Ximpledu

# Normal Distribution

See how to find the probability of a normal distribution
(z-score).
6 examples and their solutions.

## Normal Distribution: Definition

### Definition

N(x, σ2)

x: Mean
σ: Standard deviation

If the shape of a histogram looks like this,
then the values show normal distribution.
It can be found when n is big enough.
(test score, fruit size, sleeping time, ...)

## Standard Normal Distribution

### Formula

N(0, 12)

Standard normal distribution is used
to compare and analyze a normal distribution easily.
Each normal distribution has different mean and standard deviation.
But their shapes are all the same.
→ Standardized: Standard normal distribution

## Z-Score

### Formula

Z = X - xσ
The z-score is used
to change N(x, σ2) to N(0, 12).

### How to Use

P(0 ≤ Z ≤ z) = (area, known)
After changing N(x, σ2) to N(0, 12),
you can find the area (= probability) under the curve.
For each z-score,
the area under the curve is known.

(total area) = 1
(left half area) = (right half area) = 0.5
The curve is symmetric to the center axis (Z = 0).
Other values: Z-score table

P(0 ≤ Z ≤ 1) = ?

zP(0 ≤ Z ≤ z)
10.3413
20.4771
30.4987
Solution

P(Z ≥ -2) = ?

zP(0 ≤ Z ≤ z)
10.3413
20.4771
30.4987
Solution

P(1 ≤ Z ≤ 3) = ?

zP(0 ≤ Z ≤ z)
10.3413
20.4771
30.4987
Solution

## Normal Distribution

### How to Solve

N(x, σ2)

N(0, 1)
1. Standardize the given normal distribution.
N(x, σ2) → N(0, 12)
2. Use the z-score table to find the probability.

### Example

The test scores of 1,000 stuents are normally distributed.
Mean: 70. Standard deviation: 7.
About how many students score between 63 and 84?

zP(0 ≤ Z ≤ z)
10.3413
20.4771
30.4987
Solution

### Example

The test scores of 1,000 stuents are normally distributed.
Mean: 70. Standard deviation: 7.
About how many students score at or below 77?

zP(0 ≤ Z ≤ z)
10.3413
20.4771
30.4987
Solution

## Binomial Distribution to Normal Approximation

### How to Solve

B(n, p)

N(np, npq)

N(0, 12)
1. Find x = np, σ = √npq.
If n is big enough,
you can approximate B(n, p) = N(np, npq).
Binomial Distribution
2. Standardize.
N(np, npq) → N(0, 12)
3. Use the z-score table to find the probability.

### Example

A coin is tossed 400 times.
Find the probability of getting a head 185 ~ 210 times.

zP(0 ≤ Z ≤ z)
0.50.1915
10.3413
1.50.4332
20.4771
Solution