The central limit theorem

Why does this particular curve turn up in biology, metrology, finance and polling? One theorem explains it — and it is among the deepest in mathematics.

The central limit theorem

When a large number of independent random contributions of comparable size are added, the sum approximately follows a normal distribution — whatever the distribution of each contribution.

That is the remarkable point: the underlying distribution does not matter. Dice, coins, arbitrary durations: the sum tends to the same bell.

Sum of 1 die           Sum of 2 dice         Sum of 10 dice
                                                    ___
 ██████                    ▁▃▅█▅▃▁              __/   \__
 1 2 3 4 5 6              2   7    12          10   35   60

 flat (uniform)           triangular            near-normal

Why nature is full of it

Many measured quantities are sums of small independent causes:

an adult's height   = hundreds of genes + nutrition + sleep + ...
a measurement error = vibration + temperature + parallax + rounding + ...
electronic noise    = motion of a very large number of electrons

Each contribution follows an unknown, often odd distribution. It does not matter: their sum is normal. This is the deep reason the bell appears everywhere, without any mechanism having been "designed" to produce it.

Sample means

This is the most-used consequence. Drawing a sample of size n from a population with mean μ and standard deviation σ:

                                     σ
the sample mean  ~  N( μ , -------- )        for n large enough
                                    √n

Two major lessons:

• the population need not be normal; only the mean is. This is the most widespread misreading of the theorem;
• the standard deviation of the mean decreases as 1/√n — the famous factor governing survey precision.

n = 100    ->  uncertainty divided by 10 compared with one individual
n = 10,000 ->  divided by 100

The binomial approximation

A historic special case, due to de Moivre and Laplace:

B(n ; p)  ≈  N( np , np(1-p) )        if np ≥ 5 and n(1-p) ≥ 5

   B(100 ; 0.5)                          N(50 ; 25)
                                             ___
      ▁▂▃▅▇█▇▅▃▂▁              ≈         __/   \__
     35   50   65                       35   50   65

Adding 40 binomial terms becomes unnecessary: one table lookup suffices.

What the theorem does NOT say

Its conditions matter as much as its conclusion:

- the contributions must be INDEPENDENT
     -> if correlated (market panic, contagion), it does not apply

- NONE may dominate the others
     -> a dominant cause imposes its own distribution

- the VARIANCE must be FINITE
     -> some distributions (Cauchy) have none: the sum never normalises

- it concerns SUMS
     -> a PRODUCT of random factors gives a log-normal distribution, highly
        skewed: the case of incomes, firm sizes or compounded returns

This is why financial models built on the normal distribution badly underestimate crashes: extreme returns are far more frequent than the bell predicts, and panic movements are anything but independent. These are fat tails. A "10-sigma" crash, impossible under a normal law (one chance in 10²³), has occurred several times in a century.

Summary

• Central limit theorem: a sum of many independent contributions tends to a normal distribution, whatever their own laws.
• This is why the bell appears in biology, metrology and industry.
• The sample mean follows N(μ , σ/√n) — even when the population is not normal.
• Special case: B(n ; p) ≈ N(np , np(1-p)).
• It requires independence, no dominant contribution, finite variance, and a sum — not a product.
• Ignoring these conditions leads to underestimating extreme events (fat tails).