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Z-Tests in Healthcare Research: Types and Applications
Z-tests are parametric statistical tests based on the standard normal
(Z) distribution, commonly used to compare sample statistics (means or
proportions) under certain assumptions. In general, a one-sample Z-test
evaluates whether a sample mean differs from a known population mean (when
population standard deviation is known), a two-sample Z-test compares
means of two independent groups (assuming known and equal variances), and Z-tests
for proportions compare one or two sample proportions in large samples. These
tests assume normality (or large-sample normality via the central limit
theorem) and known variances[1][2]. In practice, healthcare
researchers often substitute t-tests or chi-square tests when variances are
unknown, but the Z-test framework remains important for large samples and
hypothesis tests.
Types of
Z-Tests
One-Sample Z-Test (Means)
A one-sample Z-test
compares the mean of a single sample to a specified population mean μ₀, under
the assumption that the population standard deviation (σ) is known[1][2]. The null
hypothesis is usually H₀: μ = μ₀ versus H₁: μ ≠ μ₀ (two-sided) or one-sided
alternatives. The test statistic is 
. Key conditions for validity
are: the data are approximately normal (especially for small n) and σ is known[1][2]. In reality,
knowing σ is rare; thus one-sample Z-tests are often taught as theoretical
cases. In large samples (n ≥ 30), the central limit theorem justifies normal
approximation even if σ is estimated, making the Z-test similar to the
one-sample t-test[2][3].
·
Assumptions: Independent, continuous data from a normal population (or large n);
known population variance.
·
Use case: Testing if a measured average (e.g. blood pressure, lab value) differs
from a target or known norm.
Two-Sample Z-Test (Means)
A two-sample Z-test
compares the means of two independent groups, assuming each population variance
is known (often assumed equal)[4]. It tests H₀: μ₁ =
μ₂ versus alternatives, with statistic 
. This is akin to comparing
treatment and control group means in a clinical trial if variances were known.
In practice, the two-sample t-test is more common when σ is unknown.
However, for very large samples or when σ is known from prior data, the Z-test
form applies. The NCSS reference notes the two-sample Z-test “is used in
situations such as the comparison of … the effectiveness of two drugs”[5].
·
Assumptions: Both groups’ data are (approximately) normal, variances known and
equal[4].
·
Use case: Theoretical comparisons of two independent group means. (See Applications
below for examples.)
Z-Test for Proportions
A Z-test for
proportions compares one or two sample proportions in large-sample settings. In
a one-proportion Z-test, one tests if an observed proportion (p̂) equals
a hypothesized value p₀ (H₀: p = p₀), with statistic 
. A two-proportion Z-test
examines the difference between two independent proportions (p̂₁ – p̂₂). For
large samples (typically each np≥5–10), the sampling distribution of
proportions is approximately normal by the central limit theorem[6][7]. The two-proportion
Z-test statistic is
where 
is the pooled proportion[7]. LibreTexts notes:
“The Z-test is a statistical test for comparing the proportions from two
populations. It can be used when … the samples are independent” and the usual
large-sample conditions hold[7].
·
Assumptions: Independent binomial samples, large sample sizes (np and n(1–p) ≥10)[6][7].
·
Use case: Comparing rates or percentages (e.g. infection rates, cure rates)
between groups.
Relation to Chi-Square
and T-Tests
For
categorical data, a two-proportion Z-test yields the same p-value as Pearson’s
chi-square test when sample sizes are large[8]. In
fact, when one or more expected counts are small, Fisher’s exact or chi-square
with corrections are preferred, but when the normal approximation is valid, the
Z-test “is justified”[9].
Similarly, if population variances are unknown and sample sizes are moderate,
one uses t-tests rather than Z-tests. As Investopedia explains, “Z-tests are
closely related to t-tests, but t-tests are best performed when an experiment
has a small sample size. Also, t-tests assume the standard deviation is
unknown, while Z-tests assume it is known”[2]. When n
≥ 30, the central limit theorem ensures the sampling distribution of the mean
is approximately normal, justifying the Z-test or its approximation[3].
Applications in Medical and Healthcare Research
In medical and public health research, Z-tests (especially for
proportions) frequently appear in large-sample analyses. Below are examples
from recent peer-reviewed studies illustrating the use of each major type of
Z-test.
- Two-Sample Z-Test (Proportions) in Clinical Studies: A recent phase-3 vaccine trial used a one-sided, two-sample
Z-test to compare proportions of participants reaching an immune
response endpoint between a heterologous booster group and a control group[10]. In this non-inferiority study, the difference in seroresponse
rates was tested via a two-sample Z statistic at α=0.025[10]. This reflects how modern vaccine studies may specify Z-tests
(with known significance levels) for large-sample comparisons of efficacy
endpoints.
- Two-Sample Z-Test (Proportions) in Epidemiology: A hospital epidemiology study compared proportions of patients
receiving prophylactic antibiotics in two time periods. They state: “Differences
in the proportions of … patients who received antibiotics … between the
two periods were assessed using the two-sample Z-test for proportions,
which is an equivalent of the Chi-square test”[11]. In other words, the study compared two independent proportions
(antibiotic use rates) over time in two patient groups, using the
large-sample Z-test approximation[11]. This is a common approach in outbreak investigations and
quality-improvement studies.
- One-Sample Z-Test (Proportions) for Benchmarks: Some public health studies compare an observed rate to a target
value. For example, in an England primary care study, researchers used one-proportion
Z-tests to examine whether the observed referral rate equaled a
hypothesized value (p=0.90)[12]. They describe performing “two-tailed, one-proportion Z-tests …
to determine whether referral rates … were equal across time as well as
across different regions” against a fixed proportion[12]. This shows how one-sample Z-tests for proportions can check if a
healthcare process meets a goal rate (e.g. screening or referral targets).
- One- and Two-Sample Z-Tests (Means) in Clinical Research: While t-tests are often used for comparing means (e.g. blood
pressure, biomarker levels) when variance is unknown, a Z-test form can
appear in sample-size calculations or theoretical discussions. For
instance, Tian et al. (2025) describe using a weighted Z-test
combining onsite/offsite data for decentralized trials[13]. In neurocognitive research, one study converted correlation
coefficients to Z-scores (via Fisher’s z-transformation) and used a
two-sample Z-test to compare these across groups[14]. Although not as common as t-tests, these examples illustrate the
flexibility of Z-tests in large-data contexts (e.g. large trials or
high-throughput studies) where normal-approximation methods are
appropriate.
These examples demonstrate that Z-tests are applied whenever
large-sample normal approximations hold: for comparing group means (with known
σ) or especially proportions in epidemiology and clinical trials. They often
coincide with chi-square tests for categorical outcomes when sample sizes are
sufficient[8][11].
Key Points and Considerations
·
Assumptions: Z-tests assume large samples or known variances. For means, the
population standard deviation must be known or the sample size must be large
enough (n≥30) for the CLT to apply[3][15]. For proportions, each group’s sample size should satisfy 
and 
so that the normal approximation is valid[7].
·
Relevance: In modern medical research, datasets (e.g. national registries, large
trials) are often large, making Z-tests viable for initial analysis or
sample-size planning[13][3]. For example, sample-size formulas for trials are based on Z-tests and
normal theory[13]. Even when Z-tests are not reported by name (for instance, t-tests or
chi-square tests are used), the same statistical principles underlie the
analysis in large samples.
·
Comparison to T- and Chi-Square
Tests: When variances are unknown or samples are
small, researchers typically use t-tests or exact tests instead. Nevertheless,
a Z-test with large n is asymptotically equivalent to a t-test, and Z-tests for
proportions are equivalent to chi-square tests for 2×2 tables[8]. Thus, understanding Z-tests helps interpret many common analyses in
the literature.
In summary, one-sample and two-sample Z-tests (for means or
proportions) provide the theoretical basis for many hypothesis tests in
healthcare research. They are most appropriate under large-sample conditions or
known variances[2][7]. Recent medical studies often use two-proportion Z-tests (or
equivalently chi-square tests) to compare rates between groups[11][10]. By ensuring clarity about assumptions and context, researchers and
students can correctly apply Z-tests or their equivalents in designing studies
and interpreting results.
Sources: Definitions and properties of Z-tests
are detailed in statistical references[1][4][7]. Applications are drawn from recent open-access medical studies
(2021–2024)[10][11][14][12]. These sources are cited using APA-style markers with direct URLs.
[1] One-Sample Z-Tests
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/One-Sample_Z-Tests.pdf
[2] [3] [15] What Is a Z-Test?
https://www.investopedia.com/terms/z/z-test.asp
[4] [5] Two-Sample Z-Tests Assuming Equal Variance
[6] [8] [9] Statistical notes for clinical researchers: Sample size calculation 2.
Comparison of two independent proportions
https://rde.ac/journal/view.php?number=711
[7] 9.3: Two Proportion Z-Test and Confidence Interval - Statistics
LibreTexts
[10] A Phase 3, randomized, non-inferiority study of a heterologous booster
dose of SARS CoV-2 recombinant spike protein vaccine in adults | Scientific
Reports
[11] Assessing Changes in Surgical
Site Infections and Antibiotic Use among Caesarean Section and Herniorrhaphy
Patients at a Regional Hospital in Sierra Leone Following Operational Research
in 2021 - PMC
https://pmc.ncbi.nlm.nih.gov/articles/PMC10458420/
[12] Exploring the levels of
variation, inequality and use of physical activity intervention referrals in
England primary care from 2017–2020: a retrospective cohort study - PMC
https://pmc.ncbi.nlm.nih.gov/articles/PMC11822388/
[13] Sample-size determination for decentralized clinical trials - PubMed
https://pubmed.ncbi.nlm.nih.gov/40393699/
[14] The population based cognitive
testing in subjects with SARS-CoV-2 (POPCOV2) study: longitudinal investigation
of remote cognitive and fatigue screening in PCR-positive cases and negative
controls - PMC
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