Calculating Quartiles in a Sorted Dataset
To find the lower quartile, median, and upper quartile, we first need to understand that these are measures of central tendency and dispersion for a given set of data, specifically in a sorted list.
The data provided is already sorted: 1, 2, 2, 2, 3, 3, 6, 6, 8, 8
The median (the middle value of the dataset) is found by taking the average of the two middle numbers if there is an even number of observations. Since we have 10 numbers here, the median will be the average of the 5th and 6th numbers:
(2 + 3)/2 = 5/2 = 2.5
So the median is 2.5.
The lower quartile (Q1), also known as the first quartile, is the median of the first half of the data. When determining quartiles, if the median is part of the data, for the lower quartile, we do not include the median as part of the lower half. So, for the lower quartile calculation, we only consider the first five numbers: 1, 2, 2, 2, 3. Since there is an odd number of numbers in this half, the lower quartile is the middle number, which is 2.
The upper quartile (Q3), or the third quartile, is the median of the second half of the data. Similar to calculating Q1, we will disregard the median of the entire data set and use only the last five numbers: 3, 6, 6, 8, 8. The middle number here is 6, so the upper quartile is 6.
Let's summarize:
- Lower quartile = 2
- Median = 2.5
- Upper quartile = 6
