For single data, we just sort it and find the middle position. Now, if the data is grouped in a frequency table (like test scores grouped as 70-79, 80-89, etc.), the method is slightly different. We don't know the exact value of each data point, only how many data points are in each group (class interval).
Similar to the median for grouped data, to find quartiles (Q1, , ), we also use . Essentially, we "estimate" the quartile's position within the class interval where it falls.
Q2
Q3
interpolation
We determine the position of the quartile using this formula:
The position of Q1 is the 41n-th data point
The position of Q2 is the 42n-th data point (or 21n-th)
First, we need a frequency table with a cumulative frequency column (Fk). Cumulative frequency is the sum of frequencies from the first class up to that class. This is important to know which class the quartile falls into.
First, let's find the position of the data point for the quartile.
Total data (n) is 30.
Position of Q1: the 41×30=7.5-th data point.
Look at the Fk column. Which class contains the 7.5th data point? The first class has Fk=2 (not enough). The second class has (data points through are here). So, the th data point is in the class.
Position of Q2 (Median): the 21×30=15-th data point.
Try calculating Q3 from the shoe sales data in the table above.
After getting the result, compare it with the method for finding quartiles for single data learned earlier. What's the difference, and why might the results be similar or different?
So, the value of Q3 is 45.35. This means about 75% of the shoes sold are size 45.35 or smaller (or 25% are sold in sizes larger than 45.35).
Comparison with Single Data:
Finding quartiles for grouped data uses interpolation because we don't know the exact value of each data point, only its range. The result is an estimated quartile value.
For single data, we can directly point to which data point is the quartile (or the average of two data points), so the result is more precise (if the data is indeed single). Quartiles for grouped data provide a good overview for large datasets that have already been grouped.
