Final answer:
The mean of the dataset is 11.47, the standard deviation is 4.99, the five-number summary is Min = 4, Q1 = 9, Median = 11, Q3 = 14, Max = 19, and the interquartile range is 5. The z-score for the minimum observation is -1.49 and for the maximum observation is 1.8. The box plot identifies one outlier at the value of 4. Results based on z-scores align with the outliers identified in the box plot.
Explanation:
The mean, calculated by adding all values and dividing by the number of observations, provides a central tendency measure at 11.47. The standard deviation, indicating the spread of data around the mean, is computed at 4.99. The five-number summary (minimum, Q1, median, Q3, maximum) highlights key percentiles in the dataset: Min = 4, Q1 = 9, Median = 11, Q3 = 14, and Max = 19. The interquartile range, derived from the difference between Q3 and Q1, is 5, showing the middle 50% of the data's spread.
Z-scores for the minimum and maximum values are -1.49 and 1.8 respectively, expressing how many standard deviations away these values are from the mean. The box plot visually represents the dataset's distribution and identifies outliers, with one outlier at the value of 4. Notably, the z-scores align with the outliers detected in the box plot, confirming consistency in identifying extreme values.
Analyzing data using z-scores and box plots serves as complementary approaches in detecting outliers and understanding data distribution. In this case, both methods affirm the presence of an outlier at 4. These methods validate each other, reinforcing the reliability of outlier detection and data interpretation.