Methods and formulas for the confidence intervals in One-Way ANOVA

Select the method or formula of your choice.

Mean

Formula

The average of the observations at a given factor level.

Notation

TermDescription
ni number of observations at factor level i
yij value of the j th observation at the i th factor level

Standard deviation (StDev)

Notation

TermDescription
yij observations at the i th factor level
mean of observations at the i th factor level
ni number of observations at the i th factor level

Pooled standard deviation

Formula

The common variance for all observations. The pooled variance is:
The pooled standard deviation is the square root of the above formula. An equivalent form follows:

The pooled standard deviation is equivalent to S, which is in the output and is equal to:

Notation

TermDescription
yij j th observation of the response for the i th factor level
sample mean for factor level i
ni number of observations for the ith factor level
nT total number of observations
si 2 variance of observations
r number of levels of the factor
MSMean Square

Individual confidence intervals

Formula

The confidence intervals are calculated for each factor level mean using the pooled standard deviation. The formula is:

Notation

TermDescription
nT total number of observations
r number of factor levels
S pooled standard deviation
inverse cumulative distribution function from the t distribution at 1− α/2 with nTr degrees of freedom
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