A win rate is an estimate
If you have won 55 of your last 100 games, your observed win rate is 55 percent. That is a fact about the games you played. It is only an estimate of your underlying chance of winning in a similar game, because luck affected each result. A different 100 games might have given 48 wins or 62.
A confidence interval puts a range around the estimate to show how much it could plausibly move. It is a way of saying “somewhere around here” with honest width.
The standard formula
For n games with an observed win proportion p, the standard error is:
√(p × (1 − p) / n)
A common approximate 95 percent interval is p plus or minus 1.96 times the standard error.
55 wins out of 100
- p = 0.55, so p × (1 − p) = 0.2475
- Divided by 100 is 0.002475, and the square root is about 0.0497
- Multiplying by 1.96 gives about 0.0975
The interval is about 0.55 plus or minus 0.0975, or roughly 45 percent to 65 percent. A fair-coin player, whose true rate is 50 percent, sits comfortably inside it. A 55 percent record over 100 games therefore does not show you are better than average.
550 wins out of 1,000
The same win rate with ten times the games gives a standard error of √(0.2475 / 1000), about 0.0157. Multiplying by 1.96 gives about 0.031. The interval is roughly 51.9 percent to 58.1 percent. Now 50 percent lies just outside it, which is suggestive, though still not a guarantee.
Note the pattern: ten times the games did not make the interval ten times narrower. It shrank by a factor of about 3.2, the square root of 10. To halve an interval’s width, you need roughly four times as many games.
What the interval does and does not mean
The usual reading is loose: “the true win rate is probably in this range.” The more careful statement is that if you repeated the same process many times, about 95 percent of intervals built this way would contain the true value. For practical purposes, treat the range as a sensible set of values the data cannot rule out.
It does not say each value inside is equally likely, and it does not cover mistakes. If your opponents changed halfway through or you only logged certain games, the interval is narrower than the real uncertainty.
Small samples need a better method
The simple formula works badly with few games or with win rates near 0 or 100 percent. With 10 games and 6 wins, it can give odd ranges, including values above 100 percent when the observed rate is high. A better choice is the Wilson interval, which stays within 0 to 1 and behaves more sensibly for small samples. For 6 wins in 10 games it gives about 31 percent to 83 percent, a range wide enough to show how little ten games establish.
In Python, the proportion_confint function in the statsmodels library offers method="wilson". In a spreadsheet, you can type the Wilson formula directly, or stick with the simple one while the sample is large enough that both agree.
How many games for a given width?
For a rough planning figure, the simple 95 percent margin of error at a 50 percent win rate is about 0.98 divided by √n. To land within plus or minus 5 percentage points you need about 385 games. For plus or minus 2 points, roughly 2,400. These are demanding numbers for anyone tracking personal results, and they explain why short-run results rarely match underlying skill.
Comparing two players or two decks
Two intervals that overlap do not necessarily mean no real difference, and non-overlapping ones are stronger evidence but not a formal test. If your question is whether two win rates differ, you need a test for the difference itself. That is a question of statistical significance, and how many games it takes is worked out separately.
Practical cautions
- Each game should be comparable. Mixed opponents and formats make “the” win rate ambiguous.
- Games should be roughly independent. Streaks caused by tilt or learning within a session break that assumption.
- Do not stop when the interval looks good. Choose the sample size first.
- Remember draws. If draws occur, decide whether to count them as half a win or exclude them, and say which.
An interval is not a verdict about skill. It is a reminder that any single record is one sample from a noisy process, and that the honest answer to “how good am I?” often has a wide margin attached.