How to Find Quartiles Using Mean & Standard Deviation

You can use the following formulas to find the first (Q₁) and third (Q₃) quartiles of a normally distributed dataset:

• Q₁ = μ – (.675)σ
• Q₃ = μ + (.675)σ

Recall that μ represents the population mean and σ represents the population standard deviation.

Also recall that the first quartile represents the 25th percentile of a dataset and the third quartile represents the 75th percentile of a dataset.

The following examples show how to use these formulas in practice.

Example 1: Find Quartiles Using Mean & Standard Deviation

Suppose we have a normally distributed dataset with μ = 300 and σ = 45.

We can use the following formulas to calculate the first and third quartiles of the dataset:

• Q₁ = μ – (.675)σ = 300 – (.675)*45 = 269.625
• Q₃ = μ + (.675)σ = 300 + (.675)*45 = 330.375

We interpret this to mean that 25% of all values in the dataset fall below 269.625 and 75% of all values in the dataset fall below 330.375.

Using these numbers, we could also calculate the interquartile range to be:

• IQR = Q₃ – Q₁
• IQR = 330.375 – 269.265
• IQR = 61.11

This represents the spread of the middle 50% of values in the dataset.

Example 2: Find Quartiles Using Mean & Standard Deviation

Suppose we have a normally distributed dataset with μ = 50 and σ = 2.

We can use the following formulas to calculate the first and third quartiles of the dataset:

• Q₁ = μ – (.675)σ = 50 – (.675)*2 = 48.65
• Q₃ = μ + (.675)σ = 50 + (.675)*2 = 51.35

We interpret this to mean that 25% of all values in the dataset fall below 48.65 and 75% of all values in the dataset fall below 51.35.

Using these numbers, we could also calculate the interquartile range to be:

• IQR = Q₃ – Q₁
• IQR = 51.35 – 48.65
• IQR = 2.7

This represents the spread of the middle 50% of values in the dataset.

Additional Resources

The following tutorials offer additional information about the normal distribution and quartiles:

An Introduction to the Normal Distribution
Percentile vs. Quartile vs. Quantile: What’s the Difference?
Interquartile Range vs. Standard Deviation: What’s the Difference?