Bell curve

The central limit theorem states that the sum of many random variables will have an approximately normal distribution.

In the image to the right you can see that throwing a single die results in the numbers 1-6 with equal probability (Discrete uniform distribution). But throwing 2 dice and summing the numbers results in the numbers 2-12 with a totally different distribution. As the number of dice increases the distribution approaches the normal distribution.

Many test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions.

## Equation

The probability density of the normal distribution is

f
(
x
∣
μ
,

σ
2

# )

1

2
π

σ
2

e

−

(
x
−
μ

)
2

2

σ
2

{\displaystyle
f(x \mid \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
}

where

* μ

{\displaystyle \mu}

 is the mean or expectation of the distribution (and also its median and mode),
* σ

{\displaystyle \sigma}

 is the standard deviation, and
* σ
2

{\displaystyle \sigma^2}

 is the variance.

# μ
0

{\displaystyle \mu = 0}

 and

# σ
1

{\displaystyle \sigma = 1}

 the result is the *standard normal distribution*:

f
(
x
∣
0
,
1
# )

1

2
π

e

−

x
2

2

{\displaystyle
f(x \mid 0, 1) = \frac{1}{\sqrt{2\pi} } e^{ -\frac{x^2}{2} }
}

* The factor

1

/

2
π

{\displaystyle 1/\sqrt{2\pi}}

 in this expression ensures that the total area under the curve

φ
(
x
)

{\displaystyle \varphi(x)}

 is equal to one. (For the proof see Gaussian integral.)
* The factor

1

/

2

{\displaystyle 1/2}

 in the exponent ensures that the distribution has unit variance (i.e. the variance is equal to one), and therefore also unit standard deviation.
* This function is symmetric around

# x
0

{\displaystyle x = 0}

, where it attains its maximum value

1

/

2
π

{\displaystyle 1/\sqrt{2\pi}}

 and has inflection points at

# x
1

{\displaystyle x=+1}

 and

# x
−
1

{\displaystyle x=-1}

.

Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor

σ

{\displaystyle \sigma}

 (the standard deviation) and then translated by

μ

{\displaystyle \mu}

 (the mean value):

f
(
x
∣
μ
,

σ
2

# )

1
σ

φ

(

x
−
μ

σ

)

.

{\displaystyle
f(x \mid \mu, \sigma^2) =\frac 1 \sigma \varphi\left(\frac{x-\mu} \sigma \right).
}

## Similar distributions

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:

* The binomial distribution

B
(
n
,
p
)

{\displaystyle B(n,p)}

 is approximately normal with mean

n
p

{\displaystyle np}

 and variance

n
p
(
1
−
p
)

{\displaystyle np(1-p)}

 for large

n

{\displaystyle n}

 and for

p

{\displaystyle p}

 not too close to 0 or 1.
* The Poisson distribution with parameter

λ

{\displaystyle \lambda}

 is approximately normal with mean

λ

{\displaystyle \lambda}

 and variance

λ

{\displaystyle \lambda}

, for large values of

λ

{\displaystyle \lambda}

.
* The chi-squared distribution

χ
2

(
k
)

{\displaystyle \chi^2(k)}

 is approximately normal with mean

k

{\displaystyle k}

 and variance

2
k

{\displaystyle 2k}

, for large

k

{\displaystyle k}

.
* The Student's t-distribution

t
(
ν
)

{\displaystyle t(\nu)}

 is approximately normal with mean 0 and variance 1 when

ν

{\displaystyle \nu}

 is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

## Standard deviation and coverage

Further information: Interval estimation, Coverage probability
About 68% of values drawn from a normal distribution are within one standard deviation *s* away from the mean;

About 95% of the values lie within two standard deviations;

About 99.7% are within three standard deviations.

This fact is known as the 68-95-99.7 (empirical) rule, or the *3-sigma rule*.

More precisely, the probability that a normal deviate lies in the range between

μ
−
n
σ

{\displaystyle \mu-n\sigma}

 and

μ
n
σ

{\displaystyle \mu+n\sigma}

 is given by

F
(
μ
n
σ
)
−
F
(
μ
−
n
σ
# )
Φ
(
n
)
−
Φ
(
−
n
# )
erf
⁡

(

n

2

)

.

{\displaystyle
F(\mu+n\sigma) - F(\mu-n\sigma) = \Phi(n)-\Phi(-n) = \operatorname{erf} \left(\frac{n}{\sqrt{2}}\right).
}