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# Sign Test

Updated on April 9, 2024
Article byRutan Bhattacharyya
Edited byAshish Kumar Srivastav
Reviewed byDheeraj Vaidya, CFA, FRM

## What Is A Sign Test?

A sign test refers to a non-parametric test utilized by organizations or individuals to test if a couple of groups are of equal sizes. In finance, such a test can independently specify the statistical importance of any assumption concerning the excess return population distribution from where data are accumulated.

For eg:
Source: Sign Test (wallstreetmojo.com)

Organizations can utilize this concept when they order samples in pairs with mutually exclusive bivariate random variables. The observation’s minus and plus sign directions form the basis of the basis of this kind of non-parametric test rather than the numerical magnitude. Note that the sign test in statistics is of two types — paired sample and one sample sign test.

### Key Takeaways

• Sign test refers to non-parametric tests that help in determining if differences exist in two dependent groups’ central tendency. One can use it for ranked categorical data.
• Such a statistical procedure has a couple of types, which are one-sample and paired-sample sign tests.
• A noteworthy sign test assumption is that the data are from two samples. These two samples could be from a couple of populations.
• Sign tests and Wilcoxon tests are non-parametric statistical methods. On the other hand, T-tests are parametric tests.

### Sign Test Explained

Sign test refers to a statistical technique that enables people to test whether the differences between the pairs of observations, for example, the subjects’ weights, remain consistent. Such a test can help one figure out whether a member of a certain pair, for example, pre-treatment, has a tendency to be less or greater when compared to the other member of that pair, for example, post-treatment.

Moreover, this statistical technique can also help one test whether the median of an accumulation of numbers is less or greater than a particular value by a significant margin. One can utilize this test for ordered or ranked categorical data.

Individuals must consider certain factors before using this statistical method. Some aspects are related to the type of variables and data available in their sample set, while others are associated with the type of test necessary to find answers to the study’s questions.

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When organizations have a data set for conducting this test, they must assess the situation and determine whether this technique is suitable for figuring out if the null hypothesis is correct. The circumstances and situations under which one can utilize this statistical method are as follows:

• When there’s a chance of a couple of outcomes, people can compute the test statistic to find out if they have the same probability of occurring.
• If the paired data analysis involves a couple of categorical variables and a continuous or an ordinal variable.
•  When variables of interest are interval, ratio, or ordinal, this test can help determine if the medians of those variables are higher than the hypothesized value by a significant margin.

Besides considering the situations and circumstances, one must factor in the test type necessary for the specific data and the p-values for the sign test statistics, among other things, when conducting this test.

### Assumptions

The sign test assumptions are as follows:

• The data are from a couple of samples, and the population can differ for both samples.
• This test is non-parametric. Hence, one does not assume normal data distribution takes place.
• Another key assumption is that dependent samples are matched or paired samples. The population can differ for both samples.

### Types

As noted above, such a test can be of two types. Let us understand them.

#### #1 – Paired Sample

These sample tests utilize negative and positive signs after or before the study. In paired samples, the null hypothesis is set in a manner such that the total number of positive and negative signs are equal or the sample mean is identical to the population means.

#### #2 – One Sample

This type of non-parametric test helps calculate a significance test for one data set’s hypothesized median value. It can assist in determining if a statistically important difference exists between a defined standard and a continuous population’s median. When carrying out this test, one sets the hypothesis to ensure that the negative and positive signs represent random values of identical numerical magnitude while comparing the overall observations greater or less than the data set’s hypothesized value.

### How To Calculate?

One can follow these steps to compute sign test in statistics:

1. Create a table with all the observations and subtract the second set from the first set.
2. Add another column indicating the sign of every number in the column with the differences.
3. Count the total number of negatives and positives.
4. Find the total number of items in the sample but do not consider in 0s.
5. Determine the p-value utilizing any binomial calculator distribution table.

### Examples

Let us look at a few sign test examples to understand the concept better.

#### Example #1

Suppose the table below shows two sets of sample data representing monthly sales figures of Company ABC for two consecutive financial years. The null hypothesis is that the increase in work hours has resulted in a decrease in the overall output.

H0 = No difference in the signed differences’ median

H1 = The signed differences’ median is lower than 0

Per the computation process requirement, the company subtracted Set 2 from Set 1 and added another column to note the differences. Then, it added another column to count the negative and positive signs. In this case, the total number of positives is 3, while the total number of negatives is 7. After that, the company added up the total number of samples and excluded the only 0 in the table. Thus, n = 10 in this case. After determining these values, the organization found out the p-value utilizing a binomial calculator.

For determining the p-value, Company ABC used the following data:

• Probability – .5
• Number of trials – 10
• Total Successes (the smaller of negative or positive signs) – 3

In this case, the binomial distribution is 0.117, which is higher than the alpha level of 0.05. So, in this case, we accept the null hypothesis. If the binomial distribution was below 0.05, we could reject the null hypothesis and say that there is a difference by a significant margin between the two groups.

#### Example #2

Suppose Company XYZ managed two types of investment portfolios in exchange for a certain fee. One portfolio comprised 50 stocks generating dividend income. The other one consisted of 50 growth stocks. The organization compared the average returns of both portfolios using a sign test. After conducting the test, the company realized that there was a significant difference between the portfolios’ performance.

The portfolio with growth stocks was delivering a better performance. Hence, it decided to keep the better-performing portfolio as it is and made changes to the second portfolio in terms of strategy.

### Sign Test vs T-Test vs Wilcoxon

Sign tests, Wilcoxon tests, and t-tests can be complicated for individuals unfamiliar with such statistical methods. To have a clear understanding of how these three concepts differ and avoid potential confusion, one must learn about their key differences, which are as follows:

1. When to use the sign test?

Researchers can consider utilizing this statistical technique in studies that involve utilizing repeated measures designs where the data accumulated are nominal. Such a method enables them to determine their investigation’s significance.

2. What are the limitations of the sign test?

The disadvantages of sign tests in statistics are as follows:
– One can include only the total number of negative and positive differences in the paired samples for the calculation. The differences in size are not taken into account.
– Such tests are distribution-free. Hence, they may not provide accurate answers.

3. What are the characteristics of a sign test?

The key features of such a test are as follows:
– It does not require the assumption of a specific distribution, for example, a normal distribution.
– Such a test associates the extent of a couple of groups.
– It applies to all kinds of data.
Another key characteristic of this test is that the data are always from two dependent samples.

This article has been a guide to what is Sign Test. We explain its examples, assumptions, comparison with Wilcoxon & t-test, types, and how to calculate it. You may also find some useful articles here –