A z-table is a table that tells you what percentage of values fall below a certain z-score in a standard normal distribution.

A z-score simply tells you how many standard deviations away an individual data value falls from the mean. It is calculated as:

**z-score = (x – μ) / σ**

where:

**x:**individual data value**μ:**population mean**σ:**population standard deviation

This tutorial shows several examples of how to use the z table.

**Example 1**

The scores on a certain college entrance exam are normally distributed with mean μ = 82 and standard deviation σ = 8. Approximately what percentage of students score less than 84 on the exam?

**Step 1: Find the z-score.**

First, we will find the z-score associated with an exam score of 84:

z-score = (x – μ) / σ = (84 – 82) / 8 = 2 / 8 = **0.25**

**Step 2: Use the z-table to find the percentage that corresponds to the z-score.**

Next, we will look up the value **0.25 **in the z-table:

Approximately **59.87% **of students score less than 84 on this exam.

**Example 2**

The height of plants in a certain garden are normally distributed with a mean of μ = 26.5 inches and a standard deviation of σ = 2.5 inches. Approximately what percentage of plants are greater than 26 inches tall?

**Step 1: Find the z-score.**

First, we will find the z-score associated with a height of 26 inches.

z-score = (x – μ) / σ = (26 – 26.5) / 2.5 = -0.5 / 2.5 = **-0.2**

**Step 2: Use the z-table to find the percentage that corresponds to the z-score.**

Next, we will look up the value **-0.2**** **in the z-table:

We see that 42.07% of values fall below a z-score of -0.2. However, in this example we want to know what percentage of values are *greater *than -0.2, which we can find by using the formula 100% – 42.07% = 57.93%.

Thus, aproximately **59.87% **of the plants in this garden are greater than 26 inches tall.

**Example 3**

The weight of a certain species of dolphin is normally distributed with a mean of μ = 400 pounds and a standard deviation of σ = 25 pounds. Approximately what percentage of dolphins weigh between 410 and 425 pounds?

**Step 1: Find the z-scores.**

First, we will find the z-scores associated with 410 pounds and 425 pounds

z-score of 410 = (x – μ) / σ = (410 – 400) / 25 = 10 / 25 = **0.4**

z-score of 425 = (x – μ) / σ = (425 – 400) / 25 = 25 / 25 = **1**

**Step 2: Use the z-table to find the percentages that corresponds to each z-score.**

First, we will look up the value **0.4**** **in the z-table:

Then, we will look up the value **1**** **in the z-table:

Lastly, we will subtract the smaller value from the larger value: **0.8413 – 0.6554 = 0.1859**.

Thus, approximately **18.59% **of dolphins weigh between 410 and 425 pounds.

**Additional Resources**

An Introduction to the Normal Distribution

Normal Distribution Area Calculator

Z Score Calculator