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# How to use the Z Table (With Examples)

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 â€“Â Î¼) / Ïƒ

where:

• x:Â individual data value
• Î¼:Â population mean
• Ïƒ:Â 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 meanÂ Î¼ = 82 and standard deviationÂ Ïƒ = 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 â€“Â Î¼) /Â Â Ïƒ = (84 â€“ 82) / 8 = 2 / 8 =Â 0.25

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

Next, we will look up the valueÂ 0.25Â in the z-table:

ApproximatelyÂ 59.87%Â 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 Â Î¼ = 26.5 inches and a standard deviation of Ïƒ = 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 â€“Â Î¼) /Â Â Ïƒ = (26 â€“ 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 -0.2Â 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 greaterÂ than -0.2, which we can find by using the formula 100% â€“ 42.07% =Â 57.93%.

Thus, aproximatelyÂ 59.87%Â 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 Î¼ = 400 pounds and a standard deviation of Ïƒ = 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 â€“Â Î¼) /Â Â Ïƒ = (410 â€“ 400) / 25 = 10 / 25 =Â 0.4

z-score of 425 = (x â€“Â Î¼) /Â Â Ïƒ = (425 â€“ 400) / 25 = 25 / 25 =Â 1

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

First, we will look up the valueÂ 0.4Â in the z-table:

Then, we will look up the valueÂ 1Â in the z-table:

Lastly, we will subtract the smaller value from the larger value:Â 0.8413 â€“ 0.6554 = 0.1859.

Thus, approximatelyÂ 18.59%Â of dolphins weigh between 410 and 425 pounds.