Let's look back at the example we used earlier to identify how a completed five-number summary would look. The data set used included: 6, 13, 5, 16, 8, 7, 21, 4, 9, 19, 24. Ready to learn exactly how we got each of those answers?
We will use this example in the next few sections to go into detail on exactly how to calculate each descriptive statistic within the five-number summary.
Lower Extreme (Minimum)
To identify this statistic, we must first arrange the data in ascending numerical order (least to greatest). It goes as follows:
4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24.
Now, we can easily identify the lower extreme as the very first number in our rearranged list. Therefore, the lower extreme is 4.
First Quartile (Q1)
To calculate this number, we must first make sure that the data set is ascending in numerical order (4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24). Then we will look for the number that is directly in the center of the data set.
For this specific data set, the number that is in the direct center is 9. Using this information, we can now safely split the data into two sections; the lower half (4, 5, 6, 7, 8) and the upper half (13, 16, 19, 21,24). Now we will use the lower half to find the number that represents the center of the lower half of the data.
For this specific data set the lower half of the data set includes 4, 5, 6, 7, 8 and the number that is directly in the center is 6. Therefore, the lower quartile or quartile 1 is 6.
Alternatively, we can also use a formula to help identify which data item within the data set represents the lower quartile of the data set. That formula is as follows:
{eq}(n+1)/4 {/eq},
where "n" represents the number of data items within the data set.
The data set for this specific example has 11 data items. Using the formula, we can determine that the data item that represents the lower quartile is:
{eq}(11+1)/4 = 12/4 = 3 {/eq}, so the 3rd data item.
Referring to our ordered list (4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24), we can see that the 3rd data item is 6.
Second Quartile (Median)
To calculate this number, we must first make sure that the data set is ascending in numerical order (4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24). Then we will look for the number that is directly in the center of the data set.
Looking at the data set, we can see that 9 is directly in the center. We can verify that by making sure there is the same number of data points in front of the 9 as there are behind the 9. There are 5 data points (4, 5, 6, 7, 8) that are in front of the 9 and another 5 data points (13, 16, 19, 21, 24) behind the 9. This verifies that 9 is directly in the center.
Alternatively, we can also use a formula to help identify which data item within the data set represents the median of the data set. That formula is as follows:
{eq}(n+1)/2 {/eq},
where "n" represents the number of data items within the data set.
The data set for this specific example has 11 data items. Using the formula, we can determine that the data item that represents the median is:
{eq}(11+1)/2 = 12/2 = 6 {/eq}, so the 6th data item.
Referring to our ordered list (4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24), we can see that the 6th data item is 9.
Third Quartile (Q3)
To calculate this number, we must first make sure that the data set is ascending in numerical order (4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24). Then we will look for the number that is directly in the center of the data set and separate the data into the upper and lower halves. We have already identified these halves to be: the lower half (4, 5, 6, 7, 8) and the upper half (13, 16, 19, 21,24).
Now we will use the upper half to find the number that represents the center of the upper half of the data. For this specific data set the upper half of the data set includes 13, 16, 19, 21, 24 and the number that is directly in the center is 19. Therefore, the upper quartile or quartile 3 is 19.
Alternatively, we can also use a formula to help identify which data item within the data set represents the lower quartile of the data set. That formula is as follows:
{eq}3(n+1)/4 {/eq},
where "n" represents the number of data items within the data set.
The data set for this specific example has 11 data items. Using the formula, we can determine that the data item that represents the lower quartile is:
{eq}3(11+1)/4 = 3(12)/4 = 36/4=9 {/eq}, so the 9th data item.
Referring to our ordered list (4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24), we can see that the 9th data item is 19.
Upper Extreme (Maximum)
To identify this statistic, we must first make sure that the data set is ascending in numerical order. Then we will look for the very last number.
In ascending numerical order, the data set is 4, 5, 6, 7, 8, 9, 13, 16, 19, 21, 24. Now, we can easily identify the upper extreme as 24.
