# Sampling Distribution

See how to find the mean, variance, and standard deviation of the mean of a sample.

1 example and its solution.

## Sampling Distribution

### Formula

Population

N(x, σ

↓

Sample

E(X) = x

V(X) = σ

σ(X) = σ√n

Population:
N(x, σ

^{2})↓

Sample

E(X) = x

V(X) = σ

^{2}nσ(X) = σ√n

Total group of values

N(x, σ

^{2}): Normal Distribution

Sampling:

Picking n values from the population

(= sample size: n)

X: Mean of the sample values

E(X): Expected Value of X

V(X): Variance of X

σ(X): Standard deviation of X

### Example

Books in a library

Mean: 250 pages

Standard deviation: 20 pages

Normal distribution

If you randomly pick 25 books,

P([mean of the pages of the books] ≥ 256 pages) = ?

Solution
Mean: 250 pages

Standard deviation: 20 pages

Normal distribution

If you randomly pick 25 books,

P([mean of the pages of the books] ≥ 256 pages) = ?

z | P(0 ≤ Z ≤ z) |
---|---|

0.5 | 0.1915 |

1 | 0.3413 |

1.5 | 0.4332 |

2 | 0.4771 |

N(250, 20

E(X) = 250

σ(X) = 20√25

= 205

= 4

X: N(250, 4

P(X ≥ 256)

Z = 256 - 2504

= 64

= 1.5

= P(Z ≥ 1.5) - [2]

= P(Z ≥ 0) - P(0 ≤ Z ≤ 1.5)

= 0.5 - 0.4332

= 0.0668

^{2})E(X) = 250

σ(X) = 20√25

= 205

= 4

X: N(250, 4

^{2}) - [1]P(X ≥ 256)

Z = 256 - 2504

= 64

= 1.5

= P(Z ≥ 1.5) - [2]

= P(Z ≥ 0) - P(0 ≤ Z ≤ 1.5)

= 0.5 - 0.4332

= 0.0668

[1]

The mean of the sample (X) shows a nomral distribution N(250, 4

^{2}).Close