Question

determine the best measure of variability. justify your answer. key: 10|2 represents 102
10: 2 3 5 6 8
11: 1 2 5
12: 3 5 6
13: 0 2 5 9 9
options:
iqr, because the data distribution is skewed
standard deviation, because the data distribution is skewed
standard deviation, because the data distribution is symmetric
iqr, because the data distribution is symmetric

Explanation:

Step1: Analyze data distribution

First, we list out all the data points from the stem - and - leaf plot. The stem - and - leaf plot has stems 10, 11, 12, 13.

• For stem 10: The leaves are 2, 3, 5, 6, 8. So the data points are 102, 103, 105, 106, 108.
• For stem 11: The leaves are 1, 2, 5. So the data points are 111, 112, 115.
• For stem 12: The leaves are 3, 5, 6. So the data points are 123, 125, 126.
• For stem 13: The leaves are 0, 2, 5, 9, 9, 9. So the data points are 130, 132, 135, 139, 139, 139.

To check for skewness, we can look at the shape of the data. The number of data points on the left (lower values, around stem 10 - 11) is less than the number of data points on the right (higher values, around stem 13). Also, the tail of the data (the part with the less - frequent values) is on the left? Wait, no. Wait, stem 10 has 5 data points, stem 11 has 3, stem 12 has 3, stem 13 has 6. Wait, actually, let's calculate the median. The total number of data points \(n=5 + 3+3 + 6=17\). The median is the \((\frac{n + 1}{2})\)-th term, which is the 9 - th term. Let's order the data:

102, 103, 105, 106, 108, 111, 112, 115, 123, 125, 126, 130, 132, 135, 139, 139, 139.

The 9 - th term is 123. Now, let's look at the lower half (first 8 terms: 102, 103, 105, 106, 108, 111, 112, 115) and the upper half (last 8 terms: 125, 126, 130, 132, 135, 139, 139, 139). The lower half has values that are more spread out towards the lower end and the upper half is more concentrated? Wait, no, actually, the left - hand side (lower values) has a shorter tail and the right - hand side (higher values) has a longer tail? Wait, no, stem 13 has more data points. Wait, maybe I made a mistake. Let's check the number of data points:

Stem 10: 5, Stem 11: 3, Stem 12: 3, Stem 13: 6. So the data is more concentrated on the higher end (stem 13) and less on the lower end (stem 10 - 11). So the distribution is skewed (left - skewed? Wait, no. Left - skewed means the tail is on the left (lower values), right - skewed means the tail is on the right (higher values). Wait, if we have more data on the right (stem 13) and less on the left (stem 10 - 11), the tail is on the left. So it's left - skewed.

Step2: Recall measures of variability

• The inter - quartile range (IQR) is a measure of variability that is resistant to skewness. It measures the range of the middle 50% of the data.
• The standard deviation is a measure of variability that is sensitive to skewness because it takes into account all the data points, including the outliers or the skewed part of the data.

Since the data distribution is skewed (left - skewed in this case), the best measure of variability is the IQR.

Answer:

IQR, because the data distribution is skewed