Describing a dataset
Descriptive statistics summarise the observations available: their centre, spread and distribution. The choice of summary determines how faithfully those observations are represented before conclusions are drawn about a population or treatment.
Description and the observations being summarised
Descriptive statistics describe what was observed in a sample; claims of superiority or non-inferiority go beyond description into inference. A numerical distribution needs both a representative centre and a measure of variability, also called dispersion, spread or heterogeneity.
Population and sample quantities use different notation: μ and σ denote population mean and standard deviation, while x̄ and s denote the corresponding sample summaries. For example, 150 simulated observations from a normal population with μ = €2,500 and σ = €500 produced a sample mean of €2,446 and sample SD of €480.
Mean, median and mode
The arithmetic mean uses all observations: their sum divided by their number. An extreme value can therefore shift the mean substantially, and the mean need not coincide with any observed value. The arithmetic mean also minimises the sum of squared distances from the observations to a proposed centre.
The median is the middle ordered observation; for an even number of observations it is the arithmetic mean of the middle two. Thus 5, 8, 11 and 14 have median 9.5, whereas 5, 8, 9, 11 and 14 have median 9. The median is less affected by extreme values and is useful for skewed data, together with a range or interquartile range (IQR).
The mode is the most frequent value or category. It can describe categorical data such as blood group or disease stage; the number of possible categories should be supplied when not obvious. A categorical mode need not be accompanied by a numerical dispersion measure. Several equally frequent values produce several modes, and a mode is unhelpful when every value occurs once.
Range and interquartile range
The minimum and maximum give the observed range, while their difference gives its width. For heart rates of 55, 60, 45, 62, 63, 110 and 56 beats/min, the minimum and maximum were 45 and 110, the width was 65 and the mean about 64 beats/min. Because the endpoints alone determine this range, one extreme observation can dominate its impression of variability.
Quartiles divide ordered data into four quarters, with the median at the 50th percentile; the interquartile range describes the middle 50%, from the 25th to the 75th percentile. The IQR is less influenced by extreme observations than the overall range.
A worked dataset of 16 heart rates contained 45, 50, 54, 56, 60, 62, 64, 65, 68, 68, 70, 74, 80, 84, 90 and 110 beats/min. Its median was 66.5 beats/min and its central eight observations lay between 60 and 74 beats/min, leaving the extreme 45 and 110 outside that central group.
Variance and standard deviation
Variance is the square of standard deviation. Sample variance is the sum of squared deviations from the sample mean divided by n − 1; taking its square root gives the sample SD. Variance has squared measurement units, while SD has the original units. Unlike a range based on endpoints, SD uses every observation and increases as values become more dispersed around their mean.
A ten-person body-mass example gives the calculation:
| Subject | Mass, kg | Deviation from mean, kg | Squared deviation, kg² |
|---|---|---|---|
| 1 | 88 | 12.1 | 146.41 |
| 2 | 72 | −3.9 | 15.21 |
| 3 | 110 | 34.1 | 1,162.81 |
| 4 | 65 | −10.9 | 118.81 |
| 5 | 56 | −19.9 | 396.01 |
| 6 | 71 | −4.9 | 24.01 |
| 7 | 81 | 5.1 | 26.01 |
| 8 | 52 | −23.9 | 571.21 |
| 9 | 95 | 19.1 | 364.81 |
| 10 | 69 | −6.9 | 47.61 |
| Total | 759 | 0 | 2,872.90 |
| Mean mass and sample variance | 75.9 | 0 | 319.21 |
| Sample SD | 17.87 | — | — |
Here the mean is 759/10 = 75.9 kg, the sample variance is 2,872.90/9 = 319.21 kg² and the sample SD is √319.21 = 17.87 kg. The coefficient of variation expresses SD relative to the mean as (SD/mean) × 100, giving 23.5% in this example.
A worked formulation comparison measured trough concentrations in healthy volunteers after a single 750 mg dose, to select a formulation for phase III:
| Parameter | Tablet 1 | Tablet 2 | Capsule A | Capsule B |
|---|---|---|---|---|
| Mean, mg/L | 152.5 | 163.5 | 109.2 | 97.7 |
| SD, mg/L | 18.0 | 47.0 | 37.1 | 7.0 |
| CV, % | 11.8 | 28.7 | 34.0 | 7.2 |
Tablet 2 had the highest mean concentration, while tablet 1 and capsule B had lower relative variability; choosing capsule B for consistency was therefore a different decision from choosing the highest mean.
Distribution and choice of summary
A normal distribution is symmetric and bell-shaped, with approximately 68.27%, 95.45% and 99.73% of its probability within one, two and three SDs of its mean. A central 95% normal reference interval therefore leaves about 2.5% in each tail.
Skew describes asymmetry: a long right tail tends to place the mean above the median, and a long left tail reverses that relation. Kurtosis describes the influence of extreme observations in the tails; normal-distribution kurtosis is 3, with heavier-tailed distributions above and thinner-tailed distributions below that reference. A bimodal distribution can represent subpopulations, such as slow and rapid metabolisers associated with a drug-metabolising enzyme polymorphism.
Plots should be examined before choosing summaries, and normality can also be assessed with tests such as Shapiro–Wilk or Kolmogorov–Smirnov. Clinical inclusion and exclusion cutoffs can alter a sample's distribution even when the source population is normal. The Statistical Analyses and Methods in the Published Literature (SAMPL) guidelines recommend mean (SD) for approximately normal data and median with interpercentile limits, minimum and maximum, or both for non-normal data. Standard error should not be used as a descriptive measure of observation variability; SD, interpercentile limits or range serve that purpose.
The geometric mean is the nth root of the product of n values. The type of mean should be identified clearly: a statistician cautions that readers may interpret an unspecified mean as an arithmetic mean and that the choice requires consideration for each publication.
What a group summary can conceal
A change in a group mean does not state how many participants improved or worsened. In a three-person illustration, the mean fell from 11.6 to 10, about 14%, even though two participants' values increased, because one person's large decrease dominated the average. Similarly, a reported 38% lower mean depression score could reflect broad improvement or a few unusually strong responders.
Percentages require their numerators and denominators: the same 50% reduction can mean counts falling from 2 to 1 or from 2,000 to 1,000. Sample and group sizes should be provided for each analysis, and ranges should give their lower and upper boundaries rather than only their widths.