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Math Details

This page gives the exact formulas Quantum XL uses to compute descriptive statistics. Each equation lists what it computes and where it appears in the output. (Elementary outputs — count, sum, minimum, maximum, range, mode — are omitted.)

Notation

Term	Description
\(x_{(1)} \le \dots \le x_{(n)}\)	the data values sorted ascending
\(x_i, w_i\)	value and its frequency weight (\(w_i = 1\) when no frequency column is used)
\(n\)	effective sample size, \(n = \sum_i w_i\)
\(\bar{x}\)	weighted mean, \(\bar{x} = \dfrac{\sum_i w_i x_i}{\sum_i w_i}\)
\(M_k\)	weighted central moment sum, \(M_k = \sum_i w_i (x_i - \bar{x})^k\)
\(\Phi\)	standard normal cumulative distribution function

Variance and standard deviation

\[ s^2_{\text{sample}} = \frac{M_2}{n - 1}, \qquad s^2_{\text{population}} = \frac{M_2}{n}, \qquad s = \sqrt{s^2} \]

The sample forms use the \(n-1\) (Bessel) denominator; both population and sample versions are reported.

Skewness (Fisher–Pearson standardized, \(G_1\))

\[ G_1 = \frac{n\sqrt{n - 1}}{n - 2}\cdot\frac{M_3}{M_2^{\,3/2}} \]

Requires \(n > 2\); undefined for constant data. This is the bias-adjusted (\(G_1\)) form used by most statistical packages.

Kurtosis (excess, \(G_2\))

\[ G_2 = \frac{(n + 1)\,n\,(n - 1)}{(n - 2)(n - 3)}\cdot\frac{M_4}{M_2^{\,2}} \;-\; \frac{3\,(n - 1)^2}{(n - 2)(n - 3)} \]

Excess kurtosis (a normal distribution gives \(0\)). Requires \(n > 3\).

Quantiles, quartiles, and percentiles

All percentiles (including the median and quartiles) use the Hyndman–Fan Type 8 estimator:

\[ h = \left(n + \tfrac{1}{3}\right)p + \tfrac{1}{3}, \qquad Q(p) = x_{(\lfloor h \rfloor)} + \left(h - \lfloor h \rfloor\right)\left(x_{(\lfloor h \rfloor + 1)} - x_{(\lfloor h \rfloor)}\right) \]

with the index clamped to \([1, n]\). Median \(= Q(0.5)\), \(Q_1 = Q(0.25)\), \(Q_3 = Q(0.75)\). Frequency-weighted data applies the same rank \(h = (n + \tfrac{1}{3})p + \tfrac{1}{3}\) to a virtual sample of size \(n = \sum_i w_i\), interpolating by cumulative frequency.

Confidence intervals

For the mean (t-interval) and the standard deviation (chi-square interval):

\[ \bar{x} \pm t_{\,n-1,\ 1-\alpha/2}\cdot\frac{s}{\sqrt{n}}, \qquad \left[\sqrt{\tfrac{(n-1)s^2}{\chi^2_{\,n-1,\ 1-\alpha/2}}},\; \sqrt{\tfrac{(n-1)s^2}{\chi^2_{\,n-1,\ \alpha/2}}}\right] \]

where \(\alpha = 1 - \text{(confidence level)}\). See Confidence Interval → Math Details for full detail.

Normality tests

Anderson–Darling

\[ A^2 = -n - \frac{1}{n}\sum_{i=1}^{n}(2i - 1)\left[\ln \Phi(z_{(i)}) + \ln\!\left(1 - \Phi(z_{(n+1-i)})\right)\right], \qquad z_{(i)} = \frac{x_{(i)} - \bar{x}}{s} \]

Quantum XL evaluates a frequency-weighted equivalent of this sum, then applies Stephens' finite-sample correction \(A^{*2} = A^2\left(1 + \tfrac{0.75}{n} + \tfrac{2.25}{n^2}\right)\) and obtains the p-value from Stephens' approximation.

Shapiro–Wilk

\[ W = \frac{\left(\sum_{i=1}^{n} a_i\, x_{(i)}\right)^2}{\sum_{i=1}^{n} (x_i - \bar{x})^2} = \frac{\left(\sum_i a_i\, x_{(i)}\right)^2}{M_2} \]

The constants \(a_i\) are derived from the expected values and covariances of standard-normal order statistics; Quantum XL computes them using Royston's Algorithm AS R94, and obtains the p-value from Royston's transformation.

Kolmogorov–Smirnov (with Lilliefors correction)

\[ D^{+} = \max_i\left(F_n(x_{(i)}) - \Phi(z_{(i)})\right), \quad D^{-} = \max_i\left(\Phi(z_{(i)}) - F_n(x_{(i-1)})\right), \quad D = \max(D^{+}, D^{-}) \]

where \(F_n\) is the empirical (cumulative-frequency) CDF and \(z_{(i)}\) standardizes with the sample mean and standard deviation. Because the parameters are estimated from the data, the p-value uses the Lilliefors correction.

See Also

• Confidence Interval Math Details
• Box Plot Math Details
• Options

References

• Snedecor, G. W., & Cochran, W. G. (1989). Statistical Methods (8th ed.). Ames, IA: Iowa State University Press.
• Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 47(1), 183–189.
• Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365.
• Anderson, T. W., & Darling, D. A. (1952). Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes. Annals of Mathematical Statistics, 23(2), 193–212.
• Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69(347), 730–737.
• D'Agostino, R. B., & Stephens, M. A. (Eds.). (1986). Goodness-of-Fit Techniques. New York: Marcel Dekker.
• Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591–611.
• Royston, P. (1995). Remark AS R94: A remark on Algorithm AS 181: The W test for normality. Journal of the Royal Statistical Society: Series C (Applied Statistics), 44(4), 547–551.
• Lilliefors, H. W. (1967). On the Kolmogorov–Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association, 62(318), 399–402.
• Massey, F. J. (1951). The Kolmogorov–Smirnov test for goodness of fit. Journal of the American Statistical Association, 46(253), 68–78.