Two sample z test  overview
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Two sample $z$ test  KruskalWallis test 


Independent variable  Independent variable  
One categorical with 2 independent groups  One categorical with $I$ independent groups ($I \geqslant 2$)  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One of ordinal level  
Null hypothesis  Null hypothesis  
$\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 
Alternative hypothesis  Alternative hypothesis  
Two sided: $\mu_1 \neq \mu_2$ Right sided: $\mu_1 > \mu_2$ Left sided: $\mu_1 < \mu_2$  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 
Assumptions  Assumptions  
 Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another  
Test statistic  Test statistic  
$z = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to H0. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1  \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$  $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  
Sampling distribution of $z$ if H0 were true  Sampling distribution of $H$ if H0 were true  
Standard normal  For large samples, approximately the chisquared distribution with $I  1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.  
Significant?  Significant?  
Two sided:
 For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 
$C\%$ confidence interval for $\mu_1  \mu_2$  n.a.  
$(\bar{y}_1  \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
where $z^*$ is the value under the normal curve with the area $C / 100$ between $z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval) The confidence interval for $\mu_1  \mu_2$ can also be used as significance test.    
Visual representation  n.a.  
  
Example context  Example context  
Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1$ = 2 amongst men and $\sigma_2$ = 2.5 amongst women.  Do people from different religions tend to score differently on social economic status?  
n.a.  SPSS  
  Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 
n.a.  Jamovi  
  ANOVA > One Way ANOVA  KruskalWallis
 
Practice questions  Practice questions  