n

Stat.

Proposal

Mardia’s

SJ

Royston’s

Q'

Q

K²

Z

H

H'

50

z( $\sqrt{b_1}$ )

−4.1668

−4.6032

−3.1285

−2.2595

−3.7671

−3.4225

p

<0.0001

<0.0001

0.0018

0.0239

0.0002

0.0006

z(b₂)

2.2434

2.9631

0.2025

−2.1260

1.8991

1.4236

p

0.0249

0.0030

0.8395

0.0335

0.0576

0.1546

K²

22.3948

29.9696

9.8284

9.6251

17.7975

13.7400

p

<0.0001

<0.0001

0.0073

0.0081

0.0001

0.0010

N(μ, σ)

no: − & ↑

no: − & ↑

no: −

no: − & ↓

no: − & ↑

no: −

40

z( $\sqrt{b_1}$ )

−5.0475

−5.4837

−4.4467

−1.3323

−4.2865

−4.0234

p

0.0000

0.0000

<0.0001

0.1828

<0.0001

0.0001

z(b₂)

3.7356

4.3276

3.1058

−4.0867

2.9193

2.5986

p

0.0002

0.0000

0.0019

<0.0001

0.0035

0.0094

K²

39.4326

48.7984

29.4190

18.4761

26.8963

22.9408

p

<0.0001

<0.0001

<0.0001

<0.0001

<0.0001

<0.0001

N(μ, σ)

no: − & ↑

no: − & ↑

no: − & ↑

no: ↓

no: − & ↑

no: − & ↑

10

z( $\sqrt{b_1}$ )

−1.0754

−1.2556

1.9175

−1.7229

−1.0924

−0.3804

p

0.2822

0.2093

0.0552

0.0849

0.2747

0.7037

z(b₂)

−0.0624

0.0740

1.9760

0.4832

0.6720

0.1697

p

0.9503

0.9410

0.0481

0.6290

0.5016

0.8653

K²

1.1605

1.5820

7.5815

3.2018

1.6449

0.1735

p

0.5598

0.4534

0.0226

0.2017

0.4394

0.9169

N(μ, σ)

yes

yes

no: + & ↑

yes

yes

yes