Given the data:
- High School: Minimum = 3, Maximum = 28, Q1 = 4.5, Q3 = 15, IQR = 10.5, Median = 11, Mean = 10.5, Sigma = 5.8
- College: Minimum = 6, Maximum = 50, Q1 = 8.5, Q3 = 17, IQR = 8.5, Median = 12, Mean = 15.4, Sigma = 11.7
We'll follow these steps to determine the best measure of spread:
### Step 1: Calculate the Range for Each Dataset
The range is calculated as the difference between the maximum and minimum values.
- High School Range:
[tex]\[
28 - 3 = 25
\][/tex]
- College Range:
[tex]\[
50 - 6 = 44
\][/tex]
### Step 2: Determine Interquartile Range (IQR)
IQR is given directly in the data:
- High School IQR: 10.5
- College IQR: 8.5
### Step 3: Check for Potential Outliers
Outliers can influence whether mean and standard deviation are appropriate, or if the IQR should be used instead.
- High School Data Check for Outliers:
[tex]\[
\text{Potential high outlier if } \text{Mean} > \text{Q3} + 1.5 \times \text{IQR} \quad (\text{or})
\][/tex]
[tex]\[
\text{Potential low outlier if } \text{Mean} < \text{Q1} - 1.5 \times \text{IQR}
\][/tex]
[tex]\[
\text{Q3} + 1.5 \times \text{IQR} = 15 + 1.5 \times 10.5 = 30.75 \quad (\text{Mean} = 10.5 \leq 30.75)
\][/tex]
[tex]\[
\text{Q1} - 1.5 \times \text{IQR} = 4.5 - 1.5 \times 10.5 = -11.25 \quad (\text{Mean} = 10.5 \geq -11.25)
\][/tex]
Since both conditions are not met, there are no apparent outliers in the high school data.
- College Data Check for Outliers:
[tex]\[
\text{Q3} + 1.5 \times \text{IQR} = 17 + 1.5 \times 8.5 = 29.75 \quad (\text{Mean} = 15.4 \leq 29.75)
\][/tex]
[tex]\[
\text{Q1} - 1.5 \times \text{IQR} = 8.5 - 1.5 \times 8.5 = -4.25 \quad (\text{Mean} = 15.4 \geq -4.25)
\][/tex]
Similar to the high school data, these conditions imply there are no apparent outliers.
### Step 4: Determine Best Measure of Spread
With the absence of significant outliers, the interquartile range (IQR) is a suitable measure of spread for both datasets as the data is relatively symmetrically distributed.
### Conclusion
Therefore, the best measure of spread for both the high school and college data is the IQR.
Answer: Both spreads are best described by the IQR.
