How do I Find the first and third quartiles, Q1 and Q3, of the following set of numbers?
13,4,12,16,8,14,8,3,10,8
order the data:
3,4,8,8,8,10,12,13,14,16
Quartile I ends at 8
The quartile II ends at 9.
quartile III ends at 13
The calculation of the quartiles is sometimes not quite clear (especially if the number of observations of a sample is not divisible by four). We therefore provide exact instructions how to calculate the quartiles. Assuming a sample of N observations the quartiles are defined as follows ("round" stands for the rounding to the nearest integer):
1. quartile: the value of the sorted series of observations having the position x = round(0.25*(N+1))
2. quartile (median): if N is even, Q2 is the mean of the two values at the positions N/2 and N/2+1; if N is odd, Q2 is the value at the position (N+1)/2
3. quartile: the value of the sorted series having the position x = round(0.75*(N+1))
Example:
To find the first and third quartiles of a set of numbers, you need to follow these steps:
1. First, order the numbers in the set from least to greatest. In this case, the set is: 3, 4, 8, 8, 8, 10, 12, 13, 14, 16.
2. To find the first quartile (Q1), use the formula: Q1 = round(0.25*(N+1)), where N is the number of observations in the set. In this case, N = 10, so Q1 = round(0.25*(10+1)) = round(0.25*11) = round(2.75) = 3. Therefore, the first quartile ends at the value 8.
3. To find the third quartile (Q3), use the formula: Q3 = round(0.75*(N+1)). In this case, Q3 = round(0.75*(10+1)) = round(0.75*11) = round(8.25) = 8. Therefore, the third quartile ends at the value 13.
So, in summary, for the set of numbers 13, 4, 12, 16, 8, 14, 8, 3, 10, 8, the first quartile (Q1) ends at 8 and the third quartile (Q3) ends at 13.