One way Analysis of Variance (ANOVA) is a statistical technique that is used to compare the means of more than two groups. One way ANOVA is a part of the ANOVA family. When we are comparing the means of more than two populations based on a single treatment factor, then it said to be one way ANOVA. For instance, we can use it if we have an output of three types of workers and we want to compare whether or not the mean output of the three types of workers is the same based on their working hours (40 or less vs. 41+ hours). Here, a working hour is only a single factor. This is therefore one way ANOVA. If we are comparing the means of workers based on their working hours and working conditions, then it said to be the two-way ANOVA. Before conducting the analysis, we should consider some assumptions that one way ANOVA should meet for true comparisons of means.
Assumptions:
1.In one way ANOVA, the population in which samples are drawn should be normally distributed.
2.Independent of case: In one way ANOVA, sample cases should be independent of each other.
3.Homogeneity: In one way ANOVA, Homogeneity means that the variance between the groups should be approximately equal.
Hypothesis In one way ANOVA:
Null hypothesis: In one way ANOVA, null hypothesis assumes that the means of all the groups are equal.
Alternative hypothesis: In one way ANOVA, alternative hypothesis assumes that the means of all the groups are not equal.
Grand mean in one way ANOVA:
Grand mean is calculated in one way ANOVA by dividing the sum of all the data values by the total sample size. Mathematically grand mean is written as:
Where 
= Grand mean
= sum of all data value
N = total number
Total variation:
In one way ANOVA, total variance is the sum of the squares of the differences of each mean with the grand mean. Mathematically we can write it as follows:
Where:
SS (T) = total variance
= grand mean
x = individual mean
Between group variance: In one way ANOVA, between group variance is equal to the sum of the square of the individual sample mean difference from the grand mean. In one way ANOVA, between group variance is denoted by SS (B). In one way ANOVA, there are k samples involved in analysis, and then the degree of freedom will be equal to k-1. Mathematically SS (B) can be written as:
Where
SS(B) = Between group variance
Within group variance: In one way ANOVA, within group variance is the individual variance due to the sample. In one way ANOVA, within group variance is denoted by SS (W). SS (W) can be computed by subtracting the between group variance from the total variance. In SS (W), the degree of freedom will be equal to N-k where N is the number of sample size.
F-test: F-statistics is used to compare the means in one way ANOVA. F-statistics is the ratio of MS between the group variance and MS within the group variance. The following table shows how the F-test is calculated:
Analysis of variance table:
Decision Rule: If the calculated critical value is greater than the table value, then we reject the null hypothesis and conclude that there is a difference between the group means. If the calculated critical value is less than the table value, then we will conclude that there is no difference between the group means. In one way ANOVA, most researchers consider the probability value for decision making.
One way ANOVA in SPSS: We can perform one way ANOVA in SPSS by clicking on “analysis” from menu, then selecting “compare mean,” and clicking on “one way ANOVA.”
ANOVA Resources
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