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Math Details¶

Under review: candidate for removal

The Bar Chart is primarily a visualization tool; the only non-trivial formula is the sample standard deviation option. This page is flagged for review: it may be removed if judged too thin to warrant a Math Details page.

This page gives the formulas Quantum XL uses when a Bar Chart aggregates numeric data. Each bar's height is one of the aggregation functions below (chosen in Options).

Notation¶

Term	Description
\(x_i, w_i\)	value in a category and its frequency weight (\(w_i = 1\) when no frequency column is used)
\(n\)	effective count in the category, \(n = \sum_i w_i\)
\(\bar{x}\)	weighted mean of the category

Aggregation functions¶

Function	Bar height
Sum	\(\sum_i w_i x_i\)
Count	\(n = \sum_i w_i\)
Mean	\(\bar{x} = \dfrac{\sum_i w_i x_i}{\sum_i w_i}\)
Minimum / Maximum	smallest / largest \(x_i\) in the category
Median	the \(0.5\) quantile (Hyndman-Fan Type 8)

Median¶

The median bar height is the \(0.5\) quantile of the category's values, computed with the shared Type 8 estimator applied to the frequency-weighted values:

\[ h = \left(n + \tfrac{1}{3}\right)p + \tfrac{1}{3} \]

\[ Q(p) = x_{(\lfloor h \rfloor)} + \left(h - \lfloor h \rfloor\right)\left(x_{(\lfloor h \rfloor + 1)} - x_{(\lfloor h \rfloor)}\right) \]

where \(x_{(1)} \le \cdots \le x_{(n)}\) are the \(n\) data values sorted ascending and \(\lfloor h \rfloor\) is the greatest integer not exceeding \(h\). The two indices are clamped to the range \([1, n]\) and the interpolation weight \(h - \lfloor h \rfloor\) is clamped to \([0, 1]\), so the ends behave as \(Q(0) = x_{(1)}\) (the minimum) and \(Q(1) = x_{(n)}\) (the maximum).

When each value carries a frequency \(f_i\), Quantum XL applies the same estimator to the virtual sample formed by repeating each value \(f_i\) times, without physically expanding it. The total count is \(N = \sum_i f_i\), and the position becomes:

\[ h = \left(N + \tfrac{1}{3}\right)p + \tfrac{1}{3} \]

Let \(V(t)\) be the value at position \(t\) of the sorted virtual sample, that is the smallest value whose cumulative frequency reaches \(t\). The quantile is:

\[ Q(p) = V(\lfloor h \rfloor) + \left(h - \lfloor h \rfloor\right)\left(V(\lfloor h \rfloor + 1) - V(\lfloor h \rfloor)\right) \]

with the same end clamping. This gives exactly the result of expanding the data and applying the unweighted formula.

Full details: Type 8 (Hyndman-Fan) Quantiles.

Sample standard deviation¶

The other non-elementary option:

\[ s = \sqrt{\frac{\sum_i w_i\,(x_i - \bar{x})^2}{n - 1}} \]

Uses the \(n-1\) (Bessel) denominator; returned as blank when \(n \le 1\).

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 bar height	Type 8 (Hyndman-Fan) Quantiles

References¶

Snedecor, G. W., & Cochran, W. G. (1989). Statistical Methods (8th ed.). Ames, IA: Iowa State University Press.
Montgomery, D. C. (2013). Introduction to Statistical Quality Control (7th ed.). Hoboken, NJ: John Wiley & Sons.