# Z-Scores (Normal Distribution)

How to use the z-scores to solve the normal distribution problems: formula, meaning, examples, and their solutions.

## Normal Distribution

If the shape of a histogram looks like this,

then that data shows a normal distribution.

Frequency histogram

The left side and the right side of the curve,

which are cut by the mean (*x*),

have the same shape.

## Z-Scores: Formula

Each normal distribution has

different mean and standard deviation.

But their shapes all show the same curve.

So, to analyze the normally distributed data,

you can use the z-score.

The z-score is

the coefficient of the standard deviation.

So the z-score of *x* can be found

by using this formula:*Z* = (*x* - *x*)/σ

By using the z-score,

you can find out where the value is located.

For the z-score curve,

the mean is 0,

and the standard deviation is 1.

## Z-Scores: Meaning

The z-score shows

where the value is located

in the z-score curve.

So, by finding the z-score,

you can find the percentage areas.

The area of [*Z*: 0 ~ 1]

is about 34%. (= 0.3413...)

So the area of [*Z*: -1 ~ 1]

is about 34⋅2 = 68%. (= 0.6826...)

The area of [*Z*: 1 ~ 2]

is about 13.5%. (= 0.1359...)

So the area of [*Z*: -2 ~ 2]

is about (34 + 13.5)⋅2 = 95%. (= 0.9544...)

The area of [*Z*: 2 ~ 3]

is about 2%. (= 0.0215...)

So the area of [*Z*: -3 ~ 3]

is about (34 + 13.5 + 2)⋅2 = 99%. (= 0.9974...)

To see more percentage areas,

use the z-score table.

## Example 1

The mean is 70.

And the standard deviation is 7.

Find the z-score of 56

by using the formula.*Z* = (56 - 70)/7

## Example 2

Find the percentage of 63 ~ 84 points:

P(63 ≦ *X* ≦ 84).

The test scores are normally distributed.

So find the z-scores of 63 and 84:*Z* = -1, 2.

So P(63 ≦ *X* ≦ 84)

= P(-1 ≦ *Z* ≦ 2).

Draw the z-score curve.

Mark the percentage areas between -1 and 2:

34%, 34%, and 13.5%.

So P(-1 ≦ *Z* ≦ 2) = 0.68 + 0.135

= 0.815.

Find the expected value.

E(*X*) = *x*⋅*P*

= 1000⋅0.815

= 815

So about 815 students score between 63 and 84.

## Example 3

Find the percentage of 0 ~ 77 points:

P(*X* ≦ 77).

The test scores are normally distributed.

So find the z-score of 77:*Z* = 1.

So P(*X* ≦ 77)

= P(*Z* ≦ 1).

Draw the z-score curve.

Mark the percentage areas:

50% (left half) and 34%.

So P(*Z* ≦ 1) = 0.5 + 0.34

= 0.84.

Find the expected value.

E(*X*) = *x*⋅*P*

= 1000⋅0.84

= 840

So about 840 students score at or below 77.