Sign Test Model Overview

The sign test asks whether positive and negative deviations are equally likely after observations equal to the null value are removed. For paired data, the deviation is the within-pair difference. In this analysis the difference is G3 − G2: a positive sign means the final grade increased, a negative sign means it decreased, and a zero difference is a tie.

The test deliberately ignores how large each change is. A rise of one grade point and a rise of six grade points each contribute one positive sign. This loss of magnitude information makes the sign test less powerful than methods that can safely use distances or ranks, but it also makes the method resistant to outliers, skewed difference distributions and extreme negative changes such as −9 or −8.

The Exact Research Question

Among students whose grade changed between G2 and G3, are increases and decreases equally likely? This conditional wording is essential because the 287 ties are documented but excluded from the exact binomial count.

Quick Answer: Sign Test Result

The paired sign test classified each G3 − G2 difference by direction. There were 290 increases, 72 decreases and 287 ties. After ties were excluded, 80.11% of changed pairs were increases and 19.89% were decreases.

Table of Contents

1. What the sign test means
2. Research question and hypotheses
3. When to use the sign test
4. One-sample and paired variants
5. Variables and data dictionary
6. Formula and exact calculation
7. Difference distribution and signs
8. Complete statistical results
9. Assumptions and diagnostics
10. Comparison with related tests
11. Sign test calculator workflow
12. Python chart interpretations
13. Paired R chart interpretations
14. Python, R, SPSS and Excel workflows
15. Expandable code
16. APA reporting
17. Common mistakes
18. Practice questions
19. Downloads
20. Related guides
21. Frequently asked questions
22. Conclusion

What Is the Sign Test?

The sign test is a nonparametric test based only on whether each observation lies above or below a reference value. In a paired design, the reference value is zero for the within-pair difference. In a one-sample design, each observation is compared with a hypothesised population median.

Let d_i denote the difference for pair i. The sign test converts every nonzero d_i to “+” or “−”. Ties are removed from the exact binomial calculation. If the null hypothesis is true and the distribution is continuous around zero, positive and negative signs should each occur with probability 0.50.

What Information the Sign Test Uses

What Information It Discards

The test does not use the size of each difference. The +6 change receives the same positive sign as each +1 change; similarly, a −9 change receives the same negative sign as a −1 change. Magnitudes remain valuable for descriptive interpretation, so this post reports them separately without allowing them to alter the exact sign-test statistic.

Research Question, Hypotheses and Paired Data Design

Research Question

Among students with a nonzero G3 − G2 difference, is the probability of an increase different from one half?

Why Pairing Matters

The analysis is not a comparison of two independent grade samples. Each G2 value is matched to the same student’s G3 value. Ignoring this matching would mix between-student variation with within-student change. The sign test preserves the pairing by reducing each student’s two measurements to one direction.

Good inference also depends on how the students entered the data. Review sampling methods and sampling bias before generalising beyond the observed sample.

When to Use the Sign Test

Decision Flow

For a broader method-selection discussion, see parametric versus nonparametric tests.

Sign Test Variants and Their Meanings

Current Worked Variant

The current analysis is a paired, two-sided, exact sign test. G3 is treated as the later measure, G2 as the earlier measure, and ties are G3 = G2. The direction convention matters: reversing the subtraction would exchange positive and negative counts but leave the two-sided p-value unchanged.

Relationship to the One-Sample Sign Test

A paired sign test is mathematically a one-sample sign test applied to the difference scores with null median zero. The distinction is substantive: the paired version requires valid within-unit matching, whereas the one-sample version compares independent observations with a fixed benchmark.

Variables, Coding and Data Dictionary

Descriptive Paired Summary

The final-grade mean is 0.3359 points above the second-period mean, and the median rises from 11 to 12. These magnitude summaries provide context, but none of them enters the exact sign-test probability.

Sign Test Formula and Exact Manual Calculation

For each pair, calculate d_i = G3_i − G2_i. Define X as the number of positive differences and n* as the number of nonzero differences.

Step 1: Classify the 649 Differences

Step 2: Form the Exact Binomial Sample

Step 3: Calculate the Two-Sided Exact Probability

Because the null distribution is symmetric around 181 positive signs, the equally extreme lower-tail count is 72. The exact two-sided probability is twice the probability of observing 72 or fewer positive signs under Binomial(362, 0.50).

Manual Calculation Audit

Exact Rejection Region at α = .05

The conservative equal-tail exact region is X ≤ 161 or X ≥ 201. The observed X = 290 is 89 signs beyond the upper critical boundary. The total null probability in these two tails is approximately 0.04024, reflecting the discreteness of the binomial distribution.

Exact Probability Versus Normal Approximation

The approximations lead to the same decision but are roughly two orders of magnitude larger than the exact two-sided probability. Because an exact binomial calculation is straightforward for n*=362, the exact result should be preferred. The probability of exactly 290 positives is not the p-value; the p-value includes outcomes at least as extreme in both tails.

Observed Difference Distribution and Direction Tables

Complete G3 − G2 Difference Distribution

Direction Counts and Proportions

Magnitude Components Kept for Description

The frequency table makes the robustness trade-off visible. There are a few very large decreases, including two −9 values, two −8 values and three −7 values. A magnitude-sensitive method would give those observations more influence. The sign test gives each one a single negative sign.

Concentration Around Zero

Complete Sign Test Results and Data Interpretation

Primary Inference Table

Positive-Sign Probability Interval

What the Result Establishes

• The direction of change is not balanced among non-tied pairs.
• Increases are approximately four times as frequent as decreases among changed pairs.
• The result remains overwhelming even though nearly half the sample is tied.
• The conclusion does not depend on normality of G3 − G2 or on the sizes of the −9, −8 and +6 changes.

What the Result Does Not Establish

• It does not estimate a causal effect of time or instruction.
• It does not claim every student improved.
• It does not show that the median of the observed difference scores is above zero; that median equals zero.
• It does not use or test the mean difference of 0.3359.
• It does not explain why grades changed.

Directional Pattern by Starting Grade Band

The overall sign test is the primary inference, but the raw paired data also show how the direction pattern varies with the starting G2 level. These subgroup summaries are descriptive and were not used to replace or multiply the primary exact test.

Positive signs dominate in all three broad baseline bands. The positive share among non-ties rises from 77.57% for G2 values below 10 to 84.51% for G2 values from 14 to 19. At the same time, ties become especially common in the upper band: 81 of 152 students with G2 from 14 to 19 had no grade change.

Exact Starting-Grade Audit

The exact-level table reveals two boundary patterns. All seven records with G2 = 0 were also G3 = 0 and therefore entered the tie count. At G2 = 18, only one pair increased while five decreased and eight tied; this local reversal is compatible with a ceiling effect because only a one-point increase to 19 is available, whereas several downward values are possible. The sign test does not model that ceiling directly, so the table is useful context rather than a causal explanation.

Sign Test Assumptions and Diagnostic Checks

Diagnostic Audit

Do Differences Need to Be Symmetric?

No. Symmetry is not required for the sign test because absolute magnitudes are discarded. This separates it from the Wilcoxon signed-rank test, whose usual location-shift interpretation depends on a symmetric difference distribution.

Do Differences Need to Be Normally Distributed?

No. The exact binomial null distribution governs the sign count. A normal distribution is used only if someone chooses an approximation, which is unnecessary here because the exact calculation is available.

Why Ties Matter

Ties reduce the effective exact-test sample from 649 to 362. They also change the interpretation: the test conditions on a change having occurred. When ties are common, report their number and percentage rather than presenting n*=362 as though 287 records had vanished.

Sign Test Versus Related Tests

Sign Test Versus Paired t Test

The paired t test would test the mean difference and use the actual −9 to +6 magnitudes. It can be more efficient when the mean is the scientific target and the difference distribution supports its standard error. The sign test targets directional balance and remains resistant to extreme magnitudes.

Sign Test Versus McNemar’s Test

McNemar’s test is appropriate when each paired measurement is binary, such as pass/fail before and after. G2 and G3 are ordered numeric grades, so reducing them to “up,” “down” and “tie” leads to the sign test rather than McNemar’s discordant 2 × 2 table.

Sign Test Versus Exact Binomial Test

The sign test’s computational engine is the exact binomial test. The substantive step is creating valid signs from paired or one-sample deviations; after that, the positive count is tested against probability 0.50.

Sign Test Calculator: Step-by-Step Workflow

A reliable sign test calculator needs either raw paired values or three verified counts: positive, negative and tied differences. The current worked calculation can be reproduced without hidden steps.

Calculator Inputs and Outputs

Excel Calculator Formulas

Python Sign Test Chart Interpretations

Each Python figure is interpreted from the exact worked counts. The charts support the statistical narrative but do not replace the numeric tables.

Change Directions: The Sign Count Behind the Test

Exact values

290 increases, 72 decreases and 287 ties are the complete directional decomposition of 649 pairs.

Statistical meaning

Among all records, 44.68% increased, 11.09% decreased and 44.22% were tied. The bar heights reconcile exactly to 649.

Interpretation checkpoint: The chart prevents the effective n=362 from being mistaken for the original sample size. Ties are not missing data; they are observed zero changes.

Difference Distribution: Direction Dominates Despite Asymmetric Extremes

Exact values

The distribution ranges from −9 to +6, with 287 zeros, 234 +1 changes and 56 −1 changes.

Statistical meaning

Most information lies near zero: 88.91% of pairs are tied or differ by one point. Large negative changes exist but are rare.

Interpretation checkpoint: The sign test intentionally prevents the two −9 and two −8 observations from dominating the inference; each contributes one negative sign.

Exact Null Distribution: Observed X = 290 Versus Expected 181

Exact values

Under H0, X follows Binomial(362, 0.50), with mean 181 and standard deviation 9.513.

Statistical meaning

The observed count is 109 signs above the null mean, or 11.46 null standard deviations without continuity correction.

Interpretation checkpoint: The exact tail—not the normal approximation—is the reported analysis, producing p = 3.9796 × 10⁻³².

Two-Sided Region: Observed Count Far Beyond the Critical Boundary

Exact values

At a conservative exact α=.05, the rejection region is X≤161 or X≥201.

Statistical meaning

X=290 lies 89 counts beyond the upper critical boundary. The visual distance explains why rounding the p-value to .000 hides substantial information.

Interpretation checkpoint: Report the scientific-notation probability or p<.001, never p=0.

Sign Proportions: 80.11% Positive Among Changed Pairs

Exact values

The non-tied positive proportion is 0.801105 and the negative proportion is 0.198895.

Statistical meaning

The exact 95% interval for the positive probability is 0.756208 to 0.840997, wholly above 0.50.

Interpretation checkpoint: The net directional advantage is 0.602210, and the directional odds are 4.0278 increases per decrease.

Result Summary: Strong Directional Evidence With a Zero Sample Median

Exact values

The summary combines X=290, n*=362, exact p≈3.98×10⁻³² and the reject decision.

Statistical meaning

The sample median difference is 0 because of 287 ties, while the median among nonzero differences is +1.

Interpretation checkpoint: The correct report emphasizes the conditional direction among changed pairs and retains the zero-median nuance.

R Sign Test Charts in Paired Rows

R chart pair 1: Change directions and Difference distribution

Data interpretation

Direction counts

The R direction chart reproduces 290 positive, 72 negative and 287 tied pairs. It verifies that the R workflow uses the same subtraction order and tie definition as the workbook.

Values must reconcile with 290 positive, 72 negative, 287 tied and n*=362.

Data interpretation

Difference distribution

The R distribution chart confirms the modal difference is 0 and the most common nonzero difference is +1. The rare long negative tail is descriptive rather than a weighted component of the sign statistic.

Values must reconcile with 290 positive, 72 negative, 287 tied and n*=362.

R chart pair 2: Exact binomial null and Cumulative paired difference

Data interpretation

Exact null

The exact-null figure places X=290 against Binomial(362,.5). The observed count is visually detached from the probability mass centred at 181.

Values must reconcile with 290 positive, 72 negative, 287 tied and n*=362.

Data interpretation

Cumulative difference

The cumulative sequence ends at +218 grade points, reaches a maximum of +241 near pair 621 and has a minimum of −1. This order-dependent curve is descriptive and must not be confused with the order-free sign test.

Values must reconcile with 290 positive, 72 negative, 287 tied and n*=362.

R chart pair 3: Non-tie balance and Result summary

Data interpretation

Non-tie balance

The non-tie chart removes 287 zeros and displays the 80.11% versus 19.89% direction split used in the exact test.

Values must reconcile with 290 positive, 72 negative, 287 tied and n*=362.

Data interpretation

Result summary

The final R panel reconciles counts, exact probability and decision. Agreement with Python and Excel confirms that the analysis is not software-specific.

Values must reconcile with 290 positive, 72 negative, 287 tied and n*=362.

Sign Test in Python, R, SPSS and Excel

Cross-Software Reconciliation

Software agreement depends on matching the same alternative and tie convention. Some packages label the procedure as a median test, while others expose the binomial calculation directly. The counts and null probability are the common core.

Expandable Sign Test Code

import numpy as np
from scipy.stats import binomtest

g2 = np.asarray(g2_values, dtype=float)
g3 = np.asarray(g3_values, dtype=float)

if g2.shape != g3.shape:
    raise ValueError("G2 and G3 must have the same number of paired values.")

d = g3 - g2
positive = int(np.sum(d > 0))
negative = int(np.sum(d < 0))
ties = int(np.sum(d == 0))
non_ties = positive + negative

result = binomtest(
    k=positive,
    n=non_ties,
    p=0.5,
    alternative="two-sided"
)
ci = result.proportion_ci(confidence_level=0.95, method="exact")

print({
    "positive": positive,
    "negative": negative,
    "ties": ties,
    "non_ties": non_ties,
    "p_value": result.pvalue,
    "positive_probability_ci": (ci.low, ci.high),
})

d <- G3 - G2

positive <- sum(d > 0, na.rm = TRUE)
negative <- sum(d < 0, na.rm = TRUE)
ties <- sum(d == 0, na.rm = TRUE)
non_ties <- positive + negative

fit <- binom.test(
  x = positive,
  n = non_ties,
  p = 0.5,
  alternative = "two.sided",
  conf.level = 0.95
)

list(
  positive = positive,
  negative = negative,
  ties = ties,
  non_ties = non_ties,
  p_value = fit$p.value,
  confidence_interval = fit$conf.int
)

NPAR TESTS
  /SIGN = G3 WITH G2 (PAIRED)
  /MISSING ANALYSIS.

C5: =B5-A5
Positive: =COUNTIF(C5:C653,">0")
Negative: =COUNTIF(C5:C653,"<0")
Ties: =COUNTIF(C5:C653,0)
NonTies: =Positive+Negative
TwoSidedP: =2*BINOM.DIST(MIN(Positive,Negative),NonTies,0.5,TRUE)

# Compare observations x with hypothesised median m0
d = x - m0
positive = sum(d > 0)
negative = sum(d < 0)
result = binomtest(positive, positive + negative, p=0.5)

# Python
binomtest(290, 362, p=0.5, alternative="greater")

# R
binom.test(290, 362, p=0.5, alternative="greater")

How to Interpret and Report the Sign Test

APA-Style Worked Report

Reporting Table

Common Sign Test Mistakes and Corrections

Quality-Control Checklist

Sign Test Practice Questions With Solutions

Sign Test Reports and Worked Files

The downloadable files reproduce the same G3 − G2 direction, counts and exact inference. Use the workbook to inspect every paired difference and formula.

Download Verification

Related Salar Cafe Guides

Sign Test Conclusion

The exact paired sign test provides a clear directional result for the 649 G2–G3 pairs. Of the 362 students whose grade changed, 290 increased and 72 decreased. The positive-sign proportion was 0.8011, with an exact 95% confidence interval from 0.7562 to 0.8410, and the two-sided exact probability was approximately 3.98 × 10⁻³².

The strongest interpretation is also the most precise: among changed pairs, grade increases were far more common than decreases. The analysis does not require normality or symmetry, does not allow rare extreme magnitudes to dominate, and does not conceal the 287 ties. Because those ties make the observed median difference equal to zero, a high-quality report should describe the directional imbalance rather than claiming a positive observed median.

The worked Python, R, SPSS and Excel files all reconcile to the same counts and decision. Readers can use the formulas, code and reporting templates above to reproduce the sign test for paired data or adapt it to a one-sample median benchmark.