Is a goal equally likely in any minute? The Uniform distribution

The Uniform distribution says every outcome is equally likely. It's the fairest-sounding model in statistics, and testing it against 28,016 goals shows football isn't that fair.

27 September 2026 Beginner Part 11 of Statistics Through Football

Contents

The football question
The concept
A football example
Testing it against real football
Where the Uniform really is right
Why it matters
Limitations
Try it yourself
Further reading

The football question

Is a goal just as likely in the 5th minute as the 85th?

If it were, the time of a goal would follow the simplest distribution in statistics, where every moment is equally likely. It's worth knowing, partly because it's the starting point for so much else, and partly because checking it against real matches tells you something about how football works.

The concept

The Uniform distribution gives every outcome in a range the same chance. It needs only the two ends of the range:

a, the lowest possible value.
b, the highest.

It comes in two versions:

Discrete: a fixed number of equally likely outcomes. A fair cup draw, where your team is equally likely to get any of the other 15 sides left in: a 1 in 15 chance of each.
Continuous: any value in a range, all equally likely. The minute of a goal, if goals really could come at any time with equal chance.

A football example

Suppose a goal is equally likely at any moment of the 90 minutes. Then the chance it falls in any stretch of the match is just that stretch's share of the 90:

$$\begin{aligned} &P(\text{goal between } c \text{ and } d) \\ &= \frac{d - c}{b - a} = \frac{d - c}{90} \end{aligned}$$

In plain football

b − a = 90 is the whole match.
d − c is the stretch you're asking about. The last 15 minutes is 15.
15 ÷ 90 = 1/6: under this model, one goal in six would come in the last quarter of an hour, and exactly half of all goals in each half.

The average time of a goal would be the middle of the match:

$$\text{mean} = \frac{a + b}{2} = \frac{0 + 90}{2} = 45 \text{ minutes}$$

Testing it against real football

The Uniform model makes a clear prediction: 50% of goals in the first half, 50% in the second. The results files record the half-time score, so this can be checked.

Across 10,471 Scottish Premiership and Championship matches since 2000/01, there were 28,016 goals:

	Share of goals
First half	44.3%
Second half	55.7%

That's not a small wobble. With 28,000 goals, chance alone would keep the split within about half a percentage point of 50%. The second half's 55.7% is 19 standard errors away. Goals are not equally likely in every minute.

Why? A few reasons all push the same way:

Tired legs. Defending well takes concentration and energy, and both run down.
Game state. A side that's behind throws players forward, which opens the game up at both ends.
More stoppage time. The second half usually has more added on, so it's slightly longer.

So the Uniform isn't the right model for when goals come. It's the model you start from, then test, then improve on. The Exponential distribution and Poisson distribution articles make the same simplifying assumption of a steady rate, and say so in their limitations.

Where the Uniform really is right

Random draws. A properly run cup draw gives every remaining side the same chance of every opponent.
The coin toss. Heads or tails, each with a half.
Simulation. Computers simulate football by generating random numbers, and those start life as uniform numbers between 0 and 1. When the league table myth replayed each season 400 times, every simulated result began as one. For an average side, a number below 0.38 became a win, one between 0.38 and 0.62 a draw, and anything higher a defeat.

Show the mathsThe density, mean and spread, and the 19 standard errors. Optional.

For the continuous Uniform on $[a, b]$, the density is flat:

$$f(x) = \frac{1}{b - a}, \qquad a \le x \le b$$

Its mean is $(a + b)/2$ and its variance $(b - a)^2/12$. For a 90-minute match, that's a mean of 45 minutes and a standard deviation of $90/\sqrt{12} \approx 26$ minutes.

For the goals test, if each goal independently fell in the second half with probability 0.5, the proportion in the second half would have standard error

$$\sqrt{\frac{0.5 \times 0.5}{28{,}016}} \approx 0.0030$$

The observed 0.557 is $(0.557 - 0.5)/0.0030 \approx 19$ standard errors above 0.5.

Why it matters

The Uniform distribution is the plainest statement of "no pattern at all". That makes it the natural baseline: if real data matches it, there's nothing to explain; if it doesn't, the gap is where the football is. It's also the engine room of simulation, which is how analysts explore questions too messy for a formula.

Limitations

Half-time scores are a coarse test. They show more goals after the break, but not which minutes. Minute-by-minute data would show the pattern in more detail.
Stoppage time makes the halves unequal. Part of the second-half excess is simply extra minutes.
"Random" draws aren't always uniform. Seeding, and rules that keep certain teams apart, change the chances.

Try it yourself

Take your team's last season of goals, scored and conceded, and count how many came in each 15-minute spell. Under the Uniform, each spell should hold about one sixth. Which spells hold more?