Measures of Dispersion: Standard Deviation, SEM, and IQR
Why central tendency without dispersion is incomplete: applying the 68–95–99.7% rule, separating SD from SEM, and pairing Median with IQR.
In this lesson you will learn
- 1Apply the Empirical Rule (68.3%, 95.4%, 99.7%) to interpret Standard Deviation under a normal curve.
- 2Distinguish Standard Deviation (biological spread among patients) from Standard Error of the Mean (precision of the sample mean).
- 3Enforce proper summary pairing in manuscript tables: Mean with SD versus Median with IQR.
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Pairing Rules & The SD vs SEM Distinction
This section explains how Standard Deviation, Standard Error, Interquartile Range, and Coefficient of Variation each answer a distinct clinical question.
Substituting SEM for SD in baseline Table 1 to make group spread look artificially tight is a classic reporting flaw flagged by medical journal editors.
Reporting a measure of central tendency without a companion measure of dispersion hides how widely individual patients vary around the centre. Two strict pairing conventions govern biomedical tables:
- Parametric Pair —
Mean (SD): Used for symmetric, normally distributed continuous variables. Under the empirical rule,Mean ± 2 SDencloses roughly 95% of individual patient values. - Non-Parametric Pair —
Median [IQR]orMedian [Q1, Q3]: Used for skewed continuous or ordinal variables. Captures the 50th percentile alongside the 25th and 75th percentiles. - SD versus SEM:
SDdescribes individual biological spread and does not shrink as sample size grows.SEM = SD / √Nmeasures how precisely the sample mean estimates the population mean, and is used to construct confidence intervals (Mean ± 1.96 × SEM).
Dispersion Lab: SD vs SEM & The 68–95–99.7% Rule
Adjust Mean, SD, and Sample Size (N) to separate individual patient spread from mean precision
Clinical Example
A clinical laboratory validates two glucose analysers on the same control serum (Mean = 100 mg/dL) and compares dispersion against body weight (Mean = 75 kg, SD = 15 kg).
Interpretation: Because mg/dL and kg are different physical units, raw SDs cannot be compared directly. Dividing SD by the Mean gives the unitless Coefficient of Variation (CV%), showing that body weight varies 4 times more relatively than glucose.
Common Pitfall
Cross-pairing a Median with a Standard Deviation (e.g. 'Median = 110, SD = 24') in manuscript tables.
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