Table II. Equations of test statistical parameters used in circular and linear tests.

Where θ_i = angular data for the ith observation as i= 1,…, n and n = number of observations. Tables providing critical values of each test statistic are available from Batschelet (Batschelet, 1981) or from circular statistical software programs such as Oriana. Refer to Table I for additional definitions of variables.

Test	Data type	Null hypothesis	Alternate hypothesis	Formula for test statistic	Notes
CIRCULAR ONE-SAMPLE TESTS
Rao’s Spacing	Continuous	Uniform	Any alternative	θ1 ≤ θ2 ≤ … ≤ θn	n ≥ 5
				Tn−1 = θn − θn−1	where Ti is the arclength between consecutive angles
				Tn = 360 + θ1 − θn
				$U=\frac{1}{2}{\displaystyle {\sum }_i=1^N(|T_i-\frac{360}{n}|)}$
Kuiper’s	Continuous	Uniform	Any alternative	Vn = D+ + D−	where D is the deviation from (uniform) theoretical distribution
				$K=\sqrt{n}{V}_n$	for small n, more powerful than χ2 test
					circular goodness-of-fit test corresponding to Kolmogorov-Smirnov
Rayleigh’s	Continuous	Uniform	Unimodal	$Z=\frac{R}{n}$	unimodal von Mises alternative
Watson’s U2	Continuous	Uniform	Unimodal	θ1 ≤ θ2 ≤ … ≤ θn	where F(θ) is the function of (uniform) theoretical distribution
				${\nu }_i=F({\theta }_i),\overline{\nu }=\frac{1}{n}{\displaystyle \sum {\nu }_i,c_i=2i-1}$	suitable for unimodal and multimodal distributions
				$U^2={\displaystyle \sum {\nu }_i^2-{\displaystyle \sum \left(\frac{c_i{\nu }_i}{n}\right)}+n\left[\frac{1}{3}-{\left(\overline{\nu }-\frac{1}{2}\right)}^2\right]}$
V-test	Continuous	Uniform	Unimodal	$V=R\text{cos}({\mu }_c-{\mu }_0),u=V\sqrt{\frac{2}{n}}$	external direction (µ0) specified a priori
CIRCULAR MULTISAMPLE TESTS
Mardia-Watson-Wheeler	Continuous	Two samples have identical distributions	Two samples have different distributions	$W=2{\displaystyle {\sum }_{i=1}^k\left[\frac{C_i^2+S_i^2}{n_i}\right]}$	data is ungrouped
Watson’s U2	Continuous	Two samples have same distributions	Two samples have different distributions	$U^2=\frac{n_1{n}_2}{N^2}\left[{\displaystyle {\sum }_{k=1}^N{d}_k^2-\frac{({\displaystyle {\sum }_{k=1}^N{d}_k{)}^2}}{N}}\right]$	where N = n1 + n2
					no ties between samples
LINEAR TESTS
χ2	Grouped	Uniform	Non-uniform	${\chi }^2={\displaystyle {\sum }_i^k=1^k\frac{{({\theta }_i-e_i)}^2}{e_i}}$	where ei = expected value for ith term
					based on approximation, need large n
					suitable for both unimodal and multimodal distributions
Kolmogorov-Smirnov	Grouped	Uniform	Non-uniform	Dmax = max(|di|)	where F̄i is the function of (uniform) theoretical distribution
				where |di| = |Fi − F̄i|	more powerful than χ2 when n is small
					dependent on starting point, more sensitive near median than at tails
Student’s t-test	Continuous	Two samples have same means	Two samples have different means	$t=\frac{{\overline{x}}_1-{\overline{x}}_2}{s_{{\overline{x}}_1-{\overline{x}}_2}}$	where x̄ is the sample mean and s is the sample standard deviation
				where $s_{{\overline{x}}_1-{\overline{x}}_2}=\sqrt{\frac{s_1^2+s_2^2}{n}}$	assuming equal variances