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

Example

P(0 ≤ Z ≤ 1) = ?

z	P(0 ≤ Z ≤ z)
1	0.3413
2	0.4771
3	0.4987

Solution

Example

P(Z ≥ -2) = ?

z	P(0 ≤ Z ≤ z)
1	0.3413
2	0.4771
3	0.4987

Solution

Example

P(1 ≤ Z ≤ 3) = ?

z	P(0 ≤ Z ≤ z)
1	0.3413
2	0.4771
3	0.4987

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 students are normally distributed.
Mean: 70. Standard deviation: 7.
About how many students score between 63 and 84?

z	P(0 ≤ Z ≤ z)
1	0.3413
2	0.4771
3	0.4987

Solution

Example

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

z	P(0 ≤ Z ≤ z)
1	0.3413
2	0.4771
3	0.4987

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.

z	P(0 ≤ Z ≤ z)
0.5	0.1915
1	0.3413
1.5	0.4332
2	0.4771

Solution