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A percentile can be calculated from a mean and standard deviation by using the z-score formula, which is (x – mean) / standard deviation. This formula can be used to calculate the percentile rank of a given score, which is the position of that score relative to the other scores in the data set. This can be useful for understanding how an individual’s score compares to the rest of the group.
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