*25*

In statistics, a z-score tells us how many standard deviations away a given value lies from a population mean.

We use the following formula to calculate a z-score for a given value:

**z = (x â€“ Î¼) / Ïƒ**

where:

**x**: Individual data value**Î¼**: Mean of population**Ïƒ**: Standard deviation of population

To find the area under a normal distribution that lies to the left of a given z-score, we can use one of two methods:

**1.** Use the z table.

**2.** Use the Area to the Left of Z-Score Calculator.

The following examples show how to use each of these methods in practice.

**Example 1: Area to the Left of Negative Z-Score**

The weight of a certain species of turtles is normally distributed with mean Î¼ = 300 pounds and standard deviation Ïƒ = 15 pounds. Approximately what percentage of turtles weigh less than 284 pounds?

The z-score for a weight of 284 pounds would be calculated as z = (284 â€“ 300)Â / 15 = **-1.07**

We can use one of two methods to find the area to the left of this z-score:

**Method 1: Use z table.**

To find the area to the left of the z-score, we can simply look up the value **-1.07 **in the z-table:

The area to the left of z = -1.07 is **0.1423**.

Applied to our scenario, this means approximately **14.23% **of turtles weight less than 284 pounds.

**Method 2: Use Area to the Left of Z-Score Calculator**

We can also use the Area to the Left of Z-Score Calculator to find that the area to the left of z = -1.07 is **0.1423**.

**Example 2: Area to the Left of Positive Z-Score**

The scores on a certain exam are normally distributed with mean Î¼ = 85 and standard deviation Ïƒ = 8. Approximately what percentage of students score less than 87 on the exam?

The z-score for an exam score of 87 would be calculated as z = (87 â€“ 85)Â / 8 = **0.25**

We can use one of two methods to find the area to the left of this z-score:

**Method 1: Use z table.**

To find the area to the left of the z-score, we can simply look up the value **0.25Â **in the z-table:

The area to the left of z = 0.25 is **0.5987**. Applied to our scenario, this means approximately **59.87%Â **of students score less than 87 on this exam.

**Method 2: Use Area to the Left of Z-Score Calculator**

We can also use the Area to the Left of Z-Score Calculator to find that the area to the left of z = 0.25 is **0.5987**.

**Additional Resources**

The following tutorials provide additional information on how to work with z-scores:

How to Find Area to the Right of Z-Score

How to Find Z-Scores Given Area

What is Considered a Good Z-Score?

How to Calculate a P-Value from a Z-Score by Hand