Understanding Quartiles
Quartiles are values that divide a data set into four equal parts. The first quartile (Q1) represents the 25th percentile of the data set, meaning 25% of the data points fall below this value.
Step 1: Organize the Data
To find Q1, we first need to arrange the data in ascending order.
- Original data: 15, 18, 10, 20, 23, 28, 12, 16
- Sorted data: 10, 12, 15, 16, 18, 20, 23, 28
Step 2: Calculate the Position of Q1
The formula to find the position of Q1 in a sorted list is:
- Position of Q1 = (n + 1) * 0.25
Where n is the number of observations. In this case, n = 8.
- Position of Q1 = (8 + 1) * 0.25 = 2.25
Step 3: Determine Q1 Value
Since the position of Q1 is 2.25, we need to find the value between the 2nd and 3rd observations.
- The 2nd value in sorted data is 12.
- The 3rd value in sorted data is 15.
To find Q1, we calculate the average of these two numbers:
- Q1 = (12 + 15) / 2 = 13.5
Conclusion
Thus, the value of the first quartile (Q1) for the given observations is 13.5. This indicates that 25% of the data points fall below this value, providing a measure of the lower range of the data set.