Questions: Content Statistics Five-number summary and interquartile range The following are the ages of 14 physics teachers in a school district. 24,27,28,35,40,45,45,50,51,54,54,56,57,58 Notice that the ages are ordered from least to greatest. Give the five-number summary and the interquartile range for the data - Five-number summary - Minimum: - Lower quartile: - Median: - Upper quartile: - Maximum: - Interquartile range:

Solution Steps

The minimum age of the physics teachers is given by: \[ \text{Minimum} = 24 \] The maximum age is: \[ \text{Maximum} = 58 \]

To find the median, we use the formula for the rank of the median: \[ \text{Rank} = Q \times (N + 1) = 0.5 \times (14 + 1) = 7.5 \] Since the rank is not an integer, we average the values at ranks 7 and 8: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{45 + 50}{2} = 47.5 \] Thus, the median is: \[ \text{Median} = 47.5 \]

For the lower quartile, we calculate the rank: \[ \text{Rank} = Q \times (N + 1) = 0.25 \times (14 + 1) = 3.75 \] Again, since the rank is not an integer, we average the values at ranks 3 and 4: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{28 + 35}{2} = 31.5 \] Thus, the lower quartile is: \[ \text{Lower quartile} = 31.5 \]

For the upper quartile, we calculate the rank: \[ \text{Rank} = Q \times (N + 1) = 0.75 \times (14 + 1) = 11.25 \] Since the rank is not an integer, we average the values at ranks 11 and 12: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{54 + 56}{2} = 55.0 \] Thus, the upper quartile is: \[ \text{Upper quartile} = 55.0 \]

The interquartile range (IQR) is calculated as: \[ \text{IQR} = \text{Upper quartile} - \text{Lower quartile} = 55.0 - 31.5 = 23.5 \]

Final Answer