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?
The information is very useful and easy to understand. Thank you so much.
what about finding the second quartile?
Hi nana…The second quartile (Q2), also known as the **median**, divides a dataset into two equal halves. It represents the middle value in the dataset when it is ordered from smallest to largest.
To find the second quartile (Q2):
1. **Order the data**: Arrange the data points in ascending order.
2. **Find the median**:
– If the number of data points (\(n\)) is **odd**, the second quartile is the middle value, located at position \(\frac{n + 1}{2}\).
– If \(n\) is **even**, the second quartile is the average of the two middle values, located at positions \(\frac{n}{2}\) and \(\frac{n}{2} + 1\).
### Example
For the dataset: 3, 7, 8, 5, 12, 14, 21, 13, 18
1. **Order the data**: 3, 5, 7, 8, 12, 13, 14, 18, 21
2. Since \(n = 9\) (odd), the second quartile is the middle value, at position \(\frac{9 + 1}{2} = 5\).
Thus, \(Q2 = 12\).