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2. The table below shows data from a survey about the amount of time students spend doing homework each week. The students were either in college or in high school:

\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline & High & Low & Q1 & Q3 & IQR & Median & Mean & [tex]$\sigma$[/tex] \\
\hline College & 20 & 6 & 8 & 18 & 10 & 14 & 13.3 & 5.2 \\
\hline High School & 20 & 3 & 5.5 & 16 & 10.5 & 11 & 11 & 5.4 \\
\hline
\end{tabular}

Which of the choices below best describes how to measure the spread of this data? (Hint: Use the minimum and maximum values to check for outliers.)

A. Both spreads are best described with the IQR.
B. Both spreads are best described with the standard deviation.
C. The college spread is best described by the IQR. The high school spread is best described by the standard deviation.
D. The college spread is best described by the standard deviation. The high school spread is best described by the IQR.



Answer :

GinnyAnswer
To determine which measure best describes the spread of the data, let's analyze the statistics provided for both College and High School students.

### Given Data:

#### College:
- High: 20
- Low: 6
- Q1: 8
- Q3: 18
- IQR: 10
- Median: 14
- Mean: 13.3
- Standard Deviation (σ): 5.2

#### High School:
- High: 20
- Low: 3
- Q1: 5.5
- Q3: 16
- IQR: 10.5
- Median: 11
- Mean: 11
- Standard Deviation (σ): 5.4

### Step-by-Step Analysis:

1. Identifying Outliers for College:
- Minimum value (Low): 6
- Maximum value (High): 20
- IQR = Q3 - Q1 = 18 - 8 = 10

2. Identifying Outliers for High School:
- Minimum value (Low): 3
- Maximum value (High): 20
- IQR = Q3 - Q1 = 16 - 5.5 = 10.5

### Comparing Measures of Spread (IQR and Standard Deviation):

- Range: The range (difference between high and low) for both College (20 - 6 = 14) and High School (20 - 3 = 17) indicates potential variability.

- IQR (Interquartile Range):
- For College: IQR = 10
- For High School: IQR = 10.5
- IQR is a measure that indicates the range of the middle 50% of the data, and it is not affected by outliers.

- Standard Deviation:
- For College: σ = 5.2
- For High School: σ = 5.4
- Standard deviation is a measure of how spread out the values are around the mean, and it can be influenced by outliers.

### Outlier Check:

To check for outliers, traditionally, one might use the formula [tex]\(1.5 \times IQR\)[/tex] to obtain outlier bounds:
- College:
- Lower bound: Q1 - (1.5 × IQR) = 8 - (1.5 × 10) = 8 - 15 = -7 (which is below the minimum)
- Upper bound: Q3 + (1.5 × IQR) = 18 + (1.5 × 10) = 18 + 15 = 33 (which is above the maximum)

- High School:
- Lower bound: Q1 - (1.5 × IQR) = 5.5 - (1.5 × 10.5) = 5.5 - 15.75 = -10.25 (which is below the minimum)
- Upper bound: Q3 + (1.5 × IQR) = 16 + (1.5 × 10.5) = 16 + 15.75 = 31.75 (which is above the maximum)

Since the lower bounds are both below the minimums and the upper bounds are above the maximums, the minimum and maximum values are not considered outliers.

### Conclusions:
- Both datasets do not have outliers based on the IQR test.
- Both spreads have similar IQRs and standard deviations, but given that neither dataset shows outliers and that the IQR is less influenced by data range extremes, IQR is often considered more robust for spread measurement in such contexts.

Therefore, the best description for how to measure the spread of this data is:

Both spreads are best described with the IQR.

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