Statistics: Data, distributions and descriptive statistics

📊 Data, distributions and descriptive statistics. Every paper, guideline and blood test result rests on a few statistical ideas: what kind of data you have, how it is distributed, and how to summarise it honestly. The MRCP Part 1 and the UKMLA test these concepts with calculation questions that need only simple arithmetic. This chapter builds the foundation for hypothesis testing, diagnostic tests and critical appraisal in the chapters that follow.
Your four questions for any dataset: What type of variable is it? What shape is the distribution? Which average and which measure of spread are appropriate? How much uncertainty is there (standard error and confidence interval)?

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Types of Variable

Type	Subtype	Definition	Examples
Categorical (qualitative)	Nominal	Categories with no order	Blood group, sex, ethnicity, diagnosis
	Binary (dichotomous)	Two categories	Alive/dead, smoker/non-smoker
	Ordinal	Ordered categories, unequal intervals	Pain score, NYHA class, cancer stage, Likert scale
Numerical (quantitative)	Discrete	Counts - whole numbers	Number of admissions, number of children
	Continuous	Any value within a range	Height, BP, haemoglobin, age

• Interval data have equal intervals but no true zero (temperature in °C); ratio data have a true zero (weight, creatinine), so ratios are meaningful.
• The data type decides the statistics: proportions and chi-squared for categories; means/medians and t-tests or non-parametric tests for numbers.
• Dichotomising a continuous variable (e.g. "hypertensive yes/no") loses information and statistical power.
• Censored data (survival times where the event has not happened yet) need survival methods (Kaplan–Meier, Cox regression).

Measurement, Accuracy and Precision

• Accuracy (validity) - closeness to the true value; inaccuracy is systematic error (bias).
• Precision (reliability, repeatability) - closeness of repeated measurements to each other; imprecision is random error, reduced by larger samples or repeated measurements.
• Inter-observer and intra-observer agreement for categorical judgements is measured with Cohen's kappa (0 = chance agreement, 1 = perfect; >0.6 substantial).
• Agreement between two methods of measuring the same continuous quantity (e.g. two BP devices) is assessed with a Bland–Altman plot - not a correlation coefficient (two methods can correlate perfectly yet disagree systematically).
• Validity of a questionnaire: face, content, construct and criterion validity; reliability by test–retest and internal consistency (Cronbach's alpha).

Measures of Central Tendency

• Mean - sum ÷ number of observations. Uses all data; sensitive to outliers; best for symmetrical (normal) data.
• Median - the middle value when ordered (average of the two middle values if n is even). Robust to outliers; best for skewed data and ordinal data.
• Mode - the most frequent value. The only average for nominal data; a distribution may be bimodal.
• Relationship by skew:
  • Symmetrical: mean = median = mode.
  • Positive (right) skew (long tail to the right - e.g. length of stay, income, triglycerides): mean > median > mode.
  • Negative (left) skew (tail to the left - e.g. gestational age at delivery, age at death in high-income countries): mean < median < mode.
• Geometric mean - for log-normally distributed data (e.g. antibody titres, some hormone levels).
• Weighted mean - combines groups of different sizes (e.g. pooled estimates in meta-analysis).

Measures of Spread

• Range - maximum minus minimum; unstable, depends on extremes.
• Interquartile range (IQR) - the 25th to 75th centiles; contains the middle 50%; paired with the median for skewed data.
• Variance - the average squared deviation from the mean: Σ(x − mean)² ÷ (n − 1) for a sample.
• Standard deviation (SD) = √variance; in the units of the data; paired with the mean for normally distributed data.
• Coefficient of variation = SD ÷ mean × 100% - compares variability between measures on different scales, and describes assay precision (e.g. CV 3%).
• SD describes the spread of individual observations; the standard error describes the precision of an estimate (such as a mean). Confusing them is a classic exam trap.

The Normal Distribution

• Symmetrical, bell-shaped, defined entirely by its mean and SD.
• The 68–95–99.7 rule:

Range	Proportion of observations
Mean ± 1 SD	About 68%
Mean ± 1.96 SD (≈2 SD)	95%
Mean ± 2.58 SD	99%
Mean ± 3 SD	About 99.7%

• z-score (standard score) = (value − mean) ÷ SD - how many SDs a value lies from the mean. Used in bone densitometry (T-score compares with young adult mean; Z-score compares with age-matched mean), growth charts and paediatric reference ranges.
• Many biological variables are approximately normal (height, BP, haemoglobin); many are not (CRP, length of stay, most enzyme levels).
• Checking normality: histogram, Q–Q plot, comparison of mean and median, Shapiro–Wilk test. A useful exam clue: if the SD is large relative to the mean for a variable that cannot be negative (e.g. mean 4, SD 5), the data must be positively skewed.

Skewed and Other Distributions

• Log-normal - becomes normal after log transformation (e.g. triglycerides, viral load, many hormone levels).
• Binomial - number of "successes" in a fixed number of independent trials with the same probability (e.g. number of patients responding out of 20).
• Poisson - counts of rare events in a fixed time or space (e.g. cases per week); mean = variance.
• Transformation (log, square root) can make skewed data approximately normal so that parametric tests can be used; results are back-transformed (giving geometric means and ratios).
• Non-parametric methods avoid distributional assumptions (see the next chapter).

Reference Ranges and "Abnormal" Results

• A reference range is usually the central 95% of values in a healthy reference population - mean ± 1.96 SD for normal data, or the 2.5th–97.5th centiles otherwise.
• By definition, 5% of healthy people fall outside the reference range (2.5% at each end).
• Multiple tests: with 20 independent tests, a healthy person has a 1 − 0.95²⁰ ≈ 64% chance of at least one "abnormal" result.
• A value outside the reference range is not the same as disease, and a value inside it does not exclude disease - interpret with pre-test probability (see the Diagnostic Tests chapter).
• Reference ranges vary by age, sex, ethnicity, pregnancy and laboratory method (e.g. eGFR, haemoglobin, troponin sex-specific 99th centiles).

Samples, Populations and Sampling

• Population - everyone of interest; sample - those studied. Statistics from the sample estimate parameters of the population.
• Random sampling - every member has a known (often equal) chance of selection; minimises selection bias.
• Stratified sampling - random samples within strata (e.g. age groups) to ensure representation.
• Cluster sampling - random selection of groups (practices, schools); cheaper but needs larger samples (design effect).
• Systematic sampling - every kth person from a list.
• Convenience sampling - whoever is available; prone to bias and limits generalisability.
• External validity (generalisability) - whether results apply to other populations; internal validity - whether results are true for the population studied (free from bias and confounding).

Standard Error and the Central Limit Theorem

• Standard error of the mean (SEM) = SD ÷ √n.
• It measures how precisely the sample mean estimates the population mean - it falls as sample size rises (to halve the SE, quadruple n).
• Central limit theorem: the distribution of sample means is approximately normal for large samples (n ≥ about 30), even if the underlying data are not.
• Standard error of a proportion = √[p(1 − p) ÷ n].
• Exam trap: papers sometimes present mean ± SEM because it looks smaller - SEM does not describe variability between patients.

Confidence Intervals

• 95% CI for a mean = mean ± 1.96 × SE (for large samples; use the t-distribution for small samples).
• Interpretation: if the study were repeated many times, 95% of the intervals calculated would contain the true population value. Informally, it is the range of plausible values for the true effect.
• Width depends on variability (SD), sample size (n) and confidence level (99% CIs are wider than 95%).
• CIs are more informative than p-values - they show the size of the effect and its precision.
• Link to significance: a 95% CI for a difference that excludes 0, or for a ratio (RR, OR, HR) that excludes 1, corresponds to p < 0.05.
• Clinical significance: compare the CI with the minimal clinically important difference - a statistically significant result can be clinically trivial, and a non-significant result with a wide CI is inconclusive rather than negative.

Presenting Data

• Bar chart - categorical data; histogram - distribution of continuous data (bars touch; area = frequency).
• Box-and-whisker plot - median, IQR (box), range or 1.5 × IQR (whiskers), outliers.
• Scatter plot - relationship between two continuous variables (see correlation).
• Kaplan–Meier curve - survival over time with censoring marks; steps at each event.
• Forest plot - effect estimates with CIs from several studies (meta-analysis) or subgroups.
• Funnel plot - study effect versus precision; asymmetry suggests publication bias.
• Misleading graphs: truncated axes, 3D effects, inconsistent scales, relative changes presented without absolute values.

Descriptive Statistics at a Glance

Data	Average	Spread
Normal (symmetrical)	Mean	Standard deviation
Skewed	Median	Interquartile range
Ordinal	Median	IQR or range
Nominal	Mode / proportion	-
Precision of a mean	-	Standard error (SD ÷ √n)

Worked Calculations

• Values: 2, 3, 3, 4, 13. Mean = 25 ÷ 5 = 5; median = 3; mode = 3. Mean > median → positive skew (the outlier 13 pulls the mean).
• Haemoglobin in a population: mean 140 g/L, SD 10. 95% of people lie between 140 ± 1.96 × 10 = 120.4 to 159.6 g/L. A value of 160 has a z-score of +2.0.
• SEM: sample of 100, SD 20 → SE = 20 ÷ 10 = 2; 95% CI for the mean = mean ± 3.92.
• To halve the width of a CI you need four times the sample size.
• SE of a proportion: 40 of 100 respond (p = 0.4) → SE = √(0.4 × 0.6 ÷ 100) = 0.049; 95% CI ≈ 0.4 ± 0.096 = 30% to 50%.

Exam-style Scenarios

• Scenario. A study reports length of hospital stay as mean 9 days (SD 12). What does this tell you about the data?
  Answer: Length of stay cannot be negative, and an SD larger than the mean implies a long right tail - the data are positively skewed. Median and IQR would be more appropriate.
• Scenario. A new point-of-care HbA1c device is compared with the laboratory method. The authors report r = 0.96 and conclude the methods agree.
  Answer: Correlation measures association, not agreement. A Bland–Altman plot (mean difference and 95% limits of agreement) is needed.
• Scenario. Two trials of the same drug report mean BP reductions of 8 mmHg; trial A has 95% CI 6 to 10, trial B has 95% CI −1 to 17. Interpret.
  Answer: Same point estimate; trial A is precise and significant; trial B is imprecise (small or variable) and non-significant - it neither confirms nor excludes a clinically important effect.
• Scenario. A healthy volunteer has 10 independent blood tests. What is the probability that at least one is outside its 95% reference range?
  Answer: 1 − 0.95¹⁰ ≈ 0.40 (40%).

Key Points

• Identify the data type before choosing a summary or a test.
• Normal data: mean and SD; skewed data: median and IQR.
• SD describes individuals; SE describes the precision of an estimate.
• 95% of normal data lie within 1.96 SD of the mean; 5% of healthy people are "abnormal" on any test.
• A 95% CI excluding 0 (difference) or 1 (ratio) means p < 0.05.

Guidelines and Further Reading

• Kirkwood BR, Sterne JAC. Essential Medical Statistics
• Petrie A, Sabin C. Medical Statistics at a Glance
• Altman DG. Practical Statistics for Medical Research
• BMJ "Statistics Notes" series (Altman and Bland)
• Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet 1986