Statistics help us make sense of the world, especially in sports. Imagine a striker's goal record over their last five games: 2, 0, 1, 1, 3. How do we summarise this performance with a single number? We can use measures of central tendency, which describe the 'typical' value in a set of data.

To find our striker's mean goals per game, we add up all the goals and divide by the number of games played.

The mean is useful, but it can be skewed by unusually high or low numbers, known as outliers. If our striker had scored 7 goals in one of those games, the mean would jump, which might not accurately reflect their typical performance.

That's where the median comes in.

Let's arrange our striker's goals in order: 0, 1, 1, 2, 3. The number in the middle is 1. So, the median is 1 goal.

What if there's an even number of games? Say the record was 0, 1, 1, 2, 3, 4. Now there are two middle numbers (1 and 2). In this case, we find the mean of those two numbers: $(1 + 2) / 2 = 1.5$. The median would be 1.5.

Finally, we have the mode.

In our striker's record (0, 1, 1, 2, 3), the number 1 appears more than any other. So, the mode is 1. A dataset can have more than one mode or no mode at all if every value appears only once.

Knowing the 'average' performance is only half the story. Consider two players who both have a mean of 1 goal per game. Player A scores 1, 1, 1, 1, 1. Player B scores 5, 0, 0, 0, 0. They have the same mean, but their performances are completely different. Player A is consistent, while Player B is erratic.

Measures of dispersion tell us how spread out the data is. A small dispersion means the data points are clustered together, indicating consistency. A large dispersion means they are far apart.

The simplest measure of spread is the range.

For Player A (1, 1, 1, 1, 1), the range is $1 - 1 = 0$. For Player B (5, 0, 0, 0, 0), the range is $5 - 0 = 5$. This simple calculation already shows us that Player B's performance is much more spread out.

While easy to calculate, the range only uses two data points and can be misleading. For a more complete picture, we use variance and standard deviation.

Variance measures the average of the squared differences from the mean. A small variance indicates that the data points tend to be very close to the mean, while a large variance indicates they are spread out.

Let's calculate the variance for Player B (scores: 5, 0, 0, 0, 0; mean: 1).

1. Find the difference from the mean for each score: (5-1), (0-1), (0-1), (0-1), (0-1), which gives us 4, -1, -1, -1, -1.
2. Square each difference: $4^2 = 16$, $(-1)^2 = 1$, $(-1)^2 = 1$, $(-1)^2 = 1$, $(-1)^2 = 1$.
3. Average these squared differences: $(16 + 1 + 1 + 1 + 1) / 5 = 20 / 5 = 4$. The variance is 4.

Because we squared the units (goals), the variance is in 'squared goals', which isn't very intuitive. To get back to our original units, we take the square root of the variance. This gives us the standard deviation.

For Player B, the standard deviation is $\sqrt{4} = 2$ goals. This number gives us a much better sense of the typical spread of their performance around the mean of 1 goal. For Player A, the variance and standard deviation are both 0, confirming their perfect consistency.

Probability is the measure of how likely an event is to occur. In football, we constantly deal with probability. What are the chances of a penalty being scored? What is the likelihood of a team winning a match?

Probability is expressed as a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means it is certain.

Let's say a striker has taken 20 penalties in their career and scored 16 of them. We can estimate the probability of them scoring their next penalty based on this historical data.

This is a simple example of experimental probability, where we use past data to predict future outcomes. The more data we have, the more reliable our probability estimate becomes.

These fundamental concepts, from averages and spreads to basic probability, are the building blocks for all sports analytics. They allow us to move beyond gut feelings and analyse performance with objective data.