# Z Distribution

The **Z table** gives us the probability of getting any z-score on a standardized normal bell curve.

Here's an overview about z-scores and how to use the Z-Table.

## What is a Z Score?

A **z-score **is a standardized value of a data point in our dataset.

The actual value of the data point is *"standardized"* or adjusted to express how many standard deviations it is from the mean of the dataset.

To convert a value in your dataset to a z-score, you can use the following equation:

**x **= the data point

**μ **= mean

**σ** = standard deviation

Subtracting the mean from a data point (x - μ) gives us the distance between the data point and the mean of the data set.

**Dividing **a point's distance from the mean **(x - μ) **by the standard deviation** (σ) **of the dataset **tells us how many standard deviations it is away from the mean.**

## Probability

We can find out what the chances are of getting a value in a dataset using it's z-score. We can find the probability or p-value of a z-score by either looking at a Z-table or using a calculator.

## Example

##### Say **we want to know what the likelihood is of getting a score of 6** in our data set of student test scores.

##### We can use our mean and standard deviation to** find the z-score** for the actual value of 6:

**x = 6 μ = 7.8 σ = 1.72**

The **Z-score **for our raw score of 6 is approximately** -1.05**.

Therefore, **a test score of 6 is 1.05 standard deviation below the mean **of our data set.

Remember - we can find the probability of getting any value in our dataset as long as we know how many standard deviations it is away from the mean (it's z-score):

Here's how:

## Empirical Rule

According to the Empirical Rule,

**68% **of the data falls within **1 standard deviation **of the mean

**95%** of the data falls within **2 standard deviations** of the mean

**99.7%** of the data falls within **3 standard deviations** of the mean

Therefore, the bell curve breaks down into the following areas of probability (out of 100%):

These distances from the mean can be represented as **z-scores:**

So in our example, our raw score of 6 points:

Has a z-score of -1.05:

So using our knowledge of the bell curve we can estimate the probability of obtaining a score of 6 or less:

Conversely, we can also find the probability of scoring more than 6 using the rules of normal probability:

## Z-Table

Rather than estimating, we can get a precise probability value for our z-score of 6 using a Z-Table.

You can look up your z-score in two steps:

** Your z-score should be rounded to two decimal places.*

**Step 1.** Find the value of the ** first two digits** (immediately before and after the decimal point) of your z-score along the

**left column of the table.**

**Step 2. **Find the corresponding** row** that contains the **last digit ***(in the hundredths place)* of

your z-score.

For example, our we can look up our **z-score of -1.05** by finding the** -1.0 **value in the **column **of the Z-table, and then finding the **row** that contains** .05**.

The **probability** value that we find for our z-score is** .1469 or 14.69%**.

Z-table gives you the **area to the** **left** of the Z-score. (The probability of getting a score less than or equal to that value.)

(Remember, all of the values inside of the Z table add up to 1 or 100%)

Our estimate of less than 16% using the empirical rule was very close!

However, the table's probability value of .1469 or about 15% is exact!

To find the **area to the right of the Z-score** (the probability of scoring higher more than a given value), you must subtract the probability from one **(1 - probability)**.

By subtracting our probability from 1, (**1-.1469) we find there is a .8531** or **85.31%** probability of scoring greater than 6 on the test.

This is close to our approximation of greater than 84%. (We're on a roll!)

## Calculators

The easiest way to calculate a z score, and find it's probability (p-value) is to use a statistics calculator.

**Graphing calculators**, **statistical software programs** (such as Excel, SPSS, R), or **free online calculators** are all able to Z-Tests, which give you the z-score and resulting probability for a value in a dataset.

**Next, let's learn more about populations, samples, and hypothesis testing.**

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