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