Two ways to say “typical”
The mean is the familiar average: add the values and divide by how many there are, or, for a random process, weight each outcome by its probability. The median is the middle value: half of outcomes are at or below it and half are at or above it.
In a symmetric situation, such as the total of two dice, the two agree. When the results are lopsided, they come apart, and then it matters which one you quote.
A simple loot example
Ten chests are opened. Nine contain an item worth 1 coin each, and one contains an item worth 100 coins.
- Mean: (9 × 1 + 100) / 10 = 109 / 10 = 10.9 coins.
- Median: sorted, the middle values are both 1, so the median is 1 coin.
Saying “a chest is worth about 11 coins on average” is arithmetically true, yet nine out of ten chests are worth 1. The mean is pulled upward by a single large value. The median describes the ordinary chest more faithfully, while the mean describes the total value spread across all chests. Both are correct, and they answer different questions.
Waiting times show it clearly
Take an item with a 10% drop chance per attempt, where each attempt is independent. How many attempts until the first drop?
The mean number of attempts is 1 / 0.10 = 10. That is the figure often quoted as “the average grind”.
The median asks a different question: how many attempts until there is at least a 50% chance you have already succeeded? The chance of failing n times in a row is 0.9^n.
- After 6 attempts: 0.9^6 is about 0.531, so there is still more than a 50% chance of having no drop.
- After 7 attempts: 0.9^7 is about 0.478, so the chance of no drop is below 50%.
So the median is 7 attempts. Half of players are done within 7 tries, yet the mean is 10 because some unlucky players need 20, 30 or more tries, which pulls the average up. This shape, a long tail on the right, is typical of waiting time problems. An honest summary says both: about half of players get the drop within 7 tries, and the average across all players is 10.
Why the mean and median split
The gap appears when a distribution is skewed, meaning one side has a long tail. With a right tail, the mean sits above the median. Rare large wins, rare long streaks of failure and rare huge costs all stretch the tail.
In symmetric cases there is no gap. Sums of several dice, for example, are symmetric around the middle, so the mean and median coincide. The same holds for many bell-shaped outcomes. A quick check: if the biggest possible values are much further from the centre than the smallest ones, expect the mean and median to differ.
Which one should you use
- Use the mean for totals and long-run budgets. If you will open thousands of chests and want to know total value, the mean multiplied by the count is the right tool. Totals depend on the extreme values the mean includes.
- Use the median for a single player’s typical experience. If you want to know what happens to the ordinary player in one go, the median is often closer to the truth.
- Report both, or the full spread, when the numbers differ. A sentence such as “the average is 10 attempts but half of players finish in 7 or fewer” is more helpful than either number alone.
This is also why the average outcome trap is such a common mistake. A game’s expected value can be driven by rare events that the median player never sees.
Practical cautions
Medians can be misleading too. They ignore how extreme the extremes are, so a median alone would call the chest example “worth 1 coin” and miss the 100-coin chest entirely. Percentiles, such as the 90th percentile, give a fuller picture of a long tail, and a drawing of the full outcome histograms is more informative still.
When you track your own results, watch for small samples. With only a few observations, both the mean and median can bounce around, and one unusual result can move the mean a lot. It is wise not to read much into either until you have a reasonable amount of data.
For players, the useful takeaway is modest: when a published average sounds off compared to your experience, check whether the distribution is skewed. If it is, your experience and the average can both be accurate at the same time.