Statistical Tests

Describe the appropriate selection of non-parametric and parametric tests and tests that examine relationships (e.g. correlation, regression)

Parametric Tests

Parametric tests are used when data is:

  • Continuous and numerical
  • Normally distributed
    • Remember that due to the central limit theorem - large data sets (n > 100) are typically amenable to parametric analysis, as sample means will follow a normal distribution
    • Non-normal data can be transformed so that they follow a normal distribution
  • Samples are taken randomly
  • Samples have the same variance
  • Observations within the group are independent
    Independent results are those when one value is not expected to influence another value.
    • A common example is repeated measures: when serial measures are taken from a patient or a hospital, the results cannot be treated as independent
    • Paired tests are used when two dependent samples are compared
    • Unpaired test are used when two independent samples are compared

Tests may be one-tailed or two-tailed:

  • A two-tailed test evaluates whether the sample mean is significantly greater or less than the population mean
  • A one-tailed test only evaluates the relationship in one direction
    This doubles the power of the test to detect a difference, but should only be performed if there is a very good reason that the effect could only occur in one direction.

Common parametric tests include:

Z test

Used to test whether the mean of a particular sample (x̄) differs from the population mean (μ) by random variation.


  • Large sample
    n > 100.
  • Data is normally distributed
  • Population standard deviation is known

Student's T Test

This is a variant of the Z test, used when the population standard deviation is not known.

  • The results from T test approximate the results of the Z test when n > 100

F Test

Compares the ratio of variances () for two samples. If F deviates significantly from 1, then there is a significant difference in group variances.

Analysis of Variance (ANOVA)

ANOVA tests for significant differences between means of multiple groups, in a more efficient manner than multiple comparisons (doing lots of T tests).

There are several types of ANOVA tests used in different situations.

Non-Parametric Tests

Non-parametric tests are used when the assumptions for parametric tests are not met. Non-parametric tests:

  • Do not assume the data follows any particular distribution
    This is required when:
    • Non-normality is obvious
      e.g. Multiple observations of 0
    • Possible non-normality
      Typically small sample sizes.
    • Data is ordinal
  • Are not as powerful as parametric tests (a larger sample size is required to achieve the same error rate)
  • Are more broadly applicable than parametric tests as they do not require the same assumptions

Non-parametric tests still require that data:

  • Is continuous or ordinal
  • Within-group observations are independent
  • Samples are taken randomly

In general, non-parametric tests;

  • Take each result and rank them
  • Calculations are then performed on each rank to find the test statistic

Common non-parametric tests include:

Mann-Whitney U Test/Wilcoxon Rank Sum Test

Alternative to the unpaired T-test for non-parametric data.


  • Data from both groups are combined, ordered, and given ranks
    • Tied data are given identical ranks, where that rank is equal to the average rank of the tied observations
  • The data are then separated into their original group
  • Ranks in each group are added to give a test statistic for each group
  • A statistical test is performed to see if the sum of ranks in one group is different to another

Wilcoxon Signed Ranks Test

Alternative to the paired T-test for non-parametric data.


  • As above (for the Wilcoxon Rank Sum Test), except absolute difference between paired observations are ranked
    The sign (i.e. positive or negative) is preserved.
  • The sum of positive ranks is then compared with the sum of negative ranks
  • If there is no difference between groups, we would expect the net value to be 0


  1. Myles PS, Gin T. Statistical methods for anaesthesia and intensive care. 1st ed. Oxford: Butterworth-Heinemann, 2001.
Last updated 2017-09-22

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