Question

1. a. what is the five - number summary for the data 0, 2, 2, 4, 5, 5, 5, 5, 7, 11?

min q1 med q3 max

1. base your answers on the following set of data below:

25, 50, 50, 60, 70, 85, 85, 90, 90, 180
a. calculate the mean:
b. calculate the median:
c. what effect does eliminating the highest value, 180, from the data set have on the mean and median? explain your answer.
the mean will (circle one) increase / decrease / stay the same
because
the median will (circle one) increase / decrease / stay the same
because

Explanation:

Step1: Find five - number summary for 0, 2, 2, 4, 5, 5, 5, 5, 7, 11

• Min: The minimum value in the data - set is 0.
• Q1: First, find the median of the lower half. The data set has 10 values. The lower half is 0, 2, 2, 4, 5. The median of this lower half is $\frac{2 + 2}{2}=2$.
• Med: The median of the whole data set (since $n = 10$, an even - numbered data set) is $\frac{5+5}{2}=5$.
• Q3: Find the median of the upper half. The upper half is 5, 5, 5, 7, 11. The median of this upper half is 5.
• Max: The maximum value in the data - set is 11.

Step2: Calculate the mean for 25, 50, 50, 60, 70, 85, 85, 90, 90, 180

The mean formula is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here, $n = 10$ and $\sum_{i=1}^{10}x_{i}=25 + 50+50 + 60+70+85+85+90+90+180=785$. So, $\bar{x}=\frac{785}{10}=78.5$.

Step3: Calculate the median for 25, 50, 50, 60, 70, 85, 85, 90, 90, 180

Since $n = 10$ (even), the median is $\frac{70 + 85}{2}=77.5$.

Step4: Analyze the effect of eliminating 180

• Mean:

The new sum of the data set (after eliminating 180) is $785-180 = 605$ and $n = 9$. The new mean is $\frac{605}{9}\approx67.22$. The mean will decrease because the sum of the data values decreases while the number of data points decreases by 1, and the value 180 was a large out - lier that was increasing the mean.

• Median:

The new data set is 25, 50, 50, 60, 70, 85, 85, 90, 90. Since $n = 9$ (odd), the median is 70. The median will decrease because the middle value of the new ordered data set is smaller than the median of the original data set.

Answer:

1. a. Min: 0, Q1: 2, Med: 5, Q3: 5, Max: 11
2. a. 78.5

b. 77.5
c. The mean will decrease because the sum of the data values decreases while the number of data points decreases by 1, and 180 was a large out - lier increasing the mean.
The median will decrease because the middle value of the new ordered data set is smaller than the median of the original data set.