Descriptive Statistics

Descriptive statistics are statistical methods used to summarize, organize, and describe the main features of a dataset without making inferences about the broader population from which the sample was drawn. They include measures of central tendency (mean, median, mode), measures of dispersion (range, standard deviation, variance, interquartile range), measures of position (percentiles, quartiles), and measures of distribution shape (skewness, kurtosis). Descriptive statistics provide the foundation for understanding any dataset before more complex analyses are applied.

Most quantitative research papers report descriptive statistics early in their results, typically in "Table 1," which characterizes the study sample. Reviewers and readers use these summaries to assess whether the sample is representative, whether groups are balanced at baseline, and whether the data are suitable for the planned analyses. Descriptive statistics also inform the choice of inferential statistical tests: if data are normally distributed, parametric tests may be appropriate; if data are skewed, nonparametric alternatives may be needed. Misunderstanding or misreporting descriptive statistics can lead to inappropriate analytical choices, misleading conclusions, and flawed evidence synthesis.

Measures of central tendency describe the typical value in a dataset. The mean (arithmetic average) is most useful for normally distributed data, the median (middle value) is preferred for skewed data or when outliers are present, and the mode (most frequent value) is used for categorical data or when identifying the most common response is important.

Measures of dispersion describe how spread out the values are. The range (difference between the highest and lowest values) is simple but sensitive to outliers. The standard deviation, the square root of the average squared deviation from the mean (for a sample, the sum of squared deviations is divided by the sample size minus one), describes how far observations typically lie from the mean, in the same units as the original data. The variance is the squared standard deviation and is used in many statistical formulas. The interquartile range (IQR), the difference between the 25th and 75th percentiles, is more resistant to outliers than the range and is the preferred dispersion measure for skewed data.

Frequency distributions show how often each value or category occurs, either as raw counts or as proportions. They can be displayed as tables, histograms, or density plots. The shape of a distribution, whether it is symmetric, positively skewed (tail extending to the right), negatively skewed (tail extending to the left), or multimodal (having multiple peaks), provides important information about the data and influences the choice of subsequent analyses.

In research reports, descriptive statistics are typically presented in tables organized by group (such as treatment and control) and by variable type (continuous variables summarized with mean and SD or median and IQR; categorical variables summarized with counts and percentages). The CONSORT statement for clinical trials and the STROBE statement for observational studies provide specific guidance on how descriptive statistics should be reported.

Descriptive statistics should be used in quantitative studies to characterize the sample, examine data distributions, identify outliers and missing patterns, and assess assumptions for subsequent analyses. They are the appropriate primary analysis for purely descriptive studies (surveys, surveillance data) and the necessary first step before any inferential analysis. They are not sufficient when the goal is to draw conclusions about population parameters, test hypotheses, or quantify uncertainty, which require inferential statistics such as confidence intervals and hypothesis tests.

A frequent error is reporting the mean and standard deviation for skewed data, where the median and interquartile range would be more informative and less misleading. Reporting standard error of the mean instead of standard deviation when the goal is to describe variability (rather than precision of the mean estimate) misrepresents the spread of the data. Using percentages without reporting the denominator makes it impossible for readers to assess whether the percentage is based on a large or small sample. Failing to report both a measure of central tendency and a measure of dispersion provides an incomplete picture; a mean without a standard deviation (or a median without an IQR) tells the reader where the center is but nothing about how much variability exists. Finally, performing statistical tests (such as t tests) to compare baseline characteristics between randomized groups is inappropriate: when randomization has been carried out properly, any differences are due to chance by design, so these p values test a null hypothesis already known to be true, and what matters is whether an imbalance is large enough to affect the results.

A clinical trial of an exercise intervention for knee osteoarthritis reports baseline characteristics for 200 participants. Continuous variables with normal distributions (age, BMI) are summarized as mean (SD): age 63.4 (8.2) years, BMI 29.7 (5.1) kg/m². A skewed variable (symptom duration) is summarized as median (IQR): 4.2 (2.1 to 7.8) years. Categorical variables are summarized as n (%): female 128 (64.0%), Kellgren Lawrence grade III 86 (43.0%), grade IV 114 (57.0%). The table is organized by intervention and control group, with no p values for baseline comparisons between randomized groups.