Elitist Jerks Statistical error on PPM measurements

Summarizing experimental tests and statistical analyses of game mechanics
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# Statistical error on PPM measurements

Posted 03/05/09 at 12:27 PM by Rezarel
I wanted to nail down some proc rates and verify how they behave with haste rating and effects. In order to make the most efficient use of my time, I wanted to see how the error on a PPM measurement depends on weapon speed and other variables. Typically people recommend fast weapons for testing proc rates, but as it turns out slower weapons are slightly better for measuring PPM values.

Define:
PPM = Proc Per Minute value
WS = Weapon Speed
T = test length in seconds
P = Proc chance per swing (0<P<=1)
$\sigma_P$ = standard deviation of the distribution of measurements of P
$\sigma_{PPM}$ = standard deviation of the distribution of measurements of PPM

The true value for P is
$P= PPM \frac {WS}{60}$.
The distribution of test results will be centered around this value with a standard deviation of $\sigma_P$.

If we test for T seconds, we have
$Swings= \frac{T}{WS}$

Given the PPM value, we can predict the number of procs we observe. The expected value is
$Procs= P*Swings = PPM \frac{WS}{60} \frac{T}{WS} = PPM \frac{T}{60}$
as expected (procs per minute times minutes spent testing).

Then we can calculate our errors on the measurement.
$\sigma_{Procs}=\sqrt{Procs \frac{Swings-Procs}{Swings}}$

$\sigma_P= \sqrt{Procs \frac{Swings-Procs}{Swings} } \frac{1}{Swings}$

In the actual experiment, we won't know the true value of P, so we'll have to estimate $\sigma_P$ using our experimental value of Procs. Here however we can use the true value of P to find the standard deviation of the distribution.

$\sigma_P= \sqrt {PPM \frac{T}{60} \frac{ \frac{T}{WS} - PPM \frac{T}{60} }{\frac{T}{WS}}} \frac{WS}{T}$


To go from P to PPM:
$PPM= \frac{60}{WS} P$

so
$\sigma_{PPM}= \frac{60}{WS}\sigma_P$



For $\frac{PPM}{60} << 1$, the choice of weapon speed doesn't really matter. It slightly favors slow weapons though.

Also, the error in PPM scales like $T^{-\frac{1}{2}}$, as you might expect.

Now we can predict how much testing it will take to get 95% confidence intervals as small as we'd like. This will depend on the PPM value and our weapon speed. The 95% confidence interval is $PPM \pm 1.96\sigma_{PPM}$. If we'd like to reduce this to $\pm0.10$ or $\pm0.05$, then we have

 PPM WS 1/2 Desired Interval Width Time (hours) 1 1.3 .10 6.3 1 3.8 .10 6.0 1.2 1.3 .10 7.5 1.2 3.8 .10 7.1 15 1.3 .10 64.8 15 3.8 .10 4.8 1 1.3 .05 25.1 1 3.8 .05 24.0 1.2 1.3 .05 29.9 1.2 3.8 .05 28.4 15 1.3 .05 259.3 15 3.8 .05 19.2

Bring on the training dummies!