Descriptive statistics summarise the data collected: their type, centre, spread and shape
Inferential statistics use a sample to draw conclusions about the population it came from (see Hypothesis Testing)

## Types of data

Type	Description	Example
Categorical (qualitative)
Nominal (binary if two categories)	Named categories with no order	Sex, eye colour, presence of diabetic retinopathy
Ordinal	Ordered categories with unequal or unknown gaps	Retinopathy grade (none, mild, moderate, severe, proliferative); anterior chamber cells 0 to 4+
Numerical (quantitative)
Discrete	Whole-number counts	Number of intravitreal injections
Continuous	Any value in a range. Interval data have no true zero; ratio data have one	IOP, axial length, central corneal thickness; temperature in °C is interval

Snellen acuity fractions are not evenly spaced. For analysis, acuity is converted to logMAR, which can be treated as continuous: each line is 0.1 logMAR and each letter 0.02

## Measures of central tendency

Mean: the sum of the values ÷ their number. Uses every value but is pulled by outliers and skew
Median: the middle value when ranked. Not affected by outliers; best for skewed data
Mode: the most frequent value. The only measure for nominal data

## Measures of spread

Range: lowest to highest. Sensitive to outliers
Interquartile range (IQR): the 25th to the 75th centile, the middle 50% of the data. Reported with the median
Variance: the average squared distance from the mean. Standard deviation (SD) = √variance, in the original units. Reported with the mean

## Distributions

The normal (Gaussian) distribution is symmetrical and bell-shaped: mean = median = mode

About 68% of values lie within 1 SD of the mean, 95% within 1.96 SD (often rounded to 2) and 99.7% within 3 SD
A reference (normal) range is usually the mean ± 1.96 SD, so 5% of healthy people fall outside it by definition
Worked example: population IOP has a mean of 15.5 mmHg and an SD of 2.6 mmHg. The 95% range is 15.5 ± (1.96 × 2.6) = 10.4 to 20.6 mmHg. This is the origin of the traditional upper limit of 21 mmHg. In reality, IOP is slightly positively skewed

Positive (right) skew: a long tail to the right; mean > median > mode. Examples: IOP, income, length of hospital stay
Negative (left) skew: a long tail to the left; mean < median < mode. Example: age at death in a high-income country
Bimodal: two peaks, suggesting two populations (e.g. age at onset of Hodgkin lymphoma)
Other distributions: binomial (number of successes in a fixed number of yes/no trials) and Poisson (counts of rare events in a fixed time or space). Skewed data can often be made normal by a log transformation
Normality can be judged by a histogram, a normal plot, or tests such as Shapiro–Wilk. It decides whether parametric tests can be used (see Hypothesis Testing)

## Standard deviation and standard error

The SD describes the spread of the individual values in the sample. It does not shrink as the sample gets bigger
The standard error of the mean (SE) describes the precision of the sample mean as an estimate of the population mean

SE = SD ÷ √n. It gets smaller as the sample grows: four times the sample halves the SE

Worked example: central corneal thickness in 100 eyes has a mean of 540 µm and an SD of 30 µm. SE = 30 ÷ √100 = 3 µm. 95% CI = 540 ± (1.96 × 3) = 534.1 to 545.9 µm
Interpretation: if the study were repeated many times, 95% of such intervals would contain the true population mean. It is a range of plausible values for the true mean, not for individual patients

Worked example: a survey of 3000 people finds a blindness prevalence of 2%. SE = √(0.02 × 0.98 ÷ 3000) = 0.0026. 95% CI = 2% ± (1.96 × 0.26%) = 1.5% to 2.5%
Cluster sampling (as in population eye surveys) widens the CI further, by the design effect

Use the SD to describe variability in the data; use the SE and confidence interval to describe how precisely the mean is known
Mean ± 1.96 SD: the range containing 95% of individuals. Mean ± 1.96 SE: the 95% confidence interval for the mean
Skewed data: report the median and IQR, not the mean and SD

## Presenting data

Bar chart: categorical data. Histogram: the distribution of continuous data (the bars touch, and the area shows frequency)
Box-and-whisker plot: the median, IQR (the box), range or a set limit (the whiskers) and outliers
Scatter plot: the relationship between two continuous variables (see Hypothesis Testing for correlation)
Kaplan–Meier curve: the proportion free of an event over time

Central corneal thickness, axial length and keratometry are roughly normally distributed in adults. Refractive error is not: it clusters near emmetropia with a long myopic tail, which makes it leptokurtic (more peaked than normal)