Atlas
statminds
GLM (Single-Group Mean Model)The underlying model family class (e.g. GLM, linear model, categorical matrix, log-linear).Parametric ReferenceStatistical methods that assume a specific probability distribution family (typically normal).12-stage workflow
One-Sample T-Test
The engine for Benchmark Discovery. This model audits the distance between your sample and a fixed clinical or population standard, revealing if your data 'breaks' from the established norm.
G1
Benchmark Audit
Determine if your intervention group deviates from a known clinical threshold or historical average.
G2
Standard Validation
Verify if your sample mean aligns with a hypothesized population parameter (μ₀).
G3
Precision Discovery
Quantify the magnitude of the 'break' from the norm using standardized effect sizes.
1
What is it?
One-Sample T-Test compares the mean of a single sample against a known, pre-specified population benchmark value to see if they differ significantly.
2
When to use it
- Single Sample: You have one set of continuous measurements.
- Standard Comparison: Test against a historical norm, design spec, or neutral value.
3
Core Idea
We measure how many standard errors the observed sample mean lies away from the baseline test value:
4
Hypotheses
H0: Mean = Test Value (No difference exists)
Ha: Mean != Test Value (The sample mean differs significantly)
5
How it works
- Compute the sample mean and standard deviation.
- Calculate Standard Error: SE = SD / sqrt(N).
- Compute t-statistic: t = (Mean - Test Value) / SE.
- Compare t against the t-distribution with df = N - 1 to find p.
6
Assumptions
📊 Normality: The continuous sample outcome is normally distributed.
👤 Independence: Each subject is measured independently.
7
Effect Size
Standardized difference is measured via Cohen's d: d = (Mean - Test Value) / SD. Benchmarks suggest d = 0.2 is small, 0.5 is medium, and 0.8 is large.
8
Quick Example
| Subject Group | Observed Mean | Target Benchmark |
|---|---|---|
| Sample (N=20) | 52.4 | 50.0 |
| Difference | +2.4 (p = 0.035) | |
Interactive Sandbox
One-Sample T-Test Live Laboratory
Adjust the sample mean, test baseline value, and standard deviation to watch statistical separation.
Presets
Population Normal curve under H0 vs sample points (X axis: 20 to 80)Amber line indicates test baseline; Blue line shows sample mean
Calculations Output
| Metric | Observed Value |
|---|---|
| Sample Mean (X-bar) | 50.3216 |
| Test Baseline (mu0) | 50.0000 |
| t-statistic | 0.1503 |
| p-value (two-tailed) | 0.8821 |
Statistical Verdict
❌ Insignificant Difference
The observed mean of 50.32 is statistically indistinguishable from baseline 50.00 (p = 0.882). Fail to reject H0.
The 12-Stage Precision Workflow
01The Null Anchor
Hypotheses
We test the null of zero distance (μ = μ₀) against the discovery of a significant departure from the benchmark.
02Distributional Integrity
Assumptions
Ensuring the sample mean comes from a normally distributed population—the foundational prerequisite for T-math.
03Skewness Audit
Diagnostics
Utilizing Shapiro-Wilk or Q-Q plots to ensure that extreme outliers aren't artificially 'pulling' the mean away from the benchmark.
04focus
Comparing the average recovery time of FlowMotion participants against the industry standard of 12 weeks.
05Wilcoxon Pivot
Alternatives
Knowing when to switch to the One-Sample Wilcoxon Signed-Rank test if the data is ordinal or non-normally skewed.
06The T-Strike
Significance
Executing the T-test strike to determine if the observed difference is a true population signal or a chance artifact of sampling.
07Cohen’s d
Effect Size
Interpreting the 'Standardized Distance'—quantifying how many standard deviations your sample sits from the norm.
08Detection Power
Sample Size
Calculating the N required to ensure that even subtle 'breaks' from the benchmark are detected with mathematical authority.
09The Distance Statement
Reporting
Reporting the mean difference and 95% CI: 'Our sample mean was significantly higher than the standard, t(df) = X.XX, p = .XXX.'
10Mu Specification
Software
Executing 't.test(mu = X)'—ensuring the 'mu' parameter correctly represents the clinical anchor.
11focus
The fatal error of using the wrong benchmark value, which invalidates the entire discovery trajectory.
12focus
Tracing the model back to William Sealy Gosset ('Student') and the birth of small-sample industrial quality control.
01Hypothesis test logic
Hypotheses
Pragmatic null and alternative hypotheses defined in mathematical notation.
A hypothesis is a question sharpened to a point. Ambiguity is the enemy of inference.
Logic CoreNull · H₀
H₀: μ = μ₀ (population mean equals hypothesized value μ₀)
Alternative · Hₐ
Hₐ: μ ≠ μ₀ (population mean differs from μ₀)
Why it matters two-tailed
μ₀ is the theoretical/expected value. Can be one-tailed if justified.
02Model diagnostics
Assumptions
The core mathematical criteria needed to ensure that statistical testing remains unbiased and valid.
Build your analysis on rock, not sand. Verify the mathematical foundation before building the model.
Integrity Shield5
Assumptions
3
Critical / High Severity
How to check
Quick
Inspect variable type; create histograms & summary statistics to verify scale is not discrete/categorical
Rigorous
Formal measurement scale validation; verify measurement units are truly interval/ratio (equal distances between values); check precision of measurement
If violated
If ordinal with limited categories (e.g., Likert 1-5) → use one-sample Wilcoxon signed-rank test (tests whether median differs from μ₀). If binary outcome → use binomial test (tests proportion against expected value). If count data → use Poisson goodness-of-fit or exact binomial test. If many ordinal levels (7+) → may cautiously proceed with t-test
chi square goodness of fit
How to check
Quick
Verify μ₀ comes from: (1) published norms/standards, (2) regulatory benchmarks, (3) theoretical predictions, or (4) historical baseline. Should NOT be arbitrary
Rigorous
Check primary source for μ₀; verify normative sample characteristics match your sample; ensure μ₀ is on same scale as your data; confirm μ₀ is not derived from same data being tested (circular reasoning)
If violated
If μ₀ is arbitrary or unjustified → reconsider analysis purpose. If comparing to another sample mean → use independent t-test, not one-sample. If no meaningful μ₀ exists → use descriptive statistics and confidence intervals instead of hypothesis test. One-sample t-test requires theoretically meaningful comparison value
independent t test
How to check
Quick
Design review; check no subject ID appears multiple times; verify no clustering by site, family, or other grouping variable
Rigorous
Durbin-Watson test for autocorrelation (if sequential data); check intraclass correlation (ICC) if clustering suspected; verify sampling design ensures independence
If violated
If repeated measures (multiple observations per subject) → aggregate within subjects (e.g., compute mean per subject) then test aggregated values, OR use one-sample Wilcoxon on difference scores, OR use LMM with random intercepts. If clustered data (e.g., students in schools) → use LMM with random effects for clusters or cluster-robust SEs. If time series/autocorrelation → use time series methods
linear mixed modelgee
How to check
Quick
Boxplot; identify values >1.5 IQR beyond quartiles; check for extreme z-scores (|z| > 3 using sample M and SD)
Rigorous
Standardized residuals (|z| > 3 suspect); check if outliers are data entry errors or legitimate extreme values; examine influence on results with sensitivity analysis
If violated
First: verify data entry errors and correct if found. If legitimate outliers: (1) Report results with and without outliers (sensitivity analysis); (2) Use one-sample Wilcoxon signed-rank test (median-based, resistant to outliers); (3) Use trimmed mean test (10-20% trimming); (4) Winsorize extreme values to 5th/95th percentiles (pre-specify); (5) Bootstrap confidence intervals. Never remove outliers without justification and transparency
How to check
Quick
Q-Q plot (points should fall on diagonal line); histogram; visual inspection for severe skewness, heavy tails, or bimodality
Rigorous
Shapiro-Wilk test (n < 50; p > .05 indicates normality); Anderson-Darling test; check skewness (<|2|) and kurtosis (<|7|) values
If violated
If mild violation with moderate sample size (n ≥ 30): t-test is robust via Central Limit Theorem, proceed cautiously. If notable violation: (1) Transform data (log for right-skewed, sqrt for count-like, inverse for severe right-skew); (2) Use one-sample Wilcoxon signed-rank test (distribution-free, tests median vs μ₀); (3) Use permutation test (resampling-based); (4) Bootstrap confidence intervals. With very small samples (n < 15) and clear non-normality, strongly prefer nonparametric alternatives
03Residual Forensics
Diagnostics
Checking residual plots and indices to examine model deviations and ensure standard error integrity.
Trust, but verify. The outliers often hold more truth than the averages.
System HealthEssential checks
- Shapiro-Wilk test for normality (if n < 50)
- Q-Q plot to assess normality
- Boxplot to identify outliers
Recommended checks
- Histogram of distribution
- Descriptive statistics (M, SD, n)
- Check skewness and kurtosis values
- Visual comparison of sample mean vs μ₀
04Live Instances
Applied Minds
Review concrete study examples, data layout guidelines, and copy executable syntax scripts.
Theory is the map. Practice is the terrain. Simulation bridges the gap.
Applied WisdomExample 01
IQ Test Validation Against Population Norm (μ₀=100)
Research question: Does a new online IQ test produce scores that match the population norm (μ₀=100)? Design: n=42 adults complete new test. Outcome: IQ score (continuous, M=100, SD=15 in general population). Tests whether sample mean differs significantly from theoretical population mean of 100.
Outcome ScaleIQ score (continuous)
# One-sample t-test: IQ test validation against norm
set.seed(2025)
n <- 42
# Simulate IQ scores (slightly below norm: M=97.5)
data <- data.frame(
subject_id = 1:n,
IQ = rnorm(n, mean=97.5, sd=14.8)
)
mu_0 <- 100 # Population norm
# === STEP 1: Check Assumptions ===
# Normality
shapiro.test(data$IQ)
# p > .05: normality OK
qqnorm(data$IQ, main="Q-Q Plot of IQ Scores")
qqline(data$IQ)
# Outliers
boxplot(data$IQ, horizontal=TRUE, main="IQ Scores")
abline(v=mu_0, col="red", lwd=2, lty=2) # Reference line at μ₀
# === STEP 2: Descriptive Statistics ===
cat("Sample: M =", round(mean(data$IQ), 2), ", SD =", round(sd(data$IQ), 2), "\n")
cat("Population norm: μ₀ =", mu_0, "\n")
# === STEP 3: One-sample t-test ===
t_result <- t.test(data$IQ, mu = mu_0)
print(t_result)
# === STEP 4: Effect Size ===
cohen_d <- (mean(data$IQ) - mu_0) / sd(data$IQ)
cat("Cohen's d:", round(cohen_d, 2), "\n")
# Visualization
library(ggplot2)
ggplot(data, aes(x=IQ)) +
geom_histogram(aes(y=..density..), bins=12, fill="steelblue", alpha=0.7) +
geom_vline(xintercept=mean(data$IQ), color="blue", lwd=1.5, linetype="solid") +
geom_vline(xintercept=mu_0, color="red", lwd=1.5, linetype="dashed") +
annotate("text", x=mean(data$IQ)-5, y=0.025, label="Sample M", color="blue") +
annotate("text", x=mu_0+5, y=0.025, label="μ₀=100", color="red") +
labs(title="IQ Scores: Sample vs Population Norm") +
theme_classic()
cat("\nt(41) = -1.09, p = .28, d = -0.17\n")
cat("No significant difference from population norm.\n")Interpretation Blueprint
t(41) = -1.09, p = .28, d = -0.17 (negligible effect). No significant difference between sample mean (M=97.5) and population norm (μ₀=100). The new IQ test produces scores consistent with standardized norms.
05Tactical Pivots
Alternatives
Structured fallback pathways for choosing alternative tests when normality or slopes requirements fail.
When the path is blocked, pivot. Rigor is not rigidity; it is the intelligent adaptation to reality.
Adaptive StrategyMeasurement Precision Ladder Ideal · Ratio / Interval
Ratio
Maintain One-Sample T-Test. Provides the most efficient audit of distance from a clinical benchmark.
Peak SignalInterval
Ideal for Single-Group Scales. Ensure the 'Benchmark' (mu) is a known constant, not a sample estimate.
Standard PrecisionOrdinal
Pivot to One-Sample Wilcoxon Signed-Rank Test to audit the median rank relative to the benchmark.
Median DistortionNominal
Abandon T-Test. Use Chi-Square Goodness-of-Fit to model the alignment with expected proportions.
Information SuicideTemporal Trajectory Audit Static Benchmark Snapshot
Static Audit
Single point vs. Constant.
Stay with One-Sample T-Test. The definitive strike for validating standards.
Paired Comparison
Subjects vs. Past.
Pivot to Paired T-Test if the 'Benchmark' is actually the same group's baseline score.
Adaptive Technical Safeguards · adaptive safeguards
normality violated
- Wilcoxon Signed-Rank (1-sample) — The robust median-based alternative for skewed samples.
- Bootstrap Mean Strike — Generate significance for the mean difference using 1,000 resamples.
extreme outliers
- Trimmed Mean T-Test — Automatically exclude the extreme 5% of the distribution to protect the average.
- Robust One-Sample Test — Use M-estimators to find the robust center of the sample.
non independent samples
- One-Sample GEE — Account for clustering if observations are nested within higher-level units.
06Adjusted Comparisons
Post-hoc
Group mean comparisons and correction controls (e.g. Tukey HSD, Bonferroni) to protect against Family-Wise Error Rates.
The omnibus test opens the door; post-hoc analysis explores the room.
Forensic DetailAdjusted Comparisons
- Compare with Wilcoxon signed-rank test (nonparametric alternative)
- Bootstrap confidence intervals for mean
- Examine normality assumption via Shapiro-Wilk
- Test sensitivity to outliers (with/without trimming)
- Calculate Cohen's d effect size: d = (mean - mu0) / SD
Interpretation Guidelines
One-sample t-test compares sample mean to a single value. Post-hoc tests are not applicable.
07Standardized scale impact
Effect Size
Understanding effect sizes (e.g., Cohen's d, Partial Eta-Squared) and clinical impact benchmarks.
Significance is noise. Magnitude is the signal. Measure the impact, not just the probability.
Impact MagnitudeStandardized difference from hypothesized mean. d = (M - μ₀) / SD. Small: 0.2, Medium: 0.5, Large: 0.8 (Cohen, 1988). Use sample SD for denominator
Unstandardized difference (M - μ₀) with 95% CI. Easiest to interpret in original units (e.g., '2.5 points below IQ norm of 100')
Recommended Metric: cohens_d for standardized effect (enables comparison across studies); mean_difference for interpretability
Small
0.2
0.2
Medium
0.5
0.5
Large
0.8
0.8
0.50
cohens_d for standardized effect (enables comparison across studies); mean_difference for interpretability
Recommended Measure
2
Available Metrics
ReportUse cohens_d for standardized effect (enables comparison across studies); mean_difference for interpretability to represent clinical impact magnitude.
08Statistical Power
Sample Size
Guidelines for minimum sample requirements and power analysis parameters.
An underpowered study is an ethical failure. Respect the data by collecting enough of it.
Power ProtocolFloor Requirements
A minimum of 20 participants is required to ensure the sample mean is a stable representative of the population and resists the pull of single outliers.
| Effect Size | Parameters | Required n |
|---|---|---|
| Small Effect | d=0.20 (Small) | n ≈ 199 |
| Medium Effect | d=0.50 (Medium) | n ≈ 34 |
| Large Effect | d=0.80 (Large) | n ≈ 15 |
Key considerations
Anchor Integrity: If the population benchmark (μ₀) is an estimate rather than a known fact, the uncertainty of the anchor must be accounted for by increasing the sample size by 15%.
G*Power StrategyBenchmark: T-tests → Means: Difference from constant (one sample case). Parameters: Cohen's d, α = .05, Power = .80. Note: Power depends on the distance between the observed mean and the theoretical anchor (μ₀).
09APA narrative blueprint
Reporting
How to compile statistical results into publication prose matching APA and journal style guides.
Data does not speak for itself. It requires a translator. Be clear, be precise, be honest.
Narrative ArcWorked APA paragraph example
A one-sample t-test was conducted to test whether a new online IQ test produced scores consistent with the population norm of μ₀ = 100. Data were approximately normally distributed (Shapiro-Wilk p = .42). The sample mean (M = 97.5, SD = 14.8) did not significantly differ from the population norm, t(41) = -1.09, p = .28, d = -0.17 (95% CI [-7.1, 2.1]). This negligible effect suggests the new test produces scores consistent with established IQ norms.
Reusable template
A one-sample t-test was conducted to test whether sample description differed from description of μ₀, e.g., 'the population norm of 100'. If assumptions checked, state: 'Data were approximately normally distributed (Shapiro-Wilk p > .05).' If violated: 'Due to non-normality (Shapiro-Wilk p < .05), Wilcoxon signed-rank test was used as a sensitivity analysis.' The sample mean (M = XX.X, SD = X.X) was significantly/not significantly different from μ₀ = XX.X, t(df) = X.XX, p = .XXX, d = X.XX (95% CI X.X, X.X). Interpret effect size and practical significance in context.
Essential statistics to report
- t-statistic
- degrees of freedom (n - 1)
- p-value (exact if p > .001, otherwise p < .001)
- effect size (Cohen's d)
- 95% confidence interval for mean difference
- sample descriptive statistics (M, SD, n)
- hypothesized population mean (μ₀)
- statement about normality assumption
10Exhibit Builder
Manuscript Lab
Copy standard summary tables and forensic reporting grids to outline analysis details.
Table 1: One-Sample t-test: Performance vs. Industry Benchmark
| Sample Mean | Target | Difference | t | df | p | Cohen's d |
|---|---|---|---|---|---|---|
| 75.2 | 70.0 | 5.2 | 4.45 | 119 | < .001 | 0.41 |
Note. Test Value (Target) = 70.0. N = 120.
Difference (5.2)Identifies the 'Performance Gap'. The sample exceeded the benchmark by 7.4% (5.2 / 70).
d = 0.41Moderate Superiority. The group significantly outperformed the target, but with enough overlap to suggest individual variability remains high.
Header glossary
The Benchmark. The hypothetical or historical value the sample is being compared against (e.g., passing grade, national average).
Benchmark Distance. Measures how many standard deviations the sample mean is away from the target.
11Algorithmic Logic
Command Center
Syntax libraries and function parameters for executing calculations in stats packages.
Code is the modern laboratory. Clean execution ensures reproducible discovery.
Execution Engine# 1. Execute One-Sample T-Test
t.test(df$score, mu = 70)
# 2. Extract One-Sample Cohen's d
lsr::cohensD(df$score, mu = 70)Library stack
R
statslsr
Python
scipy.stats
Elite Forensic Strike
The 'Benchmark Trap'. Statistical significance does not mean practical significance. Always check if the 95% CI of the difference excludes your 'Minimally Important Difference'.
# Generate Narrative with CI Focus
report::report(t.test(df$score, mu = 70))12The Over-adjustment Trap
Common Mistakes
Analytical caveats and corrections to maintain modeling integrity.
Wisdom is learning from the failures of others. Anticipate the error before it occurs.
Defensive LogicWhy it's wrong
One-sample t-test requires theoretically meaningful μ₀ from published norms, regulatory standards, or theoretical predictions. Using arbitrary values (e.g., '0 because it seems neutral') makes the test meaningless. The null hypothesis H₀: μ = μ₀ is only interpretable when μ₀ has substantive meaning.
The correction
Verify μ₀ comes from: (1) Published normative data (e.g., IQ M=100, SD=15); (2) Regulatory standards (e.g., maximum contaminant level); (3) Theoretical predictions (e.g., chance performance = 50%); (4) Historical baseline. Cite source for μ₀. If no meaningful μ₀ exists, report confidence intervals instead of hypothesis test.
Why it's wrong
If you have data from TWO samples and want to compare their means, use independent t-test (tests μ₁ = μ₂), not one-sample t-test on one group against the other's mean as μ₀. Using one-sample t-test ignores uncertainty in the comparison group's mean and inflates Type I error.
The correction
If comparing two samples: use independent t-test. If testing single sample against known population parameter (from external source, not your data): use one-sample t-test. Never use one sample's mean as μ₀ for the other.
Why it's wrong
p-value only indicates statistical significance, not practical significance or magnitude. Cohen's d and 95% CI for (M - μ₀) provide crucial information about effect size. With large n, tiny trivial differences can be 'statistically significant' but meaningless.
The correction
Always report: (1) Cohen's d = (M - μ₀) / SD with interpretation (small/medium/large); (2) 95% CI for mean difference in original units; (3) Sample M and SD. Example: 'M = 97.5 (SD = 14.8) vs μ₀ = 100, d = -0.17, 95% CI [-7.1, 2.1]'.
Why it's wrong
One-sample t-test assumes approximate normality. With large n (≥30), CLT makes t-test robust. But with small n (<30) and severe non-normality (heavy skew, outliers), t-test can have inflated Type I error or reduced power. The p-value may be inaccurate.
The correction
Always check normality with Q-Q plot and Shapiro-Wilk (if n < 50). If n ≥ 30 and mild violation: proceed with t-test. If n < 30 and clear non-normality: (1) Transform data (log, sqrt); (2) Use one-sample Wilcoxon signed-rank test (tests median vs μ₀); (3) Use bootstrap CI; (4) Use permutation test.
Why it's wrong
If observations come from same subjects measured multiple times, or are clustered (e.g., students in schools), independence assumption is violated. One-sample t-test treats all observations as independent, which inflates Type I error when dependency exists. Standard errors are too small.
The correction
If repeated measures: aggregate within subjects (compute mean per subject) then test, OR use LMM with random intercepts. If clustered data: use LMM with random effects for clusters OR cluster-robust SEs. Report dependency structure and how it was handled.
Why it's wrong
One-tailed tests (H₁: μ > μ₀ or μ < μ₀) double your risk of missing effect in opposite direction and are only justified with strong theoretical reason to predict direction BEFORE seeing data. Choosing one-tailed after seeing data direction inflates Type I error.
The correction
Default to two-tailed tests. Use one-tailed only if: (1) Strong theoretical prediction of direction a priori; (2) Effect in opposite direction is impossible/meaningless; (3) Pre-registered hypothesis specifies direction. Never choose tail based on observed data.
Why it's wrong
Using part of your sample to estimate μ₀ and then testing the full sample (or another part) against that μ₀ is circular reasoning. This inflates Type I error and produces spurious significance. μ₀ must be independent of the data being tested.
The correction
μ₀ must come from external source: published norms from different sample, theoretical value, or pre-specified standard. Never compute μ₀ from your own data. If you want to test whether two subgroups differ, use independent t-test, not one-sample t-test on each.
13Academic Lineage
References
Scholarly lineage and citation keys grounding the statistical framework.
We stand on the shoulders of giants. Honor the source of the method.
Academic Lineage[1]
Mackowiak, P. A., Wasserman, S. S., & Levine, M. M. (1992). A critical appraisal of 98.6°F, the upper limit of the normal body temperature, and other legacies of Carl Reinhold August Wunderlich. JAMA, 268(12), 1578-1580.
Modern study showing normal body temperature is ~98.2°F, not 98.6°F. Basis for Example 2.
doi: 10.1001/jama.1992.03490120092034
[2]
Wechsler, D. (2008). Wechsler Adult Intelligence Scale–Fourth Edition (WAIS-IV). Pearson.
IQ tests standardized to M=100, SD=15. Normative basis for Example 1.
[3]
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.
Classic reference for effect size interpretation (d: small=0.2, medium=0.5, large=0.8) and power analysis.
[4]
Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch's t-test instead of Student's t-test. International Review of Social Psychology, 30(1), 92-101.
Argues for robust alternatives to classical t-tests. Discusses assumptions and robustness.
doi: 10.5334/irsp.82
A benchmark is a shadow of the past. Use the One-Sample test to prove your data has stepped into the light of something new.
The Interpretive Rigor Directive
statminds · One-SampleMind reference · v2.2 · updated 2026-01-1715 of 15 sections