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Math Details¶

This page gives the exact formulas Quantum XL uses to build a box plot. Each equation lists what it computes and where it appears on the chart.

Notation¶

Term	Description
\(x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}\)	the data values sorted in ascending order
\(x_i,\ f_i\)	an unsorted value and its frequency weight, when a frequency column is supplied
\(n\)	the number of data values (unweighted case)
\(N\)	the effective sample size: \(N = n\) without frequencies, or \(N = \operatorname{round}\!\left(\sum_i f_i\right)\) with a frequency column
\(p\)	a probability in \([0,1]\) (for example \(p = 0.25\) for the first quartile)
\(Q(p)\)	the sample quantile at probability \(p\) (the Hyndman-Fan Type 8 quantile; see Shared Math Details)

A box plot is produced only when \(N \ge 4\); otherwise the box is skipped as having insufficient data.

Quantile \(Q(p)\)¶

Every positional value on the chart, including the quartiles, median, custom-percentile box edges, and the minimum and maximum endpoints, comes from one quantile estimator: the Hyndman-Fan Type 8 definition. It interpolates between the two order statistics nearest a fractional rank, is approximately median-unbiased, and returns the extremes exactly, with \(Q(0) = x_{(1)} = \min\) and \(Q(1) = x_{(n)} = \max\). When a frequency column is supplied, the same estimator is applied to a virtual weighted sample of size \(N = \sum_i f_i\).

For the exact position formula, the interpolation, and the frequency-weighted form, see Type 8 (Hyndman-Fan) Quantiles.

Median, quartiles, and range¶

Quantity	Formula	Used by
First quartile	\(Q_1 = Q(0.25)\)	lower box edge (default), fences
Median	\(\text{med} = Q(0.5)\)	center line drawn across the box
Third quartile	\(Q_3 = Q(0.75)\)	upper box edge (default), fences
Minimum	\(Q(0) = x_{(1)}\)	red X marker (when Display Dataset Minimum is on)
Maximum	\(Q(1) = x_{(n)}\)	red X marker (when Display Dataset Maximum is on)

Box edges¶

The box edges depend on the Box Height option.

Box Height option	Lower edge	Upper edge
Interquartile Range (default)	\(Q(0.25)\)	\(Q(0.75)\)
Custom Percentile \(P\)	\(Q\!\left(\dfrac{P}{100}\right)\)	\(Q\!\left(1 - \dfrac{P}{100}\right)\)

\(P\) is the integer percentile entered in the dialog (5 to 45). For example \(P = 10\) gives a box from the 10th to the 90th percentile.

Interquartile range¶

\[ \text{IQR} = Q_3 - Q_1 = Q(0.75) - Q(0.25) \]

The IQR is always computed from the true quartiles, even when Box Height is set to Custom Percentile, so the fences below do not move with the box height.

Fences¶

Fences are the thresholds used to place whiskers and to classify outliers.

\[ \text{inner fences} = \left[\, Q_1 - 1.5\,\text{IQR},\ \ Q_3 + 1.5\,\text{IQR} \,\right] \]

\[ \text{outer fences} = \left[\, Q_1 - 3\,\text{IQR},\ \ Q_3 + 3\,\text{IQR} \,\right] \]

The fences are not drawn directly; they feed the whisker and outlier rules.

Whiskers¶

The whiskers depend on the Whiskers option.

Lowest/Highest Datum Within Inner Fence (default). The whiskers reach the most extreme actual values that lie within the inner fences:

\[ \text{lower whisker} = \min\{\, x_i : x_i \ge Q_1 - 1.5\,\text{IQR} \,\} \]

\[ \text{upper whisker} = \max\{\, x_i : x_i \le Q_3 + 1.5\,\text{IQR} \,\} \]

The whiskers are then clamped so they never fall inside the box: if the lower whisker exceeds the lower box edge it is set to the lower box edge, and if the upper whisker is below the upper box edge it is set to the upper box edge. If no value satisfies a condition, that whisker falls back to the corresponding box edge.

Minimum/Maximum Data Point. The whiskers reach the full data range, and no outliers are identified:

\[ \text{lower whisker} = x_{(1)} = \min, \qquad \text{upper whisker} = x_{(n)} = \max \]

Outlier classification¶

Outliers are identified only under the default Lowest/Highest Datum Within Inner Fence whisker option. A value \(x\) is classified by which fences it falls beyond:

\[ \text{suspected outlier:}\quad Q_1 - 3\,\text{IQR} \le x < Q_1 - 1.5\,\text{IQR} \quad\text{or}\quad Q_3 + 1.5\,\text{IQR} < x \le Q_3 + 3\,\text{IQR} \]

\[ \text{outlier:}\quad x < Q_1 - 3\,\text{IQR} \quad\text{or}\quad x > Q_3 + 3\,\text{IQR} \]

Suspected outliers (between the inner and outer fences) are drawn as open red circles; outliers (beyond the outer fences) are drawn as filled red circles. Each distinct value is marked once, regardless of how many times it occurs.

Mean marker¶

When Display Mean is on, an orange diamond is drawn at the frequency-weighted arithmetic mean:

\[ \bar{x} = \frac{\sum_i f_i\, x_i}{\sum_i f_i} \]

Without a frequency column, every \(f_i = 1\) and this reduces to \(\bar{x} = \frac{1}{n}\sum_i x_i\).

Shared Math Details used here¶

This tool uses shared formulas defined once in Shared Math Details. See those pages for the exact definitions.

Shared concept	Used here for	Reference
Type 8 quantiles	the median, quartiles, whisker and box percentile edges, minimum, and maximum	Type 8 (Hyndman-Fan) Quantiles

References¶

Tukey, J. W. (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley.
McGill, R., Tukey, J. W., & Larsen, W. A. (1978). Variations of box plots. The American Statistician, 32(1), 12–16.
Hoaglin, D. C., Mosteller, F., & Tukey, J. W. (Eds.). (1983). Understanding Robust and Exploratory Data Analysis. New York: John Wiley & Sons.
Frigge, M., Hoaglin, D. C., & Iglewicz, B. (1989). Some implementations of the boxplot. The American Statistician, 43(1), 50–54.
Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365.
Montgomery, D. C. (2013). Introduction to Statistical Quality Control (7th ed.). Hoboken, NJ: John Wiley & Sons.