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One way ANOVA with tukey test calculator


ANOVA is analysis of variance. There are many types of ANOVA test. This calculator is One way ANOVA calculator. ANOVA Table is provided at the end of this solution. The post hoc test we are using is tukey test. the most used post hoc test is Tukey's HSD. Every step is provided as if it is solved by hand. You can learn how to calculate a one-way ANOVA by submitting any sample values. F statistic and the p-value is calculated and shown in Table

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

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$$ \displaylines{---} $$
$$ \displaylines{} $$
$$ Treatment \; no $$
$$ 1 $$
$$ 2 $$
$$ 3 $$
$$ $$
$$ 1.0 $$
$$ 1.0 $$
$$ 3.0 $$
$$ $$
$$ 2.0 $$
$$ 3.0 $$
$$ 6.0 $$
$$ $$
$$ 3.0 $$
$$ 4.0 $$
$$ 7.0 $$
$$ $$
$$ 4.0 $$
$$ - $$
$$ - $$
$$ $$
$$ 5.0 $$
$$ - $$
$$ - $$
$$ Total $$
$$ 15.0 $$
$$ 8.0 $$
$$ 16.0 $$
$$ \displaylines{} $$
$$ \displaylines{H_{0}:\;there\;is\;no\;difference\;in\;means \\ \\ H_{a}:\;At\;least\;2\;means\;differ \\ \\ \mathbf{\color{Green}{First\;we\;have\;to\;find\;Total\;mean}} \\ \\ Total\;Mean\;= \frac{Total}{n} \\ \\ \Rightarrow \frac{15.0+8.0+16.0}{11} \\ \\ \Rightarrow \frac{39.0}{11} \\ \\ \Rightarrow 3.545455 \\ \\ \\ \\ \mathbf{\color{Green}{Now\;create\;below\;table\;}} \\ \\ } $$
$$ x $$
$$ \bar{x} $$
$$ x-\bar{x} $$
$$ [x-\bar{x}]^{2} $$
$$ 1.0 $$
$$ 3.545455 $$
$$ 2.545455 $$
$$ 6.479341 $$
$$ 2.0 $$
$$ 3.545455 $$
$$ 1.545455 $$
$$ 2.388431 $$
$$ 3.0 $$
$$ 3.545455 $$
$$ 0.545455 $$
$$ 0.297521 $$
$$ 4.0 $$
$$ 3.545455 $$
$$ -0.454545 $$
$$ 0.206611 $$
$$ 5.0 $$
$$ 3.545455 $$
$$ -1.454545 $$
$$ 2.115701 $$
$$ 1.0 $$
$$ 3.545455 $$
$$ 2.545455 $$
$$ 6.479341 $$
$$ 3.0 $$
$$ 3.545455 $$
$$ 0.545455 $$
$$ 0.297521 $$
$$ 4.0 $$
$$ 3.545455 $$
$$ -0.454545 $$
$$ 0.206611 $$
$$ 3.0 $$
$$ 3.545455 $$
$$ 0.545455 $$
$$ 0.297521 $$
$$ 6.0 $$
$$ 3.545455 $$
$$ -2.454545 $$
$$ 6.024791 $$
$$ 7.0 $$
$$ 3.545455 $$
$$ -3.454545 $$
$$ 11.933881 $$
$$ Total $$
$$ $$
$$ $$
$$ 36.727271 $$
$$ \displaylines{} $$
$$ \displaylines{\\ \\ From\;the\;table\;we\;can\;get\;SS_{Total} \\ \\ Sum\;of\;square_{Total}= \mathbf{\color{Red}{36.727271}} \\ \\ \mathbf{\color{Green}{Now\;do\;this\;separately\;for\;each\;Treatment}} \\ \\ } $$
$$ \displaylines{ \mathbf{\color{Green}{Calculating\;for\;Treatment\;1}} \\ \\ Mean\;of1\;th\;Treatment\;= \frac{Total\;of1\;th\;Treatment\;}{n} \\ \\ \Rightarrow \frac{15.0}{5} \\ \\ \Rightarrow 3.0 \\ \\ } $$
$$ x $$
$$ \bar{x} $$
$$ x-\bar{x} $$
$$ [x-\bar{x}]^{2} $$
$$ 1.0 $$
$$ 3.0 $$
$$ 2.0 $$
$$ 4.0 $$
$$ 2.0 $$
$$ 3.0 $$
$$ 1.0 $$
$$ 1.0 $$
$$ 3.0 $$
$$ 3.0 $$
$$ 0.0 $$
$$ 0.0 $$
$$ 4.0 $$
$$ 3.0 $$
$$ -1.0 $$
$$ 1.0 $$
$$ 5.0 $$
$$ 3.0 $$
$$ -2.0 $$
$$ 4.0 $$
$$ Total $$
$$ $$
$$ $$
$$ 10.0 $$
$$ \displaylines{} $$
$$ \displaylines{\\ \\ Sum\;of\;square \;of\;Treatment\;No\;1\;= 10.0 \\ \\ \mathbf{\color{Green}{Calculating\;for\;Treatment\;2}} \\ \\ Mean\;of2\;th\;Treatment\;= \frac{Total\;of2\;th\;Treatment\;}{n} \\ \\ \Rightarrow \frac{8.0}{3} \\ \\ \Rightarrow 2.666667 \\ \\ } $$
$$ x $$
$$ \bar{x} $$
$$ x-\bar{x} $$
$$ [x-\bar{x}]^{2} $$
$$ 1.0 $$
$$ 2.666667 $$
$$ 1.666667 $$
$$ 2.777779 $$
$$ 3.0 $$
$$ 2.666667 $$
$$ -0.333333 $$
$$ 0.111111 $$
$$ 4.0 $$
$$ 2.666667 $$
$$ -1.333333 $$
$$ 1.777777 $$
$$ Total $$
$$ $$
$$ $$
$$ 4.666667 $$
$$ \displaylines{} $$
$$ \displaylines{\\ \\ Sum\;of\;square \;of\;Treatment\;No\;2\;= 4.666667 \\ \\ \mathbf{\color{Green}{Calculating\;for\;Treatment\;3}} \\ \\ Mean\;of3\;th\;Treatment\;= \frac{Total\;of3\;th\;Treatment\;}{n} \\ \\ \Rightarrow \frac{16.0}{3} \\ \\ \Rightarrow 5.333333 \\ \\ } $$
$$ x $$
$$ \bar{x} $$
$$ x-\bar{x} $$
$$ [x-\bar{x}]^{2} $$
$$ 3.0 $$
$$ 5.333333 $$
$$ 2.333333 $$
$$ 5.444443 $$
$$ 6.0 $$
$$ 5.333333 $$
$$ -0.666667 $$
$$ 0.444445 $$
$$ 7.0 $$
$$ 5.333333 $$
$$ -1.666667 $$
$$ 2.777779 $$
$$ Total $$
$$ $$
$$ $$
$$ 8.666667 $$
$$ \displaylines{} $$
$$ \displaylines{\\ \\ Sum\;of\;square \;of\;Treatment\;No\;3\;= 8.666667 \\ \\ Total\;SS_{Within-treatments}\;=\; 10.0+4.666667+8.666667 \\ \\ \Rightarrow \mathbf{\color{Red}{23.333334}} \\ \\ df_{Between-treatments}=k-1= 3 -1 \\ \\ \Rightarrow 2 \\ \\ df_{Within-treatments}=N-k= 11 - 3 \\ \\ \Rightarrow 8 \\ \\ df_{Total}=N-1= 11 -1 \\ \\ \Rightarrow 10 \\ \\ SS_{Between-treatments} = 36.727271-23.333334 \\ \\ \Rightarrow 13.393937 \\ \\ MS_{Between-treatments}= \frac{SS_{Between-treatments}}{df_{Between-treatments}} \\ \\ \Rightarrow \frac{13.393937}{2} \\ \\ \Rightarrow 6.696968 \\ \\ MS_{Within-treatments}= \frac{SS_{Within-treatments}}{df_{Within-treatments}} \\ \\ \Rightarrow \frac{23.333334}{8} \\ \\ \Rightarrow 2.916667 \\ \\ F = \frac{MS_{Between-treatments}}{MS_{Within-treatments}} \\ \\ \Rightarrow \frac{6.696968}{2.916667} \\ \\ \Rightarrow \mathbf{\color{Red}{2.296103}} \\ \\ p\;value\;is\;\;\mathbf{\color{Red}{0.162912}} \\ \\ \\ \\ \mathbf{\color{Red}{ANOVA\;table\;given\;below}} } $$
$$ Sourse $$
$$ Sum\;of\;squares $$
$$ df $$
$$ MS $$
$$ F\;test $$
$$ p\;value $$
$$ Between\;SS $$
$$ 13.393937 $$
$$ 2 $$
$$ 6.696968 $$
$$ 2.296103 $$
$$ 0.162912 $$
$$ Within\;SS $$
$$ 23.333334 $$
$$ 8 $$
$$ 2.916667 $$
$$ $$
$$ $$
$$ Total $$
$$ 36.727271 $$
$$ 10 $$
$$ $$
$$ $$
$$ $$
$$ \displaylines{} $$
$$ \displaylines{\alpha\;=\;0.05 \\ \\ p\;is\;more\;than\;\alpha\;.\; So,\;Failed\;to\;Reject\;H_{0} } $$
$$ \displaylines{} $$
$$ \displaylines{} $$
$$ \displaylines{ \mathbf{\color{Red}{Post\;Hoc\;Tukey\;HSD\;test}} \\ \\ \mathbf{\color{Green}{First\;we\;have\;to\;find\;difference\;of\;means\;of\;every\;pair\;possible}} \\ \\ } $$
$$ Pairwise\;Comparisons $$
$$ Mean\;of\;pair1\;(x_{i}) $$
$$ Mean\;of\;pair2\;(x_{j}) $$
$$ (x_{i}-x_{j}) $$
$$ 1nd\;T\;:\;2nd\;T $$
$$ 3.0 $$
$$ 2.666667 $$
$$ -0.333333 $$
$$ 1nd\;T\;:\;3nd\;T $$
$$ 3.0 $$
$$ 5.333333 $$
$$ 2.333333 $$
$$ 2nd\;T\;:\;3nd\;T $$
$$ 2.666667 $$
$$ 5.333333 $$
$$ 2.666667 $$
$$ \displaylines{} $$
$$ \displaylines{ \mathbf{\color{Green}{Now\;we\;have\;to\;calculate\;confidence\;interval}} \\ \\ \alpha = 0.05 \\ \\ Formula\;for\;confidence\;interval\;=\; x_{i} - x_{j} \pm q*\sqrt{\frac{MS_{within}}{2}(\frac{1}{n_{i}}+\frac{1}{n_{j}}) } \\ \\ \\ \\ Where\;q\;=\; q_{\alpha,k,N-k}\;=\; q_{0.05,3,8}\;=\; 4.037484 \\ \\ MS_{within}\;=\; 2.916667 \\ \\ \Rightarrow x_{i} - x_{j} \pm4.037484\sqrt{\frac{2.916667}{2}(\frac{1}{n_{i}}+\frac{1}{n_{j}}) } \\ \\ \Rightarrow confidence\;interval\;=\; x_{i} - x_{j} \pm4.875725\sqrt{\frac{1}{n_{i}}+\frac{1}{n_{j}} } \\ \\ \mathbf{\color{Green}{Now\;create\;table\;for\;calculating\;confidence\;interval\;using\;above\;formula}} \\ \\ } $$
$$ Pairwise\;Comparisons $$
$$ (x_{i}-x_{j}) $$
$$ n_{i} $$
$$ n_{j} $$
$$ confidence\;interval $$
$$ 1nd\;T\;:\;2nd\;T $$
$$ -0.333333 $$
$$ 5 $$
$$ 3 $$
$$ -3.89406\;,\;3.227393 $$
$$ 1nd\;T\;:\;3nd\;T $$
$$ 2.333333 $$
$$ 5 $$
$$ 3 $$
$$ -1.227393\;,\;5.89406 $$
$$ 2nd\;T\;:\;3nd\;T $$
$$ 2.666667 $$
$$ 3 $$
$$ 3 $$
$$ -1.314346\;,\;6.64768 $$
$$ \displaylines{} $$
$$ \displaylines{ \mathbf{\color{Red}{Post\;Hoc\;Tukey\;HSD\;Table}} \\ \\ } $$
$$ Pairwise $$
$$ x_{i}\;-\;x_{j} $$
$$ Confidence\;interval $$
$$ p\;value $$
$$ Result $$
$$ 1nd\;T\;:\;2nd\;T $$
$$ -0.333333 $$
$$ -3.89406\;,\;3.227393 $$
$$ 0.9 $$
$$ Falied\;to\;Reject $$
$$ 1nd\;T\;:\;3nd\;T $$
$$ 2.333333 $$
$$ -1.227393\;,\;5.89406 $$
$$ 0.208377 $$
$$ Falied\;to\;Reject $$
$$ 2nd\;T\;:\;3nd\;T $$
$$ 2.666667 $$
$$ -1.314346\;,\;6.64768 $$
$$ 0.19668 $$
$$ Falied\;to\;Reject $$
$$ \displaylines{} $$
$$ \displaylines{} $$