Types Of Data (Categorical, Numerical, Parametric, Non-Parametric, Discrete/Continuous)

Types of Data: Categorical, Numerical, Parametric, Non-Parametric, Discrete and Continuous

For the Primary FRCA, data type is not a semantic nicety: it determines how data should be summarised, plotted and analysed. A large proportion of statistics questions can be answered by first asking three questions: what is being measured?, what scale is it measured on?, and are observations independent or paired? Errors in these early decisions lead to inappropriate use of means, standard deviations, t tests and correlation coefficients.

In anaesthetic research and audit, examples are everywhere: ASA physical status, Cormack-Lehane view, pain scores, blood pressure, plasma propofol concentration, duration of apnoea, postoperative nausea and vomiting, ICU length of stay and mortality. Each is a different form of data and requires different statistical handling.

1. The hierarchy of measurement scales

The classic classification is attributed to Stevens and divides data into nominal, ordinal, interval and ratio scales. This hierarchy is exam-relevant because the permissible mathematical operations increase as one moves from nominal to ratio data.

Key principle: nominal and ordinal data are forms of categorical data; interval and ratio data are forms of numerical data. However, the boundary is not always rigid. A pain score from 0 to 10 is often analysed as numerical in large trials, but strictly it is ordinal because the difference between 1 and 2 is not guaranteed to be physiologically equal to the difference between 8 and 9.

2. Categorical data

Nominal data

Nominal data classify observations into mutually exclusive categories without natural ranking. Examples include postoperative nausea present or absent, type of anaesthetic technique, surgical specialty, blood group and death versus survival. The appropriate descriptive statistics are counts and proportions. For example, postoperative vomiting in 18 of 120 patients is reported as 18/120 or 15%.

Nominal data may be binary or multicategory. Binary variables have two categories, such as alive/dead, smoker/non-smoker, or difficult intubation yes/no. Multicategory nominal variables have more than two unordered groups, such as blood group A/B/AB/O or anaesthetic technique inhalational/TIVA/regional/combined.

Ordinal data

Ordinal data have a natural order, but the spacing between categories is not mathematically guaranteed. This is crucial for common anaesthetic scoring systems. ASA IV is more severe than ASA II, but it is not twice ASA II. A Cormack-Lehane grade 4 view is worse than grade 2, but the difference between grades 1 and 2 is not equivalent to the difference between grades 3 and 4.

Ordinal data should usually be described using median and interquartile range, or by displaying the full distribution across categories. In small samples, reporting only the median can hide clinically important differences. For example, two groups may both have median pain score 3, but one group may contain many patients with severe pain scores of 8-10.

3. Numerical data

Numerical data arise from measurement or counting and are usually divided into discrete and continuous data. They may be interval or ratio scale. Numerical data can be summarised using measures of central tendency and dispersion, but the choice depends strongly on the distribution.

Discrete numerical data

Discrete data can take only separate values, usually whole numbers. They commonly arise from counts. Examples include number of intubation attempts, number of vasopressor boluses, number of postoperative complications, number of ICU admissions, and number of failed epidural top-ups.

Discrete count data often follow a Poisson distribution if events occur independently at a constant average rate. For a Poisson variable, the mean equals the variance. In clinical data this assumption is frequently violated by overdispersion, clustering and patient heterogeneity. For example, some patients are predisposed to repeated hypotensive episodes, so the variance exceeds the mean; a negative binomial model may then fit better.

Continuous numerical data

Continuous data can theoretically take any value within a range, limited only by measurement precision. Examples include age, weight, height, blood pressure, arterial oxygen tension, end-tidal carbon dioxide, plasma remifentanil concentration, time to eye opening, duration of surgery and cardiac output. Continuous variables may be measured to varying precision: a blood pressure displayed as 124 mmHg is still conceptually continuous, although recorded as an integer.

4. Parametric and non-parametric: what these terms really mean

The term parametric refers to statistical methods that assume the data are drawn from a population described by a finite set of parameters. In common FRCA usage, this usually means methods based on the normal distribution, described by its mean and standard deviation. Examples include the unpaired Student t test, paired t test, analysis of variance and Pearson correlation.

Non-parametric methods make fewer assumptions about the precise distribution. Many work by ranking data rather than analysing raw values. Examples include the Mann-Whitney U test, Wilcoxon signed-rank test, Kruskal-Wallis test and Spearman rank correlation. They are particularly useful for ordinal data, skewed numerical data and small samples where normality is doubtful.

However, non-parametric does not mean assumption-free. For example, the Mann-Whitney U test assumes independent observations and, if interpreted as a test of medians, similarly shaped distributions. If distributions differ in shape or spread, it tests a broader concept: the probability that a randomly selected observation from one group exceeds a randomly selected observation from another.

5. The normal distribution and parametric assumptions

A normally distributed variable has a symmetric bell-shaped distribution defined by mean μ and standard deviation σ. The mean, median and mode coincide. In a normal distribution, approximately 68.3% of observations lie within 1 SD of the mean, 95.4% within 2 SD, and 99.7% within 3 SD. For a standard normal distribution, the 2.5th and 97.5th centiles are approximately -1.96 and +1.96 SD, giving the familiar 95% interval.

Many physiological variables are approximately normal within a defined clinical population: haemoglobin concentration, serum sodium, height and some haemodynamic measurements. Others are commonly skewed: C-reactive protein, lactate, hospital stay, time to first analgesia, drug concentrations and blood loss. Skewed variables are often better described using median and interquartile range.

Core assumptions of common parametric tests

• Scale: the dependent variable is usually continuous interval or ratio data.
• Independence: observations are independent unless a paired/repeated-measures method is used.
• Normality: the outcome, or more precisely the residuals from the model, are approximately normally distributed.
• Homogeneity of variance: groups have similar variances for tests such as the conventional unpaired t test and ANOVA.
• Linearity: for Pearson correlation and linear regression, the relationship should be approximately linear.

The central limit theorem is important: for sufficiently large samples, the sampling distribution of the mean tends towards normality even if the raw data are not normal. This is why t tests may be robust in large samples with mild non-normality. Nevertheless, extreme skewness, outliers and small sample sizes remain problematic.

6. Descriptive statistics by data type

The standard deviation describes variability among individual observations. The standard error of the mean describes precision of the estimated mean and equals SD divided by the square root of n. A common exam trap is presenting SEM instead of SD to make variability appear smaller. Clinical baseline characteristics should generally use SD, not SEM.

7. Choosing a statistical test

Selection of a statistical test depends on: data type of the outcome, number of groups, whether observations are independent or paired, distributional assumptions, and whether adjustment for covariates is required.

8. Paired, unpaired and repeated data

Data type alone is insufficient. The structure of the data is equally important. Comparing pre-induction and post-induction blood pressure in the same patient requires a paired analysis because measurements are correlated within individuals. Treating them as independent wastes information and gives incorrect standard errors. Repeated measurements over time, such as serial blood pressure readings every minute after induction, require repeated-measures ANOVA, mixed-effects modelling, area-under-the-curve analysis, or clinically defined summaries such as lowest MAP or time below MAP 65 mmHg.

In perioperative research, clustering is common: patients are clustered within anaesthetists, lists, theatres and hospitals. Ignoring clustering can overstate precision because observations are not fully independent. Cluster randomised trials and multicentre studies often require intracluster correlation adjustment.

9. Dichotomisation and loss of information

Continuous variables are often converted into categories: age over 65 years, MAP below 65 mmHg, BMI over 30 kg/m2, or haemoglobin below 80 g/L. This may be clinically useful, especially where guidelines define thresholds, but it discards information and reduces statistical power. A patient with MAP 64 mmHg is treated as different from 66 mmHg, whereas a patient with MAP 40 mmHg is treated the same as 64 mmHg if the outcome is simply hypotension yes/no.

Common perioperative thresholds include MAP below 65 mmHg for intraoperative hypotension in many observational studies, SpO2 below 90% for hypoxaemia, BMI 30 kg/m2 for obesity and 40 kg/m2 for severe obesity, and haemoglobin thresholds such as 70-80 g/L for restrictive transfusion strategies in stable patients depending on clinical context. These thresholds are clinically meaningful but should not be confused with the original data type.

10. Percentages, proportions, rates and ratios

Candidates often confuse percentages with continuous data. A percentage may represent a proportion based on binary outcomes, such as 20% postoperative nausea, or a continuous derived quantity, such as percentage change in blood pressure. The analysis depends on how it was generated.

• Proportion: number with event divided by number at risk, for example 12/100 patients with desaturation.
• Rate: events per unit time, for example central line infections per 1000 catheter-days.
• Ratio: one quantity divided by another, for example PaO2/FiO2 ratio; this is continuous ratio data.
• Percentage change: continuous derived variable, but often skewed and sensitive to baseline values.

11. Transformations and log-normal data

Some clinical variables are not normally distributed but become approximately normal after transformation. Drug concentrations, inflammatory markers and length of stay may be log-normally distributed: their logarithms are approximately normal. In such cases, analyses may be performed on log-transformed data, with results back-transformed as geometric means or ratios of geometric means.

For example, plasma remifentanil concentrations or serum CRP values cannot be negative and often show right skew. Reporting a mean and SD on the raw scale may be misleading because a few high values dominate the mean. A median with IQR, or a geometric mean with multiplicative confidence intervals, is often more appropriate.

12. Time-to-event data and censoring

Time-to-event data deserve separate treatment. Examples include time to extubation, time to first analgesic request, time to discharge readiness, time to ICU discharge and survival. These are continuous ratio data but are often censored: the event has not occurred by the end of follow-up, or observation is incomplete. Standard t tests are inappropriate when censoring is substantial. Kaplan-Meier curves, log-rank tests and Cox proportional hazards models are used.

A patient not yet discharged from PACU at 2 hours is not simply a 120-minute discharge time; the true time is unknown but exceeds 120 minutes. Treating censored observations as if the event occurred at the censoring time biases estimates.

13. Distributional assessment

In practice, distribution should be assessed visually and clinically, not by mechanical reliance on a normality test. Histograms, Q-Q plots and box plots are more informative than a Shapiro-Wilk p value alone. With very small samples, normality tests have low power; with very large samples, trivial deviations become statistically significant. The question is whether the chosen method is robust and whether the summary statistic reflects the clinical pattern.

14. Examination-level examples

Example 1: Pain score after caesarean section

A numeric rating scale from 0 to 10 is collected at 6 hours. Strictly, this is ordinal. If the sample is small or distribution is skewed, use median and IQR and compare groups with Mann-Whitney U or Kruskal-Wallis. In a large randomised trial, it may be pragmatically treated as continuous, but the assumption should be defensible.

Example 2: Propofol induction dose

Dose in mg/kg is continuous ratio data. If approximately normally distributed, summarise using mean and SD and compare two independent groups using an unpaired t test. If elderly frail patients create a skewed distribution, median and IQR or transformation may be preferable.

Example 3: Incidence of difficult intubation

Difficult intubation yes/no is binary nominal data. Present as n/N and percentage. Compare independent groups using chi-square test if expected cell counts are adequate, or Fisher exact test when expected counts are small, classically when any expected cell count is below about 5.

Example 4: Number of laryngoscopy attempts

This is discrete count data, usually skewed with many patients having one attempt and few having multiple attempts. Reporting mean attempts may be less clinically informative than the proportion requiring more than one attempt or the distribution across 1, 2, 3 and more than 3 attempts.

15. Parametric versus non-parametric is not the same as numerical versus categorical

A common misconception is that numerical data are automatically parametric and categorical data are automatically non-parametric. Parametric refers to the statistical model and assumptions, not merely to the appearance of numbers. ASA class is coded numerically I-VI but remains ordinal categorical data. Conversely, binary outcomes may be analysed with parametric models such as logistic regression, which has parameters but is not a normal-theory t test.

16. Reporting standards and modern practice

Current reporting guidance, including CONSORT for randomised trials, STROBE for observational studies and SAMPL statistical reporting recommendations, emphasises reporting estimates with 95% confidence intervals, not just p values. The conventional p value threshold of 0.05 is arbitrary and should not be interpreted as proof of effect or no effect. A p value is the probability of observing data at least as extreme as those obtained, assuming the null hypothesis and model assumptions are true. It is not the probability that the null hypothesis is true.

For exam purposes, remember that a 95% confidence interval for a mean difference, risk difference or odds ratio gives a range of values compatible with the data under the model. For ratio measures such as odds ratio, risk ratio and hazard ratio, the null value is 1. For difference measures, the null value is 0.

17. Summary classification table for Primary FRCA

18. Viva-level discussion points

• Mean versus median: the mean is efficient for symmetric normal data but sensitive to outliers; the median is robust for skewed data but may be less statistically efficient under normality.
• SD versus IQR: SD describes spread around the mean for approximately normal data; IQR describes the middle 50% and is robust to extremes.
• Ranks: non-parametric rank tests reduce sensitivity to outliers but discard information about magnitude.
• Clinical versus statistical significance: a statistically significant 1 mmHg difference in MAP may be clinically trivial in a huge dataset; a non-significant mortality difference may still be clinically important if the study is underpowered.
• Composite scores: total scores may conceal different patterns. A GCS of 8 may arise from different combinations of eye, verbal and motor responses.
• Repeated measurements: multiple blood pressure readings are not independent; analysing every minute as an independent observation inflates sample size artificially.