The sign test is a simple inferential test used in psychology to judge whether a pattern in paired results is likely to reflect a real difference rather than chance variation.

Statistical testing and the sign test

In psychological research, statistical testing helps researchers decide whether the results they found are strong enough to be considered significant. This means asking whether the pattern in the data is unlikely to have happened just by chance.

This figure plots the binomial distribution (with $p=0.5$) and highlights the tail regions that represent ‘rare’ outcomes under the null hypothesis. It helps students see why a large imbalance of pluses vs minuses is treated as statistically significant: it falls into the low-probability tails of the distribution. Source

The sign test is one of the simplest inferential tests students need to know, because it focuses only on the direction of differences between paired scores.

The sign test does not use the size of the difference between scores. Instead, it checks whether one condition tends to produce higher or lower scores than the other. This makes it useful when the data can be reduced to simple categories such as positive and negative signs.

When to use the sign test

The sign test should be used when the research situation matches certain conditions.

• It is a test of difference.

  • The researcher wants to know whether there is a difference between two conditions, treatments, or occasions.

• The data are related or paired.

  • This usually means a repeated measures design, where the same participants take part in both conditions.

  • It can also be used when participants are paired in a matched pairs design.

• The data can be treated as nominal after comparison.

  • Each pair of scores is turned into a sign:

    • + if one condition produced the higher score

    • - if the other condition produced the higher score

• The researcher is interested in the direction of change, not the size of change.

For example, the test is suitable when each participant produces two scores and the psychologist compares them pair by pair. If a participant shows no difference between the two scores, that pair does not help show direction and is removed from the calculation.

A good way to remember the sign test is that it asks: Are there enough pluses or minuses in one direction to suggest a real difference?

How to calculate the sign test

To calculate the sign test, the researcher compares each pair of related scores and records the direction of the difference.

This worked example shows paired measurements (e.g., “Before” and “After”) and the computed difference for each participant, making it clear how each pair is converted into a direction of change. It visually reinforces that the sign test reduces paired data to whether the change is positive or negative (and that ties would be treated separately). Source

The key idea is that the observed value for the sign test is the smaller of the two sign totals, not the larger one.

Step-by-step process

• State the hypotheses.

  • The null hypothesis says there is no significant difference between the two conditions and any pattern is due to chance.

  • The alternative hypothesis says there is a significant difference.

• Compare each pair of scores.

  • Decide whether the second score is higher or lower than the first, or vice versa, depending on how the data are arranged.

• Assign a sign to each pair.

  • Use + for a difference in one direction.

  • Use - for a difference in the opposite direction.

• Ignore ties.

  • If the two scores in a pair are the same, record 0 and remove that pair from the final totals.

• Count the signs.

  • Add up the total number of pluses and the total number of minuses.

• Find the observed value.

  • Choose the smaller of the two totals. This is the value compared with the critical value.

• Work out $N$.

  • This is the number of pairs left after any zeros have been removed.

• Use a sign test critical values table.

  • Find the row for the correct value of $N$.

  • Use the correct level of significance required by the question, usually 0.05.

  • Make sure the correct tail is used if the hypothesis is directional or non-directional.

Interpreting the result

Once the observed value and the critical value have been identified, the researcher compares them.

• If the observed value is equal to or less than the critical value, the result is significant.

  • The null hypothesis is rejected.

• If the observed value is greater than the critical value, the result is not significant.

  • The null hypothesis is retained.

This matters because the sign test is designed to show whether the pattern of pluses and minuses is unlikely to have happened by chance alone. A significant result suggests that one condition consistently produced higher or lower scores than the other.

However, the sign test does not show:

• how large the difference is

• how important the difference is in practical terms

• why the difference occurred

Common mistakes and exam tips

Students often lose marks on the sign test because of small procedural errors.

• Do not include zeros in the final total for $N$.

• Do not use the larger sign total as the observed value.

• Keep the sign system consistent throughout the calculation.

• Check the hypothesis carefully before selecting the correct critical value.

• Use the reduced sample size after ties are removed, not the original number of participants.

• Interpret the result correctly:

  • smaller or equal to the critical value = significant

  • larger than the critical value = not significant

In exam answers, it is important to show clear understanding of when the sign test is appropriate, how signs are assigned, and how significance is decided from the critical value table.