The goal of an ANOVA is to be able to report that you compared how different independent variables affected a dependent variable, under specific conditions. You’ll end up stating your statistical evidence in this way:
“A one-way between-subjects ANOVA was conducted to compare the effect of (Ind. Variable)______________ on (Dependent Variable)_______________ in ______ (list the conditions).”
E.g. “A one-way ANOVA was conducted to compare the effect of average streaming distance (in phons) between two unpulsed streams segregated by loudness, on participants’ confidence in hearing a pulsed high or low stream segregated by pitch. The conditions were .8 phons, 2.5 phons, and 4.2 phons [F(1,20) = 25.65, p <.0001].”
I constructed that assertion from the following steps. My data consisted of a list of the loudness distances of 12 stimuli (12 rows, each one with the number .8, the number 2.5, or the number 4.2 in it), and 12 “average participant responses” (12 rows, each with an average of the 5 participants’ subjective 1-5 rankings of confidence in it.
Eager to understand the significance of these numbers, I navigated to http://vassarstats.net, and sure enough, their server was online. (I think it helps to keep your fingers crossed, though I can’t demonstrate that statistically.)
1. Preparation: I selected ANOVA in the left column, entered “2” in “Number of samples…”, selected “Independent Samples,” and chose a “Weighted” analysis. (These are the most common procedures; I’ll refer you to literature on their distinctions, and special cases, in our individual meetings, if these become important.)
2. Data: Then I entered the list of loudness distances in the column marked “Sample 1,” and the list of “average participant responses” in the column marked “Sample 2.” (These are the two columns whose variance you hope will be correlated.) Then I clicked “calculate.”
To report the analysis:
1. I stated my “degrees of freedom” in parentheses after the enigmatic letter “F.” The first degree is 1 (number of samples, minus 1); the second is 20 (the number of scores in all samples, minus the number of samples), so it reads “F(1,20).”
2. I stated the “F” value (F = the ratio of the squares of the treatment variance to the squares of the effect variance^1), which is listed under “F” in the “Treatment” row: = 25.65.
3. The “p” value(<.0001, which is the chance arrangement of numbers could have happened randomly) —Vassar shows it as a “less than” statement if it’s low enough to be considered “significant,” or as an “equals” statement if it’s not. That’s the value that counts!
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^1. The F value is MSR/MSE where MSR (the “model mean square”) is the sum of the squares of the deviations of the independent variable from its mean, and MSE (the “error mean square”) is the same measurement of the deviations of the dependent variable from its mean. In other words, it’s how much variance you find in the conditions you’re applying to the situation, divided by how much variance you find that isn’t related to those conditions. A high number suggests that the dependent variable is responding more to those variable conditions that you’re testing (the independent), than to all those unrelated, and unknown factors, that make the relationship less than perfect. It’s called an F value because it’s named after Ronald Fisher, the genetic and evolutionary biologist who invented it.