What is the significance of the interquartile range in descriptive statistics?

The Interquartile Range (IQR) is a non-parametric measure of statistical dispersion, quantifying the spread of the central 50% of a dataset. Defined as IQR=Q3−Q1, where Q1 and Q3 denote the first and third quartiles respectively, it isolates the middle half of the data, effectively filtering out the influence of extreme values and skewed distributions. Unlike variance or standard deviation, the IQR is robust to outliers and does not assume a normal distribution, making it an essential tool in exploratory data analysis (EDA) for understanding data variability.

É comum perder algum detalhe ao tentar resumir um conjunto de dados a um único número com as medidas-resumos, como média, mediana e moda. Dito isso, o IQR tem aplicação prática no nosso dia a dia, sempre que desejamos expressar uma medida de centralidade resistente a valores extremos como a mediana, é importante atrelar o IQR junto para tornar claro qual a variabilidade dos dados que compõe a mediana.

O intervalo interquartil (IQR) é a diferença entre o 3º quartil (Q3) e o 1º quartil (Q1), representando os 50% centrais dos dados. Ele é uma medida de dispersão robusta, pouco sensível a outliers, sendo muito usado para: ● Identificar outliers: Valores fora do intervalo [Q1−1,5×IQR,Q3+1,5×IQR] são considerados potenciais outliers; ● Analisar variabilidade: Mostra a dispersão dos dados em torno da mediana; ● Visualização gráfica: Forma a "caixa" do boxplot, facilitando a leitura da distribuição e dos valores extremos. O IQR é útil principalmente quando há distribuições assimétricas ou dados com outliers, sendo uma alternativa mais confiável que o desvio padrão nesses casos.

The calculation of IQR involves the use of order statistics. A dataset is first arranged in non-decreasing order, and the quartiles are determined based on the position of the data points. The first quartile (Q1) represents the 25th percentile, indicating the value below which 25% of the data lies, while the third quartile (Q3) corresponds to the 75th percentile. The IQR is computed as the difference between these two quartiles. For continuous data distributions, interpolation methods may be used to estimate precise quartile values, ensuring accuracy even in large, unsorted datasets.

Eu gosto de usar o gráfico de BoxPlot, pois faz a divisão de forma bem visual as distribuições dos quartis, mediana, média e outlier. Para ter uma visão mais detalhada e resumida dos dados é interessante analisar dois gráficos: Histograma e BoxPlot.

Para uma análise mais simples e visual o uso do Gráfico de Box Plot é ótimo! Ele permite identificar com facilidade os outliers, se as diferenças de valores centrais são significativamente distintas, e o comportamento da variância em diferentes grupos

The IQR is foundational in outlier detection through the use of Tukey’s fences. Thresholds for outlier identification are established using: Lower Bound: Q1 − k × IQR Upper Bound: Q3 + k × IQR where k is typically set to 1.5 for standard outliers and 3 for extreme outliers. This approach provides a non-parametric mechanism to detect data points that deviate significantly from the interquartile spread. It assumes no underlying distribution, making it particularly effective in heterogeneous datasets and resistant to the masking effect where extreme outliers distort other metrics like standard deviation.

The IQR is pivotal in comparative data analysis across multiple distributions. It serves as a scale-invariant measure, allowing analysts to assess relative dispersion without being influenced by outliers or skewness. A larger IQR denotes greater heterogeneity and variability, whereas a smaller IQR indicates data concentration around the median. When used alongside measures of central tendency (mean, median) and skewness coefficients, the IQR enables multidimensional comparisons, highlighting subtle differences in data spread across groups.

In boxplot visualizations, the IQR defines the central “box” structure, encapsulating the interquartile spread from Q1 to Q3. The median is typically represented as a line within the box, providing insight into data symmetry and skewness. Whiskers extend from the quartiles to the furthest data points within the acceptable range, while points beyond Tukey’s fences are plotted as individual outliers. Boxplots effectively leverage the IQR to visualize data distribution, central tendency, spread, and potential anomalies in a compact graphical format.