I asked AI for help and this is what I got:
My Question to AI:
As a statistician, how would I determine which prognosticator is more accurate than the others? For example, there are three prognosticators trying to predict how many wins a football team will win each year over the next three years. Three years later, we see that the team won 4 games in year one, 4 games in year two, and 7 games in year three for a total of 15 games. Prognosticator #1 guessed 1 game in year one, 2 games in year two, and 12 games in year three for a total of 15 games. Prognosticator #2 guessed 5 games won in each year for a total of 15 games. Prognosticator #3 guessed 4 games, 3 games, and 8 games for a total of 15 games. Who was more accurate?
AI's Answer:
puts on statistician hat
To determine which prognosticator is more accurate in predicting the number of wins for this football team over 3 years, we need to look beyond just the total predicted wins. The distribution and variability of their predictions across the 3 years matter.
One metric I would use is the standard deviation of the prediction errors. This measures how much, on average, the predictions deviated from the actual outcomes each year.
Let's calculate the prediction errors for each prognosticator per year:
Year 1 (Actual: 4 wins)
Prognosticator 1 Error: 4 - 1 = 3
Prognosticator 2 Error: 4 - 5 = -1
Prognosticator 3 Error: 4 - 4 = 0
Year 2 (Actual: 4 wins)
Prognosticator 1 Error: 4 - 2 = 2
Prognosticator 2 Error: 4 - 5 = -1
Prognosticator 3 Error: 4 - 3 = 1
Year 3 (Actual: 7 wins)
Prognosticator 1 Error: 7 - 12 = -5
Prognosticator 2 Error: 7 - 5 = 2
Prognosticator 3 Error: 7 - 8 = -1
Now to find the standard deviation of these errors for each:
Prognosticator 1 Std Dev = sqrt((3^2 + 2^2 + (-5)^2)/3) = 3.51
Prognosticator 2 Std Dev = sqrt((-1)^2 + (-1)^2 + 2^2)/3) = 1.63
Prognosticator 3 Std Dev = sqrt(0^2 + 1^2 + (-1)^2)/3) = 0.82
The lower the standard deviation, the more accurate and consistent the predictions were overall.
So based on the standard deviation of prediction errors, Prognosticator 3 with a std dev of 0.82 was the most accurate, followed by Prognosticator 2 (1.63), then Prognosticator 1 (3.51). Even though they all predicted the same total wins, the distribution of Prognosticator 3's yearly predictions more closely matched the actual outcomes.