Summarising Numerical Data

Overview

Numerical data record counts or measurements for which arithmetic has scientific meaning. Examples include age, blood pressure, serum concentration, number of admissions and duration of hospital stay.

A useful numerical summary usually describes at least two features:

• centre, indicating a typical or central value; and
• spread, indicating how much observations vary.

The mean and standard deviation form one pair. The median and interquartile range form another. These pairs are not interchangeable labels for the same information. Each describes the data differently.

The correct summary depends on the clinical question, measurement scale, distribution, unusual observations and intended interpretation—not solely on a formal test of normality.

Why this matters

Suppose seven hospital stays are:

2, 3, 3, 4, 5, 6, 30 days

The mean is influenced strongly by the 30-day stay. The median describes the central patient more closely, but the mean remains relevant to total bed use because total resource consumption depends on every day, including long stays.

Neither summary is universally superior. The question determines which quantity matters.

Numerical summaries should also retain units. A standard deviation of 12 is incomplete; 12 mmHg describes variation in systolic blood pressure.

Measures of centre

Arithmetic mean

The arithmetic mean is the sum of observed values divided by the number of values.

The mean uses every value. It is affected by extreme observations and skewed tails.

The mean is useful when:

• totals or balances matter;
• differences from the mean are meaningful;
• the analysis targets a population mean; or
• the distribution and influence of unusual observations are acceptable for the intended interpretation.

Median

The median is the middle value after ordering observations. It is the 50th percentile.

For an odd number of observations, it is the central ranked value. For an even number, it is usually the mean of the two central ranked values.

The median is resistant to extreme values because it depends mainly on order rather than distance.

It answers a different question from the mean:

• mean: what value would result from distributing the total equally?
• median: what value divides observations into lower and upper halves?

Mode

The mode is the most frequently occurring value.

It is useful for categories and some discrete numerical variables. In continuous measurements with many distinct values, the mode may depend heavily on rounding or grouping and is often not the most informative summary.

A distribution may have:

• one mode;
• several modes; or
• no repeated observed value.

Measures of spread

Minimum, maximum and range

The minimum is the smallest observed value. The maximum is the largest.

The range is:

$$\text{maximum}-\text{minimum}$$

It is easy to interpret but depends on only two observations and tends to increase with sample size.

Minimum and maximum values are useful for data-quality review but should not replace a more stable measure of spread.

Quartiles and interquartile range

The first quartile, $Q_1$, is the 25th percentile. The third quartile, $Q_3$, is the 75th percentile.

The interquartile range, abbreviated IQR, is:

The IQR spans the middle 50% of ordered observations.

Medical reports commonly present:

median (Q1–Q3)

For example:

136 mmHg (126–149)

Some reports call 126–149 the IQR, although strictly it displays the quartile limits, while the numerical IQR width is 149 - 126 = 23 mmHg. The table heading should make the convention clear.

Quartile algorithms differ slightly across software, especially in small samples [1]. For large datasets, differences are usually small. Reproducible analysis should use one documented method.

Variance

Variance summarises squared deviations from the mean.

The denominator $n-1$ is used for the conventional unbiased estimator of population variance under random sampling assumptions.

Variance has squared units. If blood pressure is measured in mmHg, variance is measured in mmHg², which is less clinically intuitive.

Standard deviation

The standard deviation, abbreviated SD, is the square root of the variance.

Standard deviation has the same unit as the original measurement.

It describes variation among observed values around the mean. It does not measure the precision of the sample mean.

Standard error

The standard error, abbreviated SE, describes sampling variability of an estimator.

For a sample mean under simple independent sampling:

$$SE(\bar{x})=\frac{s}{\sqrt{n}}$$

Standard deviation and standard error answer different questions [2]:

• SD: how variable are participant values?
• SE: how variable would the estimated mean be across repeated samples under the sampling model?

A baseline descriptive table should usually show SD, not SE.

Percentiles and five-number summary

A percentile is a value below which a specified percentage of observations lies.

Examples include:

• 5th percentile;
• 25th percentile;
• median or 50th percentile;
• 75th percentile; and
• 95th percentile.

The five-number summary consists of:

1. minimum;
2. first quartile;
3. median;
4. third quartile; and
5. maximum.

It provides a compact description of order and spread but can still conceal multimodality or gaps. A graph should accompany it when distribution shape matters.

Data structure

Simulated systolic blood-pressure summaries

simulated data

Table 1. Systolic blood pressure among participants with observed measurements

Summary	Value
Number observed	480
Missing	20
Mean	139.2 mmHg
Standard deviation	18.4 mmHg
Median	136 mmHg
First quartile	126 mmHg
Third quartile	149 mmHg
Minimum	92 mmHg
Maximum	238 mmHg

Note: The quartile limits are shown separately; the IQR width is 23 mmHg.

Table 2. Examples of numerical variable summaries

Variable	Suitable descriptive presentation
Age, approximately symmetric	Mean (SD), years
Systolic blood pressure with several high values	Mean (SD) and median (Q1–Q3), with a distribution plot
Number of medicines, small count scale	Median (Q1–Q3) and frequency distribution
Time since diagnosis, right-skewed	Median (Q1–Q3), years
LDL cholesterol	Summary chosen after examining scale, distribution and clinical purpose

Note: LDL = low-density lipoprotein; SD = standard deviation.

Worked example

Choosing a descriptive summary

Common pairings are:

• mean with standard deviation;
• median with quartile limits or IQR;
• geometric mean with a multiplicative spread measure for positive log-scaled data;
• percentages across count categories when the count has few possible values.

The next two lectures develop distributions and the choice between mean–SD and median–IQR in greater depth.

Geometric mean for multiplicative data MMed extension

For positive values, the geometric mean is the exponentiated mean of logarithms.

It can be useful when:

• changes are naturally multiplicative;
• ratios are more meaningful than differences;
• the logarithms are approximately symmetric; or
• laboratory concentrations span several orders of magnitude.

A geometric mean is not appropriate when values are zero or negative without a justified alternative method.

If analysis is performed on a log scale, estimates and confidence intervals should be interpreted correctly after transformation [3].

Suggested reporting

Common mistakes

Knowledge check

What is the difference between the mean and median?

Answer and reasoning

The mean uses the magnitude of every value and is the equal-share average. The median is the middle ranked value. Extreme observations usually affect the mean more strongly.

What does a standard deviation of 18.4 mmHg describe?

Answer and reasoning

It describes variation of observed systolic blood-pressure values around their mean, in the same unit as the measurement. It is not the standard error of the estimated mean.

Why should the range not be the only measure of spread?

Answer and reasoning

It depends only on the minimum and maximum, is sensitive to extreme values and tends to increase with sample size. It ignores how the remaining observations are distributed.

What does the interquartile range describe?

Answer and reasoning

It describes the span of the middle 50% of ordered observations, from the first to the third quartile. It is relatively resistant to extreme values.

Can a mean remain clinically useful for a skewed variable?

Answer and reasoning

Yes. If the research question concerns totals, resource use or the population arithmetic mean, the mean may be the relevant estimand. Skewness affects presentation and analysis but does not automatically make the mean meaningless.

Key take-home messages

References

1. Altman DG, Bland JM. Quartiles, quintiles, centiles, and other quantiles. BMJ. 1994;309(6960):996. https://doi.org/10.1136/bmj.309.6960.996
2. Altman DG, Bland JM. Standard deviations and standard errors. BMJ. 2005;331(7521):903. https://doi.org/10.1136/bmj.331.7521.903
3. Bland JM, Altman DG. Transformations, means, and confidence intervals. BMJ. 1996;312(7038):1079. https://doi.org/10.1136/bmj.312.7038.1079