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Sign Test (S)

Non-parametric test for difference between two related groups.

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S = min (n_{+}, n_{-})

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Core idea

Overview

The Sign Test is a non-parametric statistical test used to determine the significance of differences between paired observations in a repeated measures design. It ignores the magnitude of differences and focuses exclusively on the direction of change, making it highly resistant to outliers.

When to use: Use the Sign Test when analyzing ordinal data or non-normally distributed interval data within a related-samples design. It is ideal for small sample sizes or situations where the only reliable information is the direction of a change rather than its specific value.

Why it matters: In psychological research, it provides a simple yet robust method to verify if a treatment or intervention consistently affects participants. It is often used as a preliminary analysis to determine if a trend exists before applying more complex parametric tests.

Symbols

Variables

S = Sign Statistic, n_+ = Plus Signs, n_- = Minus Signs

S

Sign Statistic

V a r iab l e

n_{+}

Plus Signs

V a r iab l e

n_{-}

Minus Signs

V a r iab l e

Walkthrough

Derivation

Definition: Sign Test (S)

Determines the test statistic S based on the direction of change in matched pairs.

Related samples.
Nominal or ordinal data.

Identify S:

S is the frequency of the least frequent sign after discarding ties.

S = min (n_{+}, n_{-})

Result

S = min (n_{+}, n_{-})

Source: A-Level Psychology — Research Methods / Statistics

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a step-like function where the dependent variable S represents the smaller count of positive or negative differences between two related groups. As the independent variable increases, the value of S remains constrained by the minimum of the two counts, resulting in a non-linear, bounded shape that does not follow a standard continuous curve.

Graph type: step

Why it behaves this way

Intuition

Imagine a line of paired data points, each showing a 'before' and 'after' measurement; the Sign Test visually counts how many points moved up versus down, focusing on the less frequent direction as evidence against

The calculated test statistic for the Sign Test

Represents the less frequent direction of change, which is compared against a critical value to determine statistical significance. A smaller S suggests a stronger, more consistent directional effect.

n_{+}

The number of observations where the second measurement is greater than the first (a positive difference)

Indicates how many participants showed an improvement or increase in the measured variable.

n_{-}

The number of observations where the second measurement is less than the first (a negative difference)

Indicates how many participants showed a decline or decrease in the measured variable.

Signs and relationships

\min(n_+, n_-): The Sign Test is concerned with whether the observed direction of change is significantly different from what would be expected by chance.

Free study cues

Insight

Canonical usage

The Sign Test statistic (S) is a dimensionless count representing the minimum number of positive or negative differences observed in paired data.

Common confusion

A common confusion is attempting to assign units to S or interpreting it as a magnitude rather than a simple count of directional changes.

Dimension note

The Sign Test statistic (S) is inherently dimensionless as it represents a count of observations. The input values (n+ and n-) are also counts of directional changes.

Unit systems

$n_{+}$ none · Represents a count of observations where the second measurement is greater than the first (positive difference).

$n_{-}$ none · Represents a count of observations where the second measurement is less than the first (negative difference).

$S$ none · The Sign Test statistic, which is the smaller of n+ and n-, used for comparison against critical values.

One free problem

Practice Problem

A clinical psychologist observes 15 patients after a new therapy session. 12 patients reported a decrease in anxiety levels (negative signs) while 3 reported an increase (positive signs). Calculate the Sign Test statistic S.

Plus Signs3

Minus Signs12

Solve for: $S$

Hint: The test statistic S is the minimum value between the count of positive and negative changes.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Testing if participants feel more relaxed after a therapy session (Before vs After).

Study smarter

Tips

Exclude any 'ties' or zero differences from your total sample size before starting.
The test statistic S is always the lower of the two counts.
For S to be significant, it must be less than or equal to the critical value in a distribution table.

Avoid these traps

Common Mistakes

Counting the most frequent sign instead of the least.

Common questions

Frequently Asked Questions

Determines the test statistic S based on the direction of change in matched pairs.

Use the Sign Test when analyzing ordinal data or non-normally distributed interval data within a related-samples design. It is ideal for small sample sizes or situations where the only reliable information is the direction of a change rather than its specific value.

In psychological research, it provides a simple yet robust method to verify if a treatment or intervention consistently affects participants. It is often used as a preliminary analysis to determine if a trend exists before applying more complex parametric tests.

Counting the most frequent sign instead of the least.