To determine the appropriate measure of variability for the given data, we first need to determine the range of the data, which is the difference between the largest and smallest values. The largest value in the data set is 49, and the smallest value is 17. Therefore:
Range = Largest value - Smallest value = 49 - 17 = 32
The range tells us the spread of the entire data set, but it can be heavily influenced by outliers. To get a measure of variability that is resistant to outliers, we can use the interquartile range (IQR), which measures the spread of the middle 50% of the data. To calculate the IQR, we need to find the first quartile (Q1) and the third quartile (Q3) of the data.
From the stem-and-leaf plot, we can see that the smallest value is 17 and the largest value is 49. To find Q1 and Q3, we need to identify the median of the lower half of the data and the median of the upper half of the data, respectively. The stem-and-leaf plot shows:
1 7, 9
2 1, 5, 9
3 0, 1, 2
4 6, 9
The median of the entire data set is the value that splits the data into two equal halves. Since there are 20 values in the data set, the median is theaverage of the 10th and 11th values when the data is sorted in ascending order. The 10th and 11th values are both 24, so the median is 24.
To find Q1 and Q3, we need to split the data set into two halves at the median and find the medians of each half. The lower half of the data set consists of:
17, 19, 21, 23, 24, 24, 25, 26, 29, 29
The median of this half is (24 + 23) / 2 = 23.5, which is closer to 23 than to 24. Therefore, Q1 is 23.
The upper half of the data set consists of:
31, 32, 34, 35, 36, 39, 49
The median of this half is (35 + 36) / 2 = 35.5, which is closer to 36 than to 35. Therefore, Q3 is 36.
Now we can calculate the IQR as:
IQR = Q3 - Q1 = 36 - 23 = 13
Therefore, the appropriate measure of variability for the given data is the IQR, which equals 13.