Gotta account for a null hypothesis.

The null would be that it is a fair die (average roll 10.5). Your test is whether the true result is significantly less than 10.5 based on a sample of 100 with a mu of 8.8. Let’s call it an alpha of 0.05

So we have to run a left tail one-sample t-test.

Unfortunately, this data set doesn’t tell me the standard deviation – but that could be determined in future trials. For now, we’ll have to just make an assumption that the D20 is fair. For a fair D20, the standard deviation should be be sqrt( ([20-1+1]^2 -1)/12) or roughly sqrt(33.25)

We can run that t-test in a simply python script:

import numpy as np
from scipy import stats as st

h0 = 10.5

sample = np.random.normal(loc=8.88, scale=(np.sqrt(33.25)), size=100)

t_stat, p_val = st.ttest_1samp(sample, h0)

print(f"T-statistic: {t_stat:.4f}")
print(f"P-value: {p_val:.4f}")

Of course, I had to randomize this a bit since I don’t have access to the full data from the true sample. That would make for a better bit of analysis. But at least assuming I didn’t make a serious dumb here, 100 rolls averaging 8.88 would seem to suggest that the you can reject your null hypothesis of this being a fair die at a typical alpha of 0.05.

Then again, the way I wrote this script is GUARANTEED to be an unfavorable result since the way I randomized it REQUIRES the average end up 8.88, which is, of course, less than 10.5. Your real world testing would not have this constraint.

100 rolls is a staggeringly small sample size for a set with 20 possible outcomes, I would take this with a grain of salt (or a lot of salt in a glass of water? ;) )

In any case it’s fun to do some experimenting like this, thanks for sharing!