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# 2 Way Anova Calculator with steps

ANOVA is analysis of variance. 2 way ANOVA is used when there are two independent categorical variables. 2-way ANOVA has 3 p values. 2 way ANOVA table is given at the end of the solution.

INSTRUCTION: Use ',' or new line to separate between values

 Enter alpha value:

## Group

$$\displaylines{---}$$
$$\displaylines{\;\mathbf{\color{Green}{The\;data\;you\;entered\;is\;below}} }$$

$$Column\;1$$
$$Column\;2$$
$$Column\;3$$
$$Row\;1$$
$$1, 2, 3, 4, 5$$
$$1, 3, 4, 4, 3, 7, 4$$
$$3, 6, 7, 1, 2$$
$$Row\;2$$
$$4, 5, 8, 7, 6$$
$$5, 6, 4, 7, 8$$
$$6, 4, 5, 8, 3, 4$$
$$\displaylines{}$$
$$\displaylines{\;\mathbf{\color{Green}{Table\;with\;sum\;of\;each\;group}} }$$

$$Column\;1$$
$$Column\;2$$
$$Column\;3$$
$$Row\;Sum$$
$$Row\;1$$
$$15$$
$$26$$
$$19$$
$$60$$
$$Row\;2$$
$$30$$
$$30$$
$$30$$
$$90$$
$$Column\;Sum$$
$$45$$
$$56$$
$$49$$
$$150$$
$$\displaylines{}$$
$$\displaylines{\;\mathbf{\color{Green}{Table\;with\;number\;of\;observations\;each\;group}} }$$

$$Column\;1$$
$$Column\;2$$
$$Column\;3$$
$$Row\;Total$$
$$Row\;1$$
$$5$$
$$7$$
$$5$$
$$17$$
$$Row\;2$$
$$5$$
$$5$$
$$6$$
$$16$$
$$Column\;Total$$
$$10$$
$$12$$
$$11$$
$$33$$
$$\displaylines{}$$
$$\displaylines{\;\mathbf{\color{Green}{Correction\;term\;C_{x}\;=\;}} \frac{(Total)^{2}}{Total\;number} \\ \\ \Rightarrow \frac{150.000000*150.000000}{33} \\ \\ \Rightarrow \mathbf{\color{Red}{681.818182}} = \frac{(\sum x)^{2}}{n} \\ \\ \;\mathbf{\color{Green}{Now\;we\;need\;sum\;of\;square\;of\;all\;values}} }$$
$$x$$
$$x^{2}$$
$$1$$
$$1$$
$$2$$
$$4$$
$$3$$
$$9$$
$$4$$
$$16$$
$$5$$
$$25$$
$$1$$
$$1$$
$$3$$
$$9$$
$$4$$
$$16$$
$$4$$
$$16$$
$$3$$
$$9$$
$$7$$
$$49$$
$$4$$
$$16$$
$$3$$
$$9$$
$$6$$
$$36$$
$$7$$
$$49$$
$$1$$
$$1$$
$$2$$
$$4$$
$$4$$
$$16$$
$$5$$
$$25$$
$$8$$
$$64$$
$$7$$
$$49$$
$$6$$
$$36$$
$$5$$
$$25$$
$$6$$
$$36$$
$$4$$
$$16$$
$$7$$
$$49$$
$$8$$
$$64$$
$$6$$
$$36$$
$$4$$
$$16$$
$$5$$
$$25$$
$$8$$
$$64$$
$$3$$
$$9$$
$$4$$
$$16$$
$$Total \sum x^{2}=$$
$$816$$
$$\displaylines{}$$
$$\displaylines{Sum\;of\;squares\;of\;total\;=\;SS_{T}= \sum x^{2}- \frac{(\sum x)^{2}}{n} \\ \\ \Rightarrow 816.000000-681.818182 \\ \\ \Rightarrow SS_{T}= \mathbf{\color{Red}{134.181818}} \\ \\ \;\mathbf{\color{Green}{Now,\;Make\;below\;table\;for\;finding\;Sum\;of\;squares\;of\;columns}} }$$

$$x_{c}$$
$$n_{c}$$
$$(x_{c})^{2}$$
$$\frac{(x_{c})^{2}}{n_{c}}$$
$$Column\;1$$
$$45$$
$$10$$
$$2025$$
$$202.5$$
$$Column\;2$$
$$56$$
$$12$$
$$3136$$
$$261.333333$$
$$Column\;3$$
$$49$$
$$11$$
$$2401$$
$$218.272727$$
$$Total$$



$$682.10606$$
$$\displaylines{}$$
$$\displaylines{Sum\;of\;squares\;of\;columns\;=\;SS_{C}= \sum \frac{(x_{c})^{2}}{n_{c}} - \frac{(\sum x)^{2}}{n} \\ \\ \Rightarrow 682.106060-681.818182 \\ \\ \Rightarrow SS_{C}= \mathbf{\color{Red}{0.287878}} \\ \\ \;\mathbf{\color{Green}{Now,\;Make\;below\;table\;for\;finding\;Sum\;of\;squares\;of\;Rows}} }$$

$$x_{r}$$
$$n_{r}$$
$$(x_{r})^{2}$$
$$\frac{(x_{r})^{2}}{n_{r}}$$
$$Row\;1$$
$$60$$
$$17$$
$$3600$$
$$211.764706$$
$$Row\;2$$
$$90$$
$$16$$
$$8100$$
$$506.25$$
$$Total$$



$$718.014706$$
$$\displaylines{}$$
$$\displaylines{Sum\;of\;squares\;of\;rows\;=\;SS_{R}= \sum \frac{(x_{r})^{2}}{n_{r}} - \frac{(\sum x)^{2}}{n} \\ \\ \Rightarrow 718.014706-681.818182 \\ \\ \Rightarrow SS_{R}= \mathbf{\color{Red}{36.196524}} \\ \\ \;\mathbf{\color{Green}{Now,\;Make\;below\;table\;for\;finding\;Sum\;of\;squares\;of\;groups}} }$$
$$Group\;Name$$
$$x_{r}$$
$$n_{r}$$
$$(x_{r})^{2}$$
$$\frac{(x_{r})^{2}}{n_{r}}$$
$$Row\;1,Column\;1$$
$$15$$
$$5$$
$$225$$
$$45.0$$
$$Row\;1,Column\;2$$
$$26$$
$$7$$
$$676$$
$$96.571429$$
$$Row\;1,Column\;3$$
$$19$$
$$5$$
$$361$$
$$72.2$$
$$Row\;2,Column\;1$$
$$30$$
$$5$$
$$900$$
$$180.0$$
$$Row\;2,Column\;2$$
$$30$$
$$5$$
$$900$$
$$180.0$$
$$Row\;2,Column\;3$$
$$30$$
$$6$$
$$900$$
$$150.0$$
$$Total$$



$$723.771429$$
$$\displaylines{}$$
$$\displaylines{Sum\;of\;squares\;within\;groups\;=\;SS_{G}= \sum \frac{(x_{g})^{2}}{n_{g}} - \frac{(\sum x)^{2}}{n} -SS_{R} -SS_{C} \\ \\ \Rightarrow 723.771429-681.818182-36.196524-0.287878 \\ \\ \Rightarrow SS_{G}= \mathbf{\color{Red}{5.468845}} \\ \\ \;\mathbf{\color{Green}{Error\;(Residual)\;Sum\;of\;Squares\;=\;SS_{E}=}} SS_{T}-SS_{C}-SS_{R}-SS_{G} \\ \\ \Rightarrow 134.181818-0.287878-36.196524-5.468845 \\ \\ \Rightarrow SS_{E}= \mathbf{\color{Red}{92.228571}} \\ \\ \mathbf{\color{Green}{Now\;make\;2\;way\;ANOVA\;table\;like\;below}} }$$
$$Source\;of\;variation$$
$$Sum\;of\;Squares$$
$$df$$
$$MS$$
$$F$$
$$p$$
$$Rows$$
$$SS_{R}$$
$$R-1$$
$$\frac{SS_{R}}{R-1}$$
$$\frac{MS_{R}}{MS_{E}}$$

$$Column$$
$$SS_{C}$$
$$C-1$$
$$\frac{SS_{C}}{C-1}$$
$$\frac{MS_{C}}{MS_{E}}$$

$$within\;groups$$
$$SS_{G}$$
$$(C-1)(R-1)$$
$$\frac{SS_{G}}{(C-1)(R-1)}$$
$$\frac{MS_{G}}{MS_{E}}$$

$$Error$$
$$SS_{E}$$
$$N-C*R$$
$$\frac{SS_{E}}{N-C*R}$$


$$Total$$
$$SS_{T}$$
$$N-1$$



$$\displaylines{}$$
$$\displaylines{Where,\;N=\;Number\;of\;Observations\;=\; 33 \\ \\ R=\;Number\;of\;Rows\;=\; 2 \\ \\ C=\;Number\;of\;Columns\;=\; 3 \\ \\ \mathbf{\color{Green}{Substitute\;values\;and\;make\;below\;table}} \\ \\ }$$
$$Source\;of\;variation$$
$$Sum\;of\;Squares$$
$$df$$
$$MS$$
$$F$$
$$p$$
$$Rows$$
$$36.196524$$
$$1$$
$$\frac{36.196524}{1}=36.196524$$
$$\frac{MS_{R}}{MS_{E}}$$

$$Column$$
$$0.287878$$
$$2$$
$$\frac{0.287878}{2}=0.143939$$
$$\frac{MS_{C}}{MS_{E}}$$

$$within\;groups$$
$$5.468845$$
$$2$$
$$\frac{5.468845}{2}=2.734422$$
$$\frac{MS_{G}}{MS_{E}}$$

$$Error$$
$$92.228571$$
$$27$$
$$\frac{92.228571}{27}=3.415873$$


$$Total$$
$$134.181818$$
$$32$$



$$\displaylines{}$$
$$\displaylines{F_{R}= \frac{MS_{R}}{MS_{E}} = \frac{36.196524}{3.415873} = 10.596566 \\ \\ F_{C}= \frac{MS_{C}}{MS_{E}} = \frac{0.143939}{3.415873} = 0.042138 \\ \\ F_{G}= \frac{MS_{G}}{MS_{E}} = \frac{2.734422}{3.415873} = 0.800505 \\ \\ \mathbf{\color{Red}{Final\;anova\;table}} \\ \\ }$$
$$Source\;of\;variation$$
$$Sum\;of\;Squares$$
$$df$$
$$MS$$
$$F$$
$$p$$
$$Rows$$
$$36.196524$$
$$1$$
$$36.196524$$
$$10.596566$$
$$0.003046$$
$$Column$$
$$0.287878$$
$$2$$
$$0.143939$$
$$0.042138$$
$$0.9588$$
$$within\;groups$$
$$5.468845$$
$$2$$
$$2.734422$$
$$0.800505$$
$$0.459475$$
$$Error$$
$$92.228571$$
$$27$$
$$3.415873$$


$$Total$$
$$134.181818$$
$$32$$



$$\displaylines{}$$
$$\displaylines{\;\mathbf{\color{Red}{Conclusion}} \\ \\ \;\mathbf{\color{Green}{For\;Rows}} \\ \\ p\;=\;0.003046\;\;\;, p\;is\;less\;than\;\alpha\;.\; \\ \\ So,\;Reject\;H_{R0}.\;So\;means\;of\;Rows\;are\;not\;same \\ \\ \;\mathbf{\color{Green}{For\;Column}} \\ \\ p\;=\;0.958800\;\;\;, p\;is\;more\;than\;\alpha\;.\; \\ \\ So,\;Failed\;to\;Reject\;H_{C0}\;So\;means\;of\;Columns\;are\;same \\ \\ \;\mathbf{\color{Green}{For\;Groups}} \\ \\ p\;=\;0.459475\;\;\;, p\;is\;more\;than\;\alpha\;.\; \\ \\ So,\;Failed\;to\;Reject\;H_{G0}.\;There\;is\;no\;interaction\;between\;Rows\;and\;Columns }$$