11.2 Descriptive and Inferential Statistics
Key Takeaways
- BPE and IOTN are ordinal scales, so a mean BPE score is not a valid statistic; the median is the appropriate average.
- Standard deviation describes the spread of individual observations while standard error describes the precision of the sample mean and shrinks as sample size grows.
- A p value is the probability of the observed result if the null hypothesis were true, not the probability that the null hypothesis is true.
- A 95% confidence interval for a difference between means indicates significance when it excludes zero, and for a ratio when it excludes one.
- Power is one minus the type II error rate, conventionally set at 80% or 90%, and underpowered studies are the usual reason an effective treatment appears to show no difference.
Describing Data
Types of data determine which statistic and which test may be used.
| Type | Example in dentistry | Appropriate average |
|---|---|---|
| Nominal (categorical) | Tooth type, ASA grade category, gender | Mode |
| Ordinal (ranked) | BPE code, IOTN grade, Miller mobility grade | Median |
| Discrete numerical | DMFT count, number of retained roots | Median or mean |
| Continuous numerical | Probing depth in mm, salivary flow in mL/min, HbA1c | Mean if symmetrical, median if skewed |
A common examination trap is treating an ordinal scale as though it were numerical. A BPE code of 4 is not twice a code of 2, so calculating a mean BPE score is not valid.
Measures of central tendency and spread.
- Mean — the arithmetic average. Sensitive to outliers.
- Median — the middle value. Robust to outliers and the correct choice for skewed data such as probing depths or waiting times.
- Mode — the commonest value. The only average available for nominal data.
- Standard deviation (SD) — describes the spread of individual observations around the mean. In a normal distribution, roughly 68% of observations lie within 1 SD, 95% within 1.96 SD and 99.7% within 3 SD.
- Standard error of the mean (SEM) — describes the precision of the sample mean as an estimate of the population mean, and equals SD divided by the square root of the sample size. SEM always shrinks as the sample grows; SD does not.
Distributions and Skew
A normal (Gaussian) distribution is symmetrical, with mean, median and mode coinciding. Many dental variables are not normal. Numbers of carious lesions in a population, for example, are strongly positively skewed — most people have few and a small group has many — so the mean exceeds the median and the median is the more honest summary. This skew is the statistical face of the well-documented finding that a minority of children carry most of the caries burden.
Inferential Statistics
The purpose of inference is to decide whether an observed difference is likely to reflect a real effect rather than chance.
- The null hypothesis (H₀) states that there is no difference between groups.
- The p value is the probability of observing a result at least as extreme as the one obtained if the null hypothesis were true. It is conventionally judged against 0.05.
Two misinterpretations are routinely tested:
- A p value is not the probability that the null hypothesis is true, and it is not the probability that the result occurred by chance.
- Statistical significance is not clinical significance. With a large enough sample, a 0.1 mm difference in probing depth will reach p < 0.05 while being clinically meaningless.
Type I error (alpha) is rejecting a true null hypothesis — a false positive, conventionally accepted at 5%. Type II error (beta) is failing to reject a false null hypothesis — a false negative. Power is 1 − beta, conventionally set at 80% or 90%, and is the probability of detecting a real effect of a specified size. Underpowered studies are the commonest reason a genuinely effective intervention appears to "show no difference".
Confidence Intervals
A 95% confidence interval gives the range within which the true population value is expected to lie with 95% confidence. It is more informative than a p value because it shows both significance and precision.
- For a difference between means, the result is significant if the interval excludes zero.
- For a ratio such as a relative risk or odds ratio, the result is significant if the interval excludes one.
A wide interval signals imprecision from a small sample even when the point estimate looks impressive.
Choosing a Test
| Comparison | Parametric (normal data) | Non-parametric (skewed or ordinal) |
|---|---|---|
| Two independent groups | Unpaired (independent) t-test | Mann-Whitney U test |
| Two paired measurements | Paired t-test | Wilcoxon signed-rank test |
| Three or more groups | One-way ANOVA | Kruskal-Wallis test |
| Two categorical variables | — | Chi-squared test (Fisher's exact if expected counts are small) |
| Association between two continuous variables | Pearson correlation | Spearman rank correlation |
Exam link. A trial reports that a new desensitising agent reduced mean visual analogue pain score by 4 mm, 95% CI −1 mm to 9 mm, p = 0.11. The confidence interval includes zero, so the result is not statistically significant, and the width of the interval tells you the study was too small to settle the question either way.
Reading a Result Correctly
The commonest statistical error in clinical reasoning, and the one most often examined, is treating statistical significance as clinical importance. A very large study can produce a p value below 0.05 for a difference of a tenth of a millimetre in probing depth, which no patient would notice. Conversely, a small study can fail to reach significance despite a clinically meaningful effect, simply because it was underpowered. This is why confidence intervals are more informative than p values: they show both the estimated effect and the range of effects compatible with the data, so a wide interval crossing the null signals uncertainty rather than proof of no effect.
A type I error is concluding there is a difference when there is none, and its probability is set by the significance level alpha. A type II error is missing a real difference, and its probability is beta; power is one minus beta, conventionally set at 80 or 90 per cent. Power increases with sample size, with the size of the effect being sought and with reduced variability in the data — the three levers a sample size calculation uses.
A randomised trial of a new fluoride varnish regimen reports a mean reduction of 0.30 decayed surfaces per child compared with standard care, with a 95% confidence interval of -0.05 to 0.65 and p = 0.09. Which interpretation is correct?