Probability looks forward, statistics looks back
Probability starts with a known setup, such as a fair die, and says what should happen. Statistics starts with results you have actually seen and asks what they suggest. Players use both: probability to understand how a game is built, and statistics to check what really happened across a session, a season or a thousand matches.
The tools below describe a set of results in a few numbers. They are the sample statistics most often used in game analysis.
The mean: where results centre
The mean, or average, is the sum of the results divided by how many there are. If you roll a die five times and get 2, 6, 3, 3 and 5, the total is 19 and the mean is 19/5 = 3.8.
For a fair six-sided die the long-run mean is 3.5, which is the average of 1 through 6. You will never roll a 3.5, yet your results gather around it over time. The difference between the 3.8 above and the theoretical 3.5 is just the ordinary wobble of a small sample.
The median and mode
The median is the middle value when results are sorted, and the mode is the most common one. They matter when a few extreme results distort the mean. If most monster kills in a game yield 10 coins but one rare drop yields 1,000, the mean looks generous while the median shows what a typical kill gives you. Reporting both is more honest than reporting either alone.
Variance and standard deviation: how spread out results are
Two games can have the same mean and feel completely different. One pays out a small steady amount; the other pays nothing most of the time and a lot occasionally. The difference is spread, and variance measures it.
To find variance, take each result’s distance from the mean, square it, and average those squares. For a fair die:
- Squared distances from 3.5: 6.25, 2.25, 0.25, 0.25, 2.25, 6.25
- Their sum is 17.5, and dividing by 6 gives a variance of about 2.92 (exactly 35/12)
- The standard deviation is the square root, about 1.71
Standard deviation is easier to interpret because it is in the same units as the results. Here it says a typical roll sits roughly 1.7 away from the mean of 3.5.
Samples, populations and why size matters
A population is every result that could ever occur; a sample is the subset you observed. The sample mean is an estimate of the population mean, and its reliability depends mostly on sample size.
A useful rule is that the typical error of a sample mean is the standard deviation divided by the square root of the number of observations. For the die:
- With 25 rolls, 1.71 ÷ 5 ≈ 0.34
- With 100 rolls, 1.71 ÷ 10 ≈ 0.17
So after 25 rolls, a mean anywhere from roughly 2.8 to 4.2 would not be surprising. After 100 rolls it should typically sit within about 0.34 of 3.5, using a rough two-error band. Quadrupling the sample only halves the error, which is why very large samples are needed to detect small differences.
Using these numbers on real play
When you track your own results, the pattern you notice is often noise. A run of six good sessions can come from pure chance, particularly in games with a large random component. Before concluding that a change in approach worked, ask how big the spread is compared with the change you saw. If the improvement is smaller than the typical sampling error, you cannot tell much yet.
This is also why pattern recognition needs care. Humans are quick to see streaks and cycles in sequences that have none, and small samples make it worse. Checking whether a pattern persists in a fresh set of data is more convincing than any single impressive run.
Online trends are a good example. If a character, weapon or deck appears to dominate for one weekend, the first questions are how many matches the claim rests on and how much those results usually vary. A shift supported by thousands of matches deserves more attention than one supported by fifty.
Some cautions
Statistics describe what happened; they do not guarantee what will happen next. Averages hide shape, so a mean should be read alongside the spread, and where a game is built to favour the house or the designer, repeating it many times tends to reveal that edge rather than erase it. The mean also depends on the sample being fair: results gathered only when you felt confident, or only from a skilled subset of players, will not describe the whole.
Used carefully, these few measures let you replace a gut impression with something checkable. They will not tell you which result comes next, but they will tell you how much a collection of results should be trusted, and that is usually the more valuable answer.