Question

Given the following data set: 8,7,1,4,6,6,4 5,7,6,3,0 a. Find the five-number summary and the IQR. b. Calculate \(\bar{x}\) and \(s\). c. Calculate the \(z\) -score for the smallest and largest observations. Is either of these observations unusually large or unusually small?

Short answer

Question: Calculate the five-number summary, the IQR, the sample mean, the sample standard deviation, and the z-scores for the smallest and largest observations of the given data set: 0, 1, 3, 3, 4, 4, 5, 6, 6, 7, 8.

Step-by-step solution

Organize the data set

First, we must organize the data set in ascending order: 0, 1, 3, 3, 4, 4, 5, 6, 6, 7, 8

Calculate the five-number summary

- Minimum: The smallest value in the dataset is 0. - Q1 (The first quartile): The middle value of the lower half of the dataset, excluding the median. In this case, it's the average of the middle pair: \((1 + 3)/2 = 2\) - Median: The middle value of the dataset is 4. - Q3 (The third quartile): The middle value of the upper half of the dataset, excluding the median. In this case, it's the average of the middle pair: \((6 + 7)/2 = 6.5\) - Maximum: The highest value in the dataset is 8. So, our five-number summary is: Minimum: 0, Q1: 2, Median: 4, Q3: 6.5, Maximum: 8.

Calculate the IQR

The interquartile range (IQR) is the difference between Q3 and Q1: \(IQR = Q3 - Q1 = 6.5 - 2 = 4.5\)

Calculate the mean

The sample mean (\(\bar{x}\)) is the sum of all values divided by the number of values in the data set: \(\bar{x} = \frac{0 + 1 + 3 + 3 + 4 + 4 + 5 + 6 + 6 + 7 + 8}{11} = \frac{47}{11} \approx 4.27\)

Calculate the standard deviation

To calculate the sample standard deviation (\(s\)), we first find the sum of squared differences between each data point and the mean, then divide by (n-1) where n = the number of values, and finally take the square root: $$ s = \sqrt{\frac{(0-4.27)^2 + (1-4.27)^2 + (3-4.27)^2 + \cdots + (8-4.27)^2}{11-1}} = \sqrt{\frac{(4.27)^2 + (-3.27)^2 + (-1.27)^2 + \cdots + (3.73)^2}{10}} \approx 2.44 $$

Calculate the z-scores

To calculate the z-score for the smallest and largest observation, we can use the formula: $$ z = \frac{x - \bar{x}}{s}$$ The smallest observation is 0, so: $$z_{\text{min}} = \frac{0 - 4.27}{2.44} \approx -1.75 $$ The largest observation is 8, so: $$z_{\text{max}} = \frac{8 - 4.27}{2.44} \approx 1.53$$

Analyze the z-scores

In general, a z-score with an absolute value greater than 2 is considered unusual. In this case, neither the smallest nor the largest observation is unusually small or large, since their z-scores are -1.75 and 1.53 respectively. To sum up our findings: a. The five-number summary is Minimum: 0, Q1: 2, Median: 4, Q3: 6.5, Maximum: 8 and the IQR is 4.5. b. The mean (\(\bar{x}\)) is approximately 4.27 and the standard deviation (s) is approximately 2.44. c. The z-score for the smallest observation is approximately -1.75 and for the largest observation approximately 1.53. Neither of these observations is unusually small or large.