How to Find Area to the Left of Z-Score (With Examples)


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 – μ) / σ

where:

  • x: Individual data value
  • μ: Mean of population
  • σ: 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 μ = 300 pounds and standard deviation σ = 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 – 300)  / 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 μ = 85 and standard deviation σ = 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 – 85)  / 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 0.25 in the z-table:

Example of how to read the z table

The area to the left of z = 0.25 is 0.5987. Applied to our scenario, this means approximately 59.87% 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

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