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 z table 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 z table. 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 Percentile to Z-Score Calculator 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**

Suppose the exam scores on a certain test are normally distributed with a mean of μ = 85 and standard deviation of σ = 5.

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 z table. 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 Percentile to Z-Score Calculator 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

**Additional Resources**

How to use the Z Table (With Examples)

How to Convert Between Z-Scores and Percentiles in Excel