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Simulation sim01

Mean dispersion $\theta=\overline d$ parameter estimation where

$$\overline d=\frac{\sum_{i=1}^n{|x_i-\overline x|}}{\sum_{i=1}^n1};x_i\ne \overline x, (1)$$

$$\hat{\overline d}=\sigma_0\cdot\zeta\cdot\sqrt{\frac{n-1}{n}}\cdot0.997 (2)$$

with

$$\hat\sigma_{\overline d}=\sigma\cdot\frac{1}{2\cdot\zeta\cdot\sqrt{n}}(3)$$

where

$\zeta_{\overline d}=\overline\delta=\frac{4}{5}$

(s. Schrausser, 2022, p. 33, 2025).

Table 1. Simulated mean dispersion $\overline d=D_0$ $x[0,1]$ (1) and standard deviation $\sigma_{\overline d}=sD$ over simulation cycles $M$ in comparison to estimated mean dispersion $\hat{\overline d}=D'$ (2) and estimated standard error $\hat\sigma_{\overline d}=sD'$ (3) with difference values $diff$ and sample sizes $N$.

N	n	M			D0		D'		diff		sD0		sD'		diff

5	1	45000		0,713	0,713	0,000		0,263	0,244	-0,019
10	2	45000		0,755	0,757	0,002		0,188	0,183	-0,005
20	3	45000		0,779	0,777	-0,001		0,134	0,133	-0,001
40	4	45000		0,787	0,788	0,000		0,095	0,095	0,001
80	5	45000		0,793	0,793	-0,000		0,067	0,068	0,001
160	6	45000		0,796	0,795	-0,001		0,047	0,048	0,001
        270000						0,000						-0,004

Table 2. Corresponding left- and right-sided as well as middle percent ranges on the $z$ (6) and $t$, $df=n-1$ (7) asymptotes to confidence intervals $CI\%$ of $95\%$ and $99\%$ for given simulation distribution of mean dispersion $\overline d=D$ (1) with difference values $diff$ ​​for sample sizes $N$.

            z								t(df)										
            range							range								%diff					
CI%	N		left	right	CI		100-CI	left	right	CI		100-CI		CI z	CI t	z-t

95	5		45,76	48,33	94,09	5,91	42,00	44,97	86,97	13,03		0,91	8,03	7,11
    10		46,44	48,17	94,62	5,38	44,77	46,70	91,46	8,54		0,38	3,54	3,15
    20		46,94	47,97	94,91	5,09	46,16	47,27	93,43	6,57		0,09	1,57	1,48
    40		47,27	47,81	95,07	4,93	46,90	47,46	94,36	5,64		-0,07	0,64	0,71
    80		47,55	47,60	95,15	4,85	47,37	47,42	94,80	5,20		-0,15	0,20	0,35
    160		47,58	47,46	95,03	4,97	47,49	47,37	94,86	5,14		-0,03	0,14	0,17
    320		47,60	47,63	95,23	4,77	47,56	47,59	95,14	4,86		-0,23	-0,14	0,09

99	5		48,12	49,74	97,86	2,14	44,69	47,60	92,29	7,71		1,14	6,71	5,57
    10		48,96	49,66	98,62	1,38	47,70	48,82	96,52	3,48		0,38	2,48	2,11
    20		49,33	49,62	98,95	1,05	48,85	49,25	98,10	1,90		0,05	0,90	0,84
    40		49,52	49,53	99,05	0,95	49,33	49,35	98,67	1,33		-0,05	0,33	0,37
    80		49,53	49,38	98,91	1,09	49,45	49,28	98,72	1,28		0,09	0,28	0,19
    160		49,55	49,37	98,92	1,08	49,51	49,32	98,83	1,17		0,08	0,17	0,09
    320		49,56	49,41	98,97	1,03	49,53	49,39	98,92	1,08		0,03	0,08	0,04

Difference of mean dispersion for unpaired samples $\theta=d_{\overline d}=\overline d_1-\overline d_2$, $\hat\theta=0$, with

$$z=t=\frac{d_{\overline d}}{\hat\sigma_{d_\overline d}}, (4)$$

$$\hat\sigma_{d_\overline d}=\sigma\cdot\frac{n\cdot\sqrt{n-2}}{2\cdot\zeta^2\cdot(n-1)^2}\cdot c(5)$$

with

$$F(z)=p=\int\limits_{-\infty}^z\vartheta \ dz(6)$$

and

$$F(t,df)=p=\int\limits_{-\infty}^t\frac{\mathit{\Gamma} _{(\frac{df-1}{2})}}{\mathit{\Gamma} _{(\frac{df}{2})}}⋅(df⋅\mathrm{\pi})^{-\frac{1}{2}}⋅\Bigl(1+\frac{t^2}{df}\Bigr)^{-\frac{df-1}{2}}\ dt(7)$$

where

$\vartheta= \frac{1}{{\sqrt {2⋅\mathrm{\pi}} }}⋅\mathrm{e}^{-\frac{z^2}{2}},$
$\mathit{\Gamma} = \int\limits_0^\infty {t^{x-1} } ⋅\mathrm{e}^{-t} \ dt+c.$

Table 3. Standard error of differences of the mean dispersion $\sigma_{d_\overline d}=sdD_0$ $x[0,1]$ from simulations over cycles $M$ vs. estimated standard error of the difference $\hat\sigma_{d_\overline d}=sdD'$ (5) for unpaired samples with difference values $diff$ at sample sizes $N$.

N	n	M			sdD0	sdD'	diff

2	1	69000		0,602	0,600	-0,002
3	2	69000		0,482	0,480	-0,002
4	3	69000		0,420	0,420	0,000
5	4	27000		0,375	0,376	0,000
10	5	27000		0,267	0,266	0,000
20	6	27000		0,189	0,189	-0,001
40	7	27000		0,133	0,133	0,000
80	8	27000		0,095	0,094	0,000
160	9	27000		0,067	0,067	0,000
320	10	31000		0,047	0,047	0,000
        400000						-0,001

Table 4. Actual levels ​​on the $z$ (6) and $t$, $df=2\cdot n-2$ (7) asymptotes for given one- and two-sided $1\%$ and $5\%$ levels from simulation distribution of mean dispersion differences $d_{\overline d}$, with corresponding difference values $diff$.

                        %level				
side	p<=		tail	z		diff		t(df)	diff		diff

left	0,01	2		0,64	-0,36		1,79	0,79		-1,16
                1		0,82	-0,18		1,60	0,60		-0,78
        0,05	2		4,66	-0,34		6,70	1,70		-2,04
                1		4,82	-0,18		5,91	0,91		-1,10
                                -0,27				1,00		-1,27

right	0,01	2		1,24	0,24		2,68	1,68		-1,44
                1		1,07	0,07		1,91	0,91		-0,84
        0,05	2		5,19	0,19		7,26	2,26		-2,06
                1		5,12	0,12		6,22	1,22		-1,10
                                0,16				1,52		-1,36

References

Schrausser, D. G. (2022). Mathematical-Statistical Algorithm Interpreter, SCHRAUSSER-MAT. DOI: 10.17605/OSF.IO/8XE42

———. (2025). HP_Prime_MATH: Manual. 1st ed. Zenodo. OL: OL60452436M