Semi-Interquartile Range

The semi-interquartile range (semi-IQR) is a numerical measure of the spread of a quantitative data distribution. Quantitative data, as opposed to categorical variables, comprise numerical values for which arithmetic operations (e.g., adding, dividing) can be reasonably conducted. Data analysis begins by examining data with graphics and numerical summaries. A quantitative data distribution is typically summarized by its center (e.g., mean or median) and its spread (e.g., standard deviation, IQR, semi-IQR). The center quantifies the average or middle value of the data distribution, and the spread communicates the variability or range in the data.

Numerical summaries of a data distribution are derived from two types of underlying distributions: parametric and nonparametric. Parametric distributions are probability distributions with known functional forms that accompany a fixed set of parameters (e.g., the uniform distribution of a variable $X$, $f\left(x\right)=\frac{1}{b-a}$ for $a\le x\le b$, has two parameters: $a$ and $b).$ In contrast, nonparametric distributions do not follow any known function (i.e., $f\left(x\right)$ is unknown). The center and spread of data from parametric distributions are quantified by the mean and standard deviation (e.g., for the uniform distribution, the mean is $\frac{a+b}{2}$ and the variance is $\frac{{\left(b-a\right)}^2}{12}$). Alternatively, the center and spread of data from nonparametric distributions are typically quantified by the median, IQR, and semi-IQR. Unlike parametric measures, nonparametric measures of center and spread make no assumptions of the form of the distribution underlying the data.

Parametric Numerical Summaries

If the data are assumed to follow the parametric normal distribution, the center is estimated by the average value or mean, $\overline{x}=85.4$ (solid vertical line in Figure 1A), and the spread is quantified by the standard deviation, $\alpha$. The values of $Y=\text{X}\widehat{\beta }+e$ and $s_x$ are sample estimates of the population parameters of the normal distribution, where μ is the population mean and σ is the population standard deviation. If the data are normally distributed, then 68% of the data will lie within $\overline{x}\pm s_x$ or [76.0, 94.9], 95% of the data will lie within $\overline{x}\pm 2s_x$ or [66.6, 104.3], and 99.7% of the data will lie within $\overline{x}\pm 3s_x$ or [57.1, 113.7] (Figure 1A). The observed range is calculated by taking the difference between the minimum and maximum score, $98-67=31.$ Observe that the minimum and maximum values of the intervals bounded by $\overline{x}\pm 2s_x$and $\overline{x}\pm 3s_x$ exceeds the observed values in the data, [67, 98], [Page 1482]suggesting that the mean and standard deviation may not be the best numerical summaries for these data. Numerical summaries based on parametric distributions are also sensitive to observations that are inconsistent with the assumed distribution. Stated differently, $\overline{x}$ and $s_x$ are not robust to the presence of outliers.

Figure 1 Histogram of Math Achievement Scores

Note: (A) Assuming a normal distribution, the mean is represented by the solid vertical line, and a standard deviation is represented by the width between two vertical reference lines. (B) Assuming a nonparametric distribution, the median is represented by the solid vertical line, and the other quartiles are represented by the dotted vertical lines. The IQR is represented by the longer horizontal line, spanning 50% of the distribution or the length of the box-and-whisker plot. The semi-IQR is represented by the shorter horizontal line, spanning 25% of the distribution or a half-width of the box-and-whisker plot.

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