You can use the following formulas to find the first (Q_{1}) and third (Q_{3}) quartiles of a normally distributed dataset:

**Q**= μ – (.675)σ_{1}**Q**= μ + (.675)σ_{3}

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:

**Q**= μ – (.675)σ = 300 – (.675)*45 =_{1}**269.625****Q**= μ + (.675)σ = 300 + (.675)*45 =_{3}**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 = Q
_{3}– Q_{1} - 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:

**Q**= μ – (.675)σ = 50 – (.675)*2 =_{1}**48.65****Q**= μ + (.675)σ = 50 + (.675)*2 =_{3}**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 = Q
_{3}– Q_{1} - 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?