normal distribution
The normal distribution is the famous bell-shaped curve: symmetric, peaked in the middle, tapering off smoothly on both sides. It is the distribution that appears whenever many small, independent influences add up — heights of people, measurement errors, the average of many losses. Most outcomes cluster near the centre, and extreme values in either direction grow rapidly less likely as you move away.
It is defined by two parameters: the mean μ (mu), which locates the centre, and the standard deviation σ (sigma), which sets the width. A reliable rule of thumb — the 68-95-99.7 rule — says about 68% of the probability lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Any normal can be rescaled to the standard normal (mean 0, standard deviation 1) by subtracting the mean and dividing by the standard deviation, which is why a single z-table serves every normal distribution.
The normal is central to actuarial practice mainly because of the central limit theorem: the sum or average of many independent risks tends toward a normal shape, so aggregate losses for a large, well-diversified portfolio are often approximately normal, making capital and probability calculations tractable. But there is a crucial caveat. Individual insurance losses are emphatically NOT normal — they are right-skewed with heavy tails (a few enormous claims), and the symmetric, thin-tailed normal badly understates the chance of a catastrophic loss. Using a normal where a heavy-tailed model belongs is one of the most dangerous errors in risk work.
Aggregate annual losses on a large diversified book have mean $10m and standard deviation $1m. Treating the total as normal, there is only about a 2.5% chance losses exceed $12m (two standard deviations above the mean) — a useful capital benchmark for a well-diversified pool.
The normal works for big diversified totals — but not for a single risky claim.
The normal's thin tails make it a dangerous model for individual losses or for tail risk; real catastrophic events happen far more often than a normal predicts, which is why heavy-tailed distributions exist.