I don't think Akshatha Banadka is asking how to carry out the ANOVA computations by hand. Rather, she is asking how to compute the p-value by hand. Is that right? If so, read on.

In order to compute the p-value for an F-test by hand, you would have to integrate the F-distribution (with appropriate df) from the observed F-value to positive infinity. Do you really want to do that? You could instead use an online calculator like this one:

• https://onlinestatbook.com/2/calculators/F_dist.html

And if you are using one of the major stats packages, it will no doubt have a function you can use. E.g., Stata has a function called Ftail(df1,df2,F) that returns the right tail probability. Suppose That F = 1.395, df_num=2, and df_denom=15.

. display Ftail(2,15,1.395)

.27820494

The calculator linked above returns the same result.

HTH.

PS- Although I do not like to encourage use of Excel for serious statistical computations, it also has a function that you could use:

=F.DIST.RT(1.395,2,15)

Result: 0.27820494

ANOVA can be done manually. For instance consider a one-way case. The total (corrected) sum of squares is given by the total (uncorrected) sum of square minus the correction factor. For the balanced case, the correction factor is equal to the grand total squared divided by number of observations. The sum of squares due to the treatment is equal to the sum of the treatment total squared divided by the number of replicates per treatment minus the correction factor. The sum of squares due to error is given by total (corrected) sum of squares minus the sum of squares due to treatment. The three mean squares are each the sum of squares divided by the degree of freedom. F statistic is derived by the treatment mean square divided by error mean square. The p-value may be guessed by the value of F obtained with the tabulated F with levels of significance 0.01 and 0.05.

1.Calculate the sample means for each of our samples as well as the mean for all of the sample data.

2.Calculate the sum of squares of error. Here within each sample, we square the deviation of each data value from the sample mean. The sum of all of the squared deviations is the sum of squares of error, abbreviated SSE.

2.Calculate the sum of squares of treatment. We square the deviation of each sample mean from the overall mean. The sum of all of these squared deviations is multiplied by one less than the number of samples we have. This number is the sum of squares of stimulus, abbreviated SST.

4.Calculate the degrees of freedom. The overall number of degrees of freedom is one less than the total number of data points in our sample, or n - 1. The number of degrees of freedom of treatment is one less than the number of samples used, or m - 1. The number of degrees of freedom of error is the total number of data points, minus the number of samples, or n - m.

5.Calculate the mean square of error. This is denoted MSE = SSE/(n - m).

Calculate the mean square of treatment. This is denoted MST = SST/m - `1.

6.Calculate the F statistic. This is the ratio of the two mean squares that we calculated. So F = MST/MSE.

I don't think Akshatha Banadka is asking how to carry out the ANOVA computations by hand. Rather, she is asking how to compute the p-value by hand. Is that right? If so, read on.

In order to compute the p-value for an F-test by hand, you would have to integrate the F-distribution (with appropriate df) from the observed F-value to positive infinity. Do you really want to do that? You could instead use an online calculator like this one:

• https://onlinestatbook.com/2/calculators/F_dist.html

And if you are using one of the major stats packages, it will no doubt have a function you can use. E.g., Stata has a function called Ftail(df1,df2,F) that returns the right tail probability. Suppose That F = 1.395, df_num=2, and df_denom=15.

. display Ftail(2,15,1.395)

.27820494

The calculator linked above returns the same result.

HTH.

PS- Although I do not like to encourage use of Excel for serious statistical computations, it also has a function that you could use:

=F.DIST.RT(1.395,2,15)

Result: 0.27820494

For anova, the p-value is based on the probability density function (PDF) for the F distribution for the degrees of freedom in the analysis, in comparison to the calculated F value from the analysis.

In theory, you could calculate the PDF for the F distribution of interest, and then determine the p-value from the calculated F value. (See for example, https://en.wikipedia.org/wiki/F-distribution)

In reality, someone a long time ago already did this, and no one wants to do this ever again.

Instead, we old folks used tables to look up the critical F values for, say, a p-value of 0.05 or 0.01. You can find these tables.

And now, these tables are stored to a precise degree in computer software, making the task even easier. For example, you could go to https://rdrr.io/snippets/ , and run the following in R. This will give you the p-value for a calculated F value of 5.77, compared with an F distribution with degrees of freedom 24 and 4. It will return a value of about 0.05. This will correspond to a table value of critical F values for p=0.05.

pf(5.77, 24, 4, lower.tail=FALSE)

You can also find the formulae to calculate the F-value for anova. When I was in graduate school, we used to do this with a hand calculator; it's not too difficult for simple designs. But you'll still want to look up the critical F values or the p-value by table or computer.