Descriptive Univariate Statistics

foundations

descriptive

mean

median

mode

iqr

Location, dispersion, and shape measures for a single variable

Published

April 17, 2026

Research question

Before any inferential test is run, a clear picture of each variable’s distribution is mandatory. Two biomedical scenarios: (1) In a type 2 diabetes registry, what is the typical HbA1c and its variability across 800 patients? (2) In a post-operative recovery study, is the length-of-stay distribution symmetric enough to report a mean, or is it right-skewed enough to require the median?

Assumptions

Descriptive summaries are assumption-light, but their interpretation depends on context:

Assumption	How to verify in R
Data are correctly scaled and typed	`str(df)`; `sapply(df, class)`
Missing values are handled explicitly	`sum(is.na(x))`; `na.omit()` or `na.rm = TRUE`
The chosen summary fits the distribution shape	Histogram (`ggplot2::geom_histogram`) or Q-Q plot

Reporting a mean when the distribution is strongly skewed gives a misleading sense of typicality; reporting an SD when the distribution is bimodal is even worse.

Hypotheses

Descriptive statistics are not tests. They produce point estimates and distributional summaries that feed into inferential questions.

R code

library(tidyverse)
library(rstatix)
library(gtsummary)
library(psych)

set.seed(42)

diabetes_registry <- tibble(
  patient_id  = sprintf("P%04d", 1:800),
  age         = round(rnorm(800, 58, 11)),
  hba1c       = round(rnorm(800, 7.2, 1.1), 1),
  los_days    = rpois(800, lambda = 5) + rexp(800, rate = 0.3),   # right-skewed
  smoker      = factor(sample(c("Never", "Former", "Current"), 800, replace = TRUE,
                              prob = c(0.55, 0.30, 0.15)))
)

# Tidy summary of continuous variables
diabetes_registry |>
  get_summary_stats(
    age, hba1c, los_days,
    type = "common"
  )

# Psychometric summary: mean, median, SD, skew, kurtosis
describe(diabetes_registry[, c("age", "hba1c", "los_days")])

# Publication-ready summary table
diabetes_registry |>
  tbl_summary(
    include   = c(age, hba1c, los_days, smoker),
    statistic = list(
      age        ~ "{mean} ({sd})",
      hba1c      ~ "{mean} ({sd})",
      los_days   ~ "{median} ({p25}-{p75})",
      smoker     ~ "{n} ({p}%)"
    )
  )

# Visualisation of length-of-stay to justify median over mean
diabetes_registry |>
  ggplot(aes(x = los_days)) +
  geom_histogram(bins = 40, fill = "#2A9D8F", colour = "white") +
  geom_vline(aes(xintercept = mean(los_days)),   colour = "#F4A261", linewidth = 1) +
  geom_vline(aes(xintercept = median(los_days)), colour = "#6A4C93", linewidth = 1) +
  labs(x = "Length of stay (days)", y = "Count",
       title = "Right-skewed LOS: mean (orange) exceeds median (purple)") +
  theme_minimal()

Two conventions are displayed: metric variables with approximately symmetric distributions (age, HbA1c) get mean and SD; the right-skewed length-of-stay gets median and IQR.

Interpreting the output

The get_summary_stats() table gives \(n\), mean, SD, median, Q1, Q3, min, and max in one call. describe() adds skewness and kurtosis; a skew of \(|g_1| > 1\) signals that the mean is being pulled by a tail. The histogram visually confirms the LOS skew: the mean sits above the median because the long right tail pulls the arithmetic average upward.

Effect size

Descriptive univariate statistics do not have effect sizes; they are the point estimates. For comparisons between groups, see the t-test and ANOVA pages.

Reporting (APA 7)

Across 800 patients in the registry, age was 58.2 years (SD = 11.0), HbA1c was 7.21 % (SD = 1.10), and post-operative length of stay had a median of 6.8 days (IQR 4.2-10.1). Current smoking status was reported by 119 patients (14.9 %).

Common pitfalls

Reporting a mean for a strongly skewed variable; use the median.
Reporting an SD for a bimodal variable; show the histogram instead.
Computing a mean of an ordinal variable stored as numeric codes; use the median or frequency counts.
Ignoring missing values silently; mean(x) returns NA unless na.rm = TRUE.

Parametric vs. non-parametric alternative

Summary choice depends on distribution shape, not on the parametric / non-parametric classification of downstream tests.