Find these measures of variability for the given dataset: range and IQR.
21, 24, 25, 27, 30, 32, 33, 34, 34, 36, 41, 44
The range is the difference between the largest and smallest values in the dataset. The smallest value is 21 and the largest value is 44, so the range is:
44 - 21 = 23
To find the IQR, we first need to find the median of the dataset. Since there are 12 values, the median will be the average of the 6th and 7th values when the dataset is arranged in order:
21, 24, 25, 27, 30, 32, 33, 34, 34, 36, 41, 44
The 6th and 7th values are both 32, so the median is 32.
Next, we need to find the quartiles of the dataset. The first quartile, Q1, is the median of the values below the median. The third quartile, Q3, is the median of the values above the median.
First, we find the median of the values below the median (i.e. the first half of the dataset):
21, 24, 25, 27, 30, 32
The 3rd and 4th values are both 25, so Q1 is 25.
Next, we find the median of the values above the median (i.e. the second half of the dataset):
33, 34, 34, 36, 41, 44
The 3rd and 4th values are both 34, so Q3 is 34.
The IQR is the difference between Q3 and Q1:
34 - 25 = 9
To find the measures of variability for the given dataset, let's start by arranging the data in ascending order:
21, 24, 25, 27, 30, 32, 33, 34, 34, 36, 41, 44
Range:
Range is a measure of variability that represents the difference between the largest and smallest values in a dataset. To find the range, we need to subtract the smallest value from the largest value.
Smallest value: 21
Largest value: 44
Range = Largest value - Smallest value
Range = 44 - 21
Range = 23
Therefore, the range of the dataset is 23.
IQR (Interquartile Range):
IQR is a measure of variability that represents the range of the middle 50% of the data. To find the IQR, we first need to determine the first quartile (Q1) and the third quartile (Q3).
Arranging the data in ascending order:
21, 24, 25, 27, 30, 32, 33, 34, 34, 36, 41, 44
To find Q1 and Q3, we can use the following steps:
1. Calculate the median (middle value) of the dataset.
2. If the total number of values is odd, the median is the middle value. If the total number is even, the median is the average of the two middle values.
3. Split the dataset into two halves: the lower half (values less than or equal to the median) and the upper half (values greater than or equal to the median).
4. Find the median of the lower half, which will be Q1 (the first quartile).
5. Find the median of the upper half, which will be Q3 (the third quartile).
Median: (30 + 32) / 2 = 31
Lower half: 21, 24, 25, 27, 30
Upper half: 32, 33, 34, 34, 36, 41, 44
Q1 = Median of the lower half = 25
Q3 = Median of the upper half = (34 + 34) / 2 = 34
To find the IQR, subtract Q1 from Q3:
IQR = Q3 - Q1
IQR = 34 - 25
IQR = 9
Therefore, the IQR of the dataset is 9.