What is the third quartile, Q3, of the following distribution?
10, 25, 18, 31, 16, 27, 5, 13, 32, 24, 15, 21, 18, 18, 24, 4, 8, 11, 11, 23
24
To find the third quartile (Q3) of a distribution, you need to first arrange the data in ascending order:
4, 5, 8, 10, 11, 11, 13, 15, 16, 18, 18, 18, 21, 23, 24, 24, 25, 27, 31, 32
Since there are 20 data points, the median would be the average of the 10th and 11th values:
(11 + 13) / 2 = 12
Now, we need to find the median of the upper half of the remaining data:
16, 18, 18, 18, 21, 23, 24, 24, 25, 27, 31, 32
There are 12 data points, so the median would be the average of the 6th and 7th values:
(21 + 23) / 2 = 22
Therefore, the third quartile, Q3, of the distribution is 22.
To find the third quartile, Q3, of a distribution, you need to arrange the data in ascending order and then locate the value that separates the highest 25% of the data from the rest. Here's how you can find the third quartile for the given distribution:
1. First, arrange the data in ascending order:
4, 5, 8, 10, 11, 11, 13, 15, 16, 18, 18, 18, 21, 23, 24, 24, 25, 27, 31, 32
2. Calculate the position of the third quartile, Q3. Since Q3 marks the boundary of the highest 25% of the data, you need to find the position that represents the 75th percentile.
- Formula to find the position of the nth percentile: (n/100) * (N + 1), where n is the percentile and N is the total number of data points.
- In this case, n = 75 (for the third quartile) and N = 20 (since there are 20 data points).
- So, the position of Q3 is (75/100) * (20 + 1) = 0.75 * 21 = 15.75.
3. Since the position is not a whole number, we need to use interpolation to find the precise value corresponding to the position. Interpolation involves taking the average of the values at the nearest lower and upper positions.
- The nearest lower position is 15, and the nearest upper position is 16.
- The value at position 15 is 24, and the value at position 16 is 25.
- Average the two values: (24 + 25) / 2 = 24.5.
Therefore, the third quartile, Q3, of the given distribution is 24.5.