Question

compare each set of data. determine which set shows greater variability.
set a
5, 8, 9, 10, 12, 15, 16, 20
set b
24, 34, 37, 42, 56, 57, 65
which set shows greater variability?
set b
which measure provides greater variability for the set you chose?
range
interquartile range
both range and
interquartile range

Explanation:

Step1: Calculate the range for each set

The range is calculated as \( \text{Range}=\text{Max}-\text{Min} \).
For Set A: \( \text{Max} = 20 \), \( \text{Min}=5 \), so \( \text{Range}_A=20 - 5=15 \).
For Set B: \( \text{Max} = 65 \), \( \text{Min}=24 \), so \( \text{Range}_B=65 - 24 = 41 \).

Step2: Calculate the inter - quartile range (IQR) for each set

The formula for IQR is \( \text{IQR}=Q_3 - Q_1 \).
For Set A:
First, find \( Q_1 \) and \( Q_3 \). The data set has \( n = 8 \) values. \( Q_1 \) is the value at the \( \frac{n + 1}{4}\)th position. \( \frac{8+1}{4}=2.25 \)th position. Using linear interpolation: \( Q_1=8+(9 - 8)\times0.25 = 8.25 \). \( Q_3 \) is the value at the \( \frac{3(n + 1)}{4}\)th position. \( \frac{3\times(8 + 1)}{4}=6.75 \)th position. Using linear interpolation: \( Q_3=15+(16 - 15)\times0.75=15.75 \). \( \text{IQR}_A=15.75-8.25 = 7.5 \).
For Set B:
The data set has \( n = 8 \) values. \( Q_1 \) is the value at the \( \frac{n + 1}{4}\)th position. \( \frac{8+1}{4}=2.25 \)th position. Using linear interpolation: \( Q_1=34+(37 - 34)\times0.25=34.75 \). \( Q_3 \) is the value at the \( \frac{3(n + 1)}{4}\)th position. \( \frac{3\times(8 + 1)}{4}=6.75 \)th position. Using linear interpolation: \( Q_3=56+(57 - 56)\times0.75=56.75 \). \( \text{IQR}_B=56.75-34.75 = 22 \).

Answer:

Which set shows greater variability? Set B.
Which measure provides greater variability for the set you chose? Range.