Probability and statistics, and how they get misused

The mean adds everything and divides by the count, which makes it sensitive to extreme values. The median is the middle value when ordered and is robust to them. The mode is the most frequent value and is the only average available for categories.

Income is the classic case: a few very high earners pull the mean well above the median, so the two figures support very different claims about a typical person. Whenever an average appears in an argument, ask which one and why that one was chosen.

Two datasets with identical means can be completely different. Range is the crudest measure of spread. The interquartile range, covering the middle half, resists outliers and pairs naturally with the median. Standard deviation measures typical distance from the mean and pairs with it.

In roughly normal data, about two thirds of values fall within one standard deviation of the mean and about ninety five percent within two. That rule of thumb makes it possible to judge quickly whether a value is ordinary or unusual.

• Probabilities run from 0 to 1, and the probabilities of all possible outcomes sum to 1.
• For mutually exclusive events, the probability of either is the sum.
• For independent events, the probability of both is the product.
• For events that can overlap, add the two and subtract the overlap so it is not counted twice.
• The probability of at least one occurrence is often easiest as one minus the probability of none.

The probability of an event often changes once you know something else. Conditional probability handles this, and it is where intuition fails most reliably. The probability of having a disease given a positive test is not the same as the probability of testing positive given the disease, and for rare conditions the two can differ enormously.

The reason is base rates. If a condition affects one person in a thousand and a test has a five percent false positive rate, most positive results come from the large healthy group rather than the small sick one. Working through the actual counts, rather than the percentages, makes this immediately visible.

• Correlation presented as causation, when a third factor or reverse causation explains it.
• Truncated axes that exaggerate small differences.
• Cherry-picked start dates chosen to produce a favourable trend.
• Survivorship bias, where the cases that failed are absent from the data entirely.
• Small or self-selected samples generalised to whole populations.
• Relative risk quoted without the absolute risk, so a rise from one in a million to two sounds alarming as a doubling.