You can use the following formula to calculate the percentile of a normal distribution based on a mean and standard deviation:

Percentile Value = μ + zσ

where:

• μ: Mean
• z: z-score from that corresponds to percentile value
• σ: Standard deviation

The following examples show how to use this formula in practice.

Example 1: Calculate 15th Percentile Using Mean & Standard Deviation

Suppose the weight of a certain species of otters is normally distributed with a mean of μ = 60 pounds and standard deviation of σ = 12 pounds.

What is the weight of an otter at the 15th percentile?

To answer this, we must find the z-score that is closest to the value 0.15 in the . This value turns out to be -1.04:

We can then plug this value into the percentile formula:

• Percentile Value = μ + zσ
• 15th percentile = 60 + (-1.04)*12
• 15th percentile = 47.52

An otter at the 15th percentile weighs about 47.52 pounds.

Note: We could also use the to find that the exact z-score that corresponds to the 15th percentile is -1.0364.

Pugging this value into the percentile formula, we get:

• Percentile Value = μ + zσ
• 15th percentile = 60 + (-1.0364)*12
• 15th percentile = 47.5632

Example 2: Calculate 93rd Percentile Using Mean & Standard Deviation

What is the exam score of a student who scores at the 93rd percentile?

To answer this, we must find the z-score that is closest to the value 0.93 in the . This value turns out to be 1.48:

We can then plug this value into the percentile formula:

• Percentile Value = μ + zσ
• 93rd percentile = 85 + (1.48)*5
• 93rd percentile = 92.4

A student who scores at the 93rd percentile would receive an exam score of about 92.4.

Note: We could also use the to find that the exact z-score that corresponds to the 93rd percentile is 1.4758.

Pugging this value into the percentile formula, we get:

• Percentile Value = μ + zσ
• 93rd percentile = 85+ (1.4758)*5
• 93rd percentile = 92.379

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