data is random or shows a pattern. A run is a consecutive sequence of similar items. If the number of runs is much higher or lower than expected, the data is considered non-random and may contain a detectable structure.

The test works with binary data such as + and −, or 1s and 0s. For data with two categories, the categories are first converted into binary choices. DNA sequences can be tested directly because categories like A, T, G, and C already fit the method. For numerical data, values are often turned into + and − by placing them above or below a threshold such as the mean or median.

The null hypothesis, H0, says the sequence is random. The alternative hypothesis, H1, says the data has an underlying order. The test statistic is G, which is the number of observed runs. This value is compared with critical values in a two-tailed test.

If G falls outside the critical range, it is too high or too low, and H0 is rejected. That means the data is not random and shows a specific sequence. If G stays inside the critical range, randomness is not rejected, so the data is likely random without a clear order.

When the numbers in the two groups are 20 or less and the significance level is α = 0.05, critical values come from a standard runs table. The two group counts are n1 and n2, and they should not be confused with the full sample size n. When n1 and n2 are greater than 20, or when α is not 0.05, the test uses a z value instead. That z score is calculated from formulas for μG and σG, and the critical values come from the standard z table.

If the test statistic z goes beyond the range between −z and +z, randomness is rejected. If it stays within that range, randomness is not rejected. The runs test does not depend on sample size or the population distribution, so it works well for many kinds of sequential data. However, it only tells whether a sequence is random. It does not measure how random the data is.

Runs Test for Randomness in Ordered Data

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The Wald-Wolfowitz runs test checks whether ordered data is random or shows a pattern. A run is a consecutive sequence of similar items. If the number of runs is much higher or lower than expected, the data is considered non-random and may contain a detectable structure. The test works with binary data such as + and −, or 1s and 0s. For data with two categories, the categories are first converted into binary choices. DNA sequences can be tested directly because categories like A, T, G, and C...

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key terms and concepts

Learn how the Wald-Wolfowitz runs test checks ordered data for randomness, using runs, critical values, and z scores for high school statistics.

Runs Test for Randomness in Ordered Data | JoVE K12

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Runs Test for Randomness in Ordered Data

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