Questions: A distribution of numbers has the following five-number summary: 20.2, 22.4, 30.0, 37.6, 39.8 True or False? These numbers can be used to calculate the standard deviation of the distribution. A. True B. False

Solution Steps

The five-number summary consists of the following values:

• Minimum: \( 20.2 \)
• First Quartile (Q1): \( 22.4 \)
• Median: \( 30.0 \)
• Third Quartile (Q3): \( 37.6 \)
• Maximum: \( 39.8 \)

These values provide a summary of the distribution but do not include the individual data points.

The standard deviation \( \sigma \) of a dataset is calculated using the formula: \[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \] where:

• \( N \) is the number of data points,
• \( x_i \) are the individual data points,
• \( \mu \) is the mean of the dataset.

To compute \( \sigma \), we need access to all individual data points \( x_i \), not just summary statistics.

Since the five-number summary does not provide the individual data points necessary for calculating the standard deviation, we conclude that it is not possible to calculate the standard deviation from these values alone.

Final Answer