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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

You can see a sample solution below. Enter your data to get the solution for your question

Row

Column

Group

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$$ \displaylines{---} $$
$$ \displaylines{\;\mathbf{\color{Green}{The\;data\;you\;entered\;is\;below}} } $$
$$ $$
$$ Column\;1 $$
$$ Column\;2 $$
$$ Column\;3 $$
$$ Row\;1 $$
$$ 1, 2, 3, 4, 5 $$
$$ 1, 3, 4, 4, 3, 7, 4 $$
$$ 3, 6, 7, 1, 2 $$
$$ Row\;2 $$
$$ 4, 5, 8, 7, 6 $$
$$ 5, 6, 4, 7, 8 $$
$$ 6, 4, 5, 8, 3, 4 $$
$$ \displaylines{} $$
$$ \displaylines{\;\mathbf{\color{Green}{Table\;with\;sum\;of\;each\;group}} } $$
$$ $$
$$ Column\;1 $$
$$ Column\;2 $$
$$ Column\;3 $$
$$ Row\;Sum $$
$$ Row\;1 $$
$$ 15 $$
$$ 26 $$
$$ 19 $$
$$ 60 $$
$$ Row\;2 $$
$$ 30 $$
$$ 30 $$
$$ 30 $$
$$ 90 $$
$$ Column\;Sum $$
$$ 45 $$
$$ 56 $$
$$ 49 $$
$$ 150 $$
$$ \displaylines{} $$
$$ \displaylines{\;\mathbf{\color{Green}{Table\;with\;number\;of\;observations\;each\;group}} } $$
$$ $$
$$ Column\;1 $$
$$ Column\;2 $$
$$ Column\;3 $$
$$ Row\;Total $$
$$ Row\;1 $$
$$ 5 $$
$$ 7 $$
$$ 5 $$
$$ 17 $$
$$ Row\;2 $$
$$ 5 $$
$$ 5 $$
$$ 6 $$
$$ 16 $$
$$ Column\;Total $$
$$ 10 $$
$$ 12 $$
$$ 11 $$
$$ 33 $$
$$ \displaylines{} $$
$$ \displaylines{\;\mathbf{\color{Green}{Correction\;term\;C_{x}\;=\;}} \frac{(Total)^{2}}{Total\;number} \\ \\ \Rightarrow \frac{150.000000*150.000000}{33} \\ \\ \Rightarrow \mathbf{\color{Red}{681.818182}} = \frac{(\sum x)^{2}}{n} \\ \\ \;\mathbf{\color{Green}{Now\;we\;need\;sum\;of\;square\;of\;all\;values}} } $$
$$ x $$
$$ x^{2} $$
$$ 1 $$
$$ 1 $$
$$ 2 $$
$$ 4 $$
$$ 3 $$
$$ 9 $$
$$ 4 $$
$$ 16 $$
$$ 5 $$
$$ 25 $$
$$ 1 $$
$$ 1 $$
$$ 3 $$
$$ 9 $$
$$ 4 $$
$$ 16 $$
$$ 4 $$
$$ 16 $$
$$ 3 $$
$$ 9 $$
$$ 7 $$
$$ 49 $$
$$ 4 $$
$$ 16 $$
$$ 3 $$
$$ 9 $$
$$ 6 $$
$$ 36 $$
$$ 7 $$
$$ 49 $$
$$ 1 $$
$$ 1 $$
$$ 2 $$
$$ 4 $$
$$ 4 $$
$$ 16 $$
$$ 5 $$
$$ 25 $$
$$ 8 $$
$$ 64 $$
$$ 7 $$
$$ 49 $$
$$ 6 $$
$$ 36 $$
$$ 5 $$
$$ 25 $$
$$ 6 $$
$$ 36 $$
$$ 4 $$
$$ 16 $$
$$ 7 $$
$$ 49 $$
$$ 8 $$
$$ 64 $$
$$ 6 $$
$$ 36 $$
$$ 4 $$
$$ 16 $$
$$ 5 $$
$$ 25 $$
$$ 8 $$
$$ 64 $$
$$ 3 $$
$$ 9 $$
$$ 4 $$
$$ 16 $$
$$ Total \sum x^{2}= $$
$$ 816 $$
$$ \displaylines{} $$
$$ \displaylines{Sum\;of\;squares\;of\;total\;=\;SS_{T}= \sum x^{2}- \frac{(\sum x)^{2}}{n} \\ \\ \Rightarrow 816.000000-681.818182 \\ \\ \Rightarrow SS_{T}= \mathbf{\color{Red}{134.181818}} \\ \\ \;\mathbf{\color{Green}{Now,\;Make\;below\;table\;for\;finding\;Sum\;of\;squares\;of\;columns}} } $$
$$ $$
$$ x_{c} $$
$$ n_{c} $$
$$ (x_{c})^{2} $$
$$ \frac{(x_{c})^{2}}{n_{c}} $$
$$ Column\;1 $$
$$ 45 $$
$$ 10 $$
$$ 2025 $$
$$ 202.5 $$
$$ Column\;2 $$
$$ 56 $$
$$ 12 $$
$$ 3136 $$
$$ 261.333333 $$
$$ Column\;3 $$
$$ 49 $$
$$ 11 $$
$$ 2401 $$
$$ 218.272727 $$
$$ Total $$
$$ $$
$$ $$
$$ $$
$$ 682.10606 $$
$$ \displaylines{} $$
$$ \displaylines{Sum\;of\;squares\;of\;columns\;=\;SS_{C}= \sum \frac{(x_{c})^{2}}{n_{c}} - \frac{(\sum x)^{2}}{n} \\ \\ \Rightarrow 682.106060-681.818182 \\ \\ \Rightarrow SS_{C}= \mathbf{\color{Red}{0.287878}} \\ \\ \;\mathbf{\color{Green}{Now,\;Make\;below\;table\;for\;finding\;Sum\;of\;squares\;of\;Rows}} } $$
$$ $$
$$ x_{r} $$
$$ n_{r} $$
$$ (x_{r})^{2} $$
$$ \frac{(x_{r})^{2}}{n_{r}} $$
$$ Row\;1 $$
$$ 60 $$
$$ 17 $$
$$ 3600 $$
$$ 211.764706 $$
$$ Row\;2 $$
$$ 90 $$
$$ 16 $$
$$ 8100 $$
$$ 506.25 $$
$$ Total $$
$$ $$
$$ $$
$$ $$
$$ 718.014706 $$
$$ \displaylines{} $$
$$ \displaylines{Sum\;of\;squares\;of\;rows\;=\;SS_{R}= \sum \frac{(x_{r})^{2}}{n_{r}} - \frac{(\sum x)^{2}}{n} \\ \\ \Rightarrow 718.014706-681.818182 \\ \\ \Rightarrow SS_{R}= \mathbf{\color{Red}{36.196524}} \\ \\ \;\mathbf{\color{Green}{Now,\;Make\;below\;table\;for\;finding\;Sum\;of\;squares\;of\;groups}} } $$
$$ Group\;Name $$
$$ x_{r} $$
$$ n_{r} $$
$$ (x_{r})^{2} $$
$$ \frac{(x_{r})^{2}}{n_{r}} $$
$$ Row\;1,Column\;1 $$
$$ 15 $$
$$ 5 $$
$$ 225 $$
$$ 45.0 $$
$$ Row\;1,Column\;2 $$
$$ 26 $$
$$ 7 $$
$$ 676 $$
$$ 96.571429 $$
$$ Row\;1,Column\;3 $$
$$ 19 $$
$$ 5 $$
$$ 361 $$
$$ 72.2 $$
$$ Row\;2,Column\;1 $$
$$ 30 $$
$$ 5 $$
$$ 900 $$
$$ 180.0 $$
$$ Row\;2,Column\;2 $$
$$ 30 $$
$$ 5 $$
$$ 900 $$
$$ 180.0 $$
$$ Row\;2,Column\;3 $$
$$ 30 $$
$$ 6 $$
$$ 900 $$
$$ 150.0 $$
$$ Total $$
$$ $$
$$ $$
$$ $$
$$ 723.771429 $$
$$ \displaylines{} $$
$$ \displaylines{Sum\;of\;squares\;within\;groups\;=\;SS_{G}= \sum \frac{(x_{g})^{2}}{n_{g}} - \frac{(\sum x)^{2}}{n} -SS_{R} -SS_{C} \\ \\ \Rightarrow 723.771429-681.818182-36.196524-0.287878 \\ \\ \Rightarrow SS_{G}= \mathbf{\color{Red}{5.468845}} \\ \\ \;\mathbf{\color{Green}{Error\;(Residual)\;Sum\;of\;Squares\;=\;SS_{E}=}} SS_{T}-SS_{C}-SS_{R}-SS_{G} \\ \\ \Rightarrow 134.181818-0.287878-36.196524-5.468845 \\ \\ \Rightarrow SS_{E}= \mathbf{\color{Red}{92.228571}} \\ \\ \mathbf{\color{Green}{Now\;make\;2\;way\;ANOVA\;table\;like\;below}} } $$
$$ Source\;of\;variation $$
$$ Sum\;of\;Squares $$
$$ df $$
$$ MS $$
$$ F $$
$$ p $$
$$ Rows $$
$$ SS_{R} $$
$$ R-1 $$
$$ \frac{SS_{R}}{R-1} $$
$$ \frac{MS_{R}}{MS_{E}} $$
$$ $$
$$ Column $$
$$ SS_{C} $$
$$ C-1 $$
$$ \frac{SS_{C}}{C-1} $$
$$ \frac{MS_{C}}{MS_{E}} $$
$$ $$
$$ within\;groups $$
$$ SS_{G} $$
$$ (C-1)(R-1) $$
$$ \frac{SS_{G}}{(C-1)(R-1)} $$
$$ \frac{MS_{G}}{MS_{E}} $$
$$ $$
$$ Error $$
$$ SS_{E} $$
$$ N-C*R $$
$$ \frac{SS_{E}}{N-C*R} $$
$$ $$
$$ $$
$$ Total $$
$$ SS_{T} $$
$$ N-1 $$
$$ $$
$$ $$
$$ $$
$$ \displaylines{} $$
$$ \displaylines{Where,\;N=\;Number\;of\;Observations\;=\; 33 \\ \\ R=\;Number\;of\;Rows\;=\; 2 \\ \\ C=\;Number\;of\;Columns\;=\; 3 \\ \\ \mathbf{\color{Green}{Substitute\;values\;and\;make\;below\;table}} \\ \\ } $$
$$ Source\;of\;variation $$
$$ Sum\;of\;Squares $$
$$ df $$
$$ MS $$
$$ F $$
$$ p $$
$$ Rows $$
$$ 36.196524 $$
$$ 1 $$
$$ \frac{36.196524}{1}=36.196524 $$
$$ \frac{MS_{R}}{MS_{E}} $$
$$ $$
$$ Column $$
$$ 0.287878 $$
$$ 2 $$
$$ \frac{0.287878}{2}=0.143939 $$
$$ \frac{MS_{C}}{MS_{E}} $$
$$ $$
$$ within\;groups $$
$$ 5.468845 $$
$$ 2 $$
$$ \frac{5.468845}{2}=2.734422 $$
$$ \frac{MS_{G}}{MS_{E}} $$
$$ $$
$$ Error $$
$$ 92.228571 $$
$$ 27 $$
$$ \frac{92.228571}{27}=3.415873 $$
$$ $$
$$ $$
$$ Total $$
$$ 134.181818 $$
$$ 32 $$
$$ $$
$$ $$
$$ $$
$$ \displaylines{} $$
$$ \displaylines{F_{R}= \frac{MS_{R}}{MS_{E}} = \frac{36.196524}{3.415873} = 10.596566 \\ \\ F_{C}= \frac{MS_{C}}{MS_{E}} = \frac{0.143939}{3.415873} = 0.042138 \\ \\ F_{G}= \frac{MS_{G}}{MS_{E}} = \frac{2.734422}{3.415873} = 0.800505 \\ \\ \mathbf{\color{Red}{Final\;anova\;table}} \\ \\ } $$
$$ Source\;of\;variation $$
$$ Sum\;of\;Squares $$
$$ df $$
$$ MS $$
$$ F $$
$$ p $$
$$ Rows $$
$$ 36.196524 $$
$$ 1 $$
$$ 36.196524 $$
$$ 10.596566 $$
$$ 0.003046 $$
$$ Column $$
$$ 0.287878 $$
$$ 2 $$
$$ 0.143939 $$
$$ 0.042138 $$
$$ 0.9588 $$
$$ within\;groups $$
$$ 5.468845 $$
$$ 2 $$
$$ 2.734422 $$
$$ 0.800505 $$
$$ 0.459475 $$
$$ Error $$
$$ 92.228571 $$
$$ 27 $$
$$ 3.415873 $$
$$ $$
$$ $$
$$ Total $$
$$ 134.181818 $$
$$ 32 $$
$$ $$
$$ $$
$$ $$
$$ \displaylines{} $$
$$ \displaylines{\;\mathbf{\color{Red}{Conclusion}} \\ \\ \;\mathbf{\color{Green}{For\;Rows}} \\ \\ p\;=\;0.003046\;\;\;, p\;is\;less\;than\;\alpha\;.\; \\ \\ So,\;Reject\;H_{R0}.\;So\;means\;of\;Rows\;are\;not\;same \\ \\ \;\mathbf{\color{Green}{For\;Column}} \\ \\ p\;=\;0.958800\;\;\;, p\;is\;more\;than\;\alpha\;.\; \\ \\ So,\;Failed\;to\;Reject\;H_{C0}\;So\;means\;of\;Columns\;are\;same \\ \\ \;\mathbf{\color{Green}{For\;Groups}} \\ \\ p\;=\;0.459475\;\;\;, p\;is\;more\;than\;\alpha\;.\; \\ \\ So,\;Failed\;to\;Reject\;H_{G0}.\;There\;is\;no\;interaction\;between\;Rows\;and\;Columns } $$