Sign Test Calculator

Advanced Statistical Tests

A non-parametric test to analyze the difference between paired observations by looking at the sign of the differences.

Practical Examples

Explore these scenarios to see how the Sign Test Calculator works.

Drug Effectiveness Study

medicine

Testing if a new drug effectively reduces blood pressure. Data is from 10 patients before and after treatment.

Sample 1: 145, 150, 130, 135, 160, 155, 140, 130, 150, 148

Sample 2: 135, 142, 132, 130, 155, 145, 138, 125, 140, 141

Tutoring Program Impact

education

Evaluating if a tutoring program improves student test scores. Scores are from 8 students before and after the program.

Sample 1: 75, 80, 82, 78, 88, 90, 85, 70

Sample 2: 80, 82, 85, 79, 87, 92, 88, 75

Advertisement Campaign

marketing

Assessing if a new ad campaign increases daily sales. Data shows sales for a week before and after the campaign launch.

Sample 1: 500, 550, 520, 480, 600, 580, 530

Sample 2: 520, 560, 520, 500, 610, 570, 540

Therapy for Anxiety

psychology

Measuring the effect of a new therapy on anxiety levels. Lower scores indicate less anxiety.

Sample 1: 25, 22, 28, 30, 24, 26, 20, 18

Sample 2: 22, 23, 25, 28, 21, 24, 19, 19

Other Titles
Understanding the Sign Test: A Comprehensive Guide
A detailed look into the non-parametric method for testing differences in paired data.

What is the Sign Test?

  • Core Concepts of Non-Parametric Testing
  • When to Use the Sign Test
  • Assumptions and Limitations
The Sign Test is a non-parametric statistical method used to test for consistent differences between pairs of observations. It's a versatile alternative to the paired t-test, especially when the data does not follow a normal distribution. The test gets its name because it uses the plus and minus signs of the differences between paired data points.
Core Concepts
The fundamental idea is to determine if the median of the differences between paired observations is zero. It does not make any assumptions about the underlying distribution of the data, which makes it robust. The null hypothesis (H₀) typically states that the median difference is zero, while the alternative hypothesis (H₁) can be one-sided (e.g., the median difference is greater than zero) or two-sided (the median difference is not zero).

Step-by-Step Guide to Using the Sign Test Calculator

  • Entering Your Data
  • Configuring Test Parameters
  • Interpreting the Results
1. Entering Your Data
Input your two sets of paired data into the 'Data Sample 1' and 'Data Sample 2' fields. Each value should be separated by a comma. It's crucial that both samples have the exact same number of data points, and that they correspond to each other (e.g., the first value in Sample 1 is paired with the first value in Sample 2).
2. Configuring Test Parameters
Set the Significance Level (α), which is your threshold for statistical significance (commonly 0.05). Then, choose the appropriate Hypothesis Test Type based on what you want to prove: use 'Two-Tailed' to check for any difference, 'Left-Tailed' if you hypothesize Sample 1's median is less than Sample 2's, or 'Right-Tailed' for the opposite.
3. Interpreting the Results
The calculator provides the number of positive/negative differences, the p-value, and the test statistic. The most important output is the p-value. If the p-value is less than or equal to your significance level (α), you reject the null hypothesis, concluding there is a statistically significant difference. Otherwise, you fail to reject it.

Real-World Applications of the Sign Test

  • Medical and Pharmaceutical Research
  • Business and Market Analysis
  • Psychological and Educational Studies
The Sign Test is applied in numerous fields due to its simplicity and minimal assumptions.
Example: Before-and-After Scenarios
It's perfect for studies that measure an outcome before and after an intervention, such as the effectiveness of a training program on employee performance or a new diet on weight loss. The test can determine if the change observed is consistently in one direction.

Common Misconceptions and Correct Methods

  • Sign Test vs. t-Test
  • Handling Ties (Zero Differences)
  • Power of the Sign Test
Sign Test vs. t-Test
A common point of confusion is when to use the Sign Test versus a paired t-test. The t-test is more powerful (more likely to detect a real effect) but requires the differences to be approximately normally distributed. If this assumption is violated, the Sign Test is a safer and more valid choice. However, the Sign Test has less statistical power because it ignores the magnitude of the differences.
Handling Ties
When a pair of observations has a difference of zero (a 'tie'), it is excluded from the analysis, and the sample size (n) is reduced. This is the standard procedure and our calculator handles it automatically.

Mathematical Derivation and Examples

  • The Binomial Connection
  • Calculating the P-value Manually
  • A Worked Example
The Binomial Connection
Under the null hypothesis (that there is no difference), any given difference between paired values is equally likely to be positive or negative. This is like flipping a coin, where P(Heads) = P(Tails) = 0.5. Therefore, the number of positive signs (or negative signs) follows a Binomial Distribution B(n, 0.5), where n is the number of pairs excluding ties.
Calculating the P-value
The test statistic, S, is the number of positive signs. For a two-tailed test, the p-value is the probability of observing a result as extreme as or more extreme than S. This is calculated as 2 * P(X ≤ min(N+, N-)), where X ~ B(n, 0.5), N+ is the count of positive signs, and N- is the count of negative signs. For one-tailed tests, the p-value is just the one-sided probability.