(05.01 MC, 05.02 MC, 05.04 MC)

Use the box plots comparing the number of males and number of females attending the latest superhero movie each day for a month to answer the questions.

Two box plots shown. The top one is labeled Males. Minimum at 0, Q1 at 1, median at 20, Q3 at 25, maximum at 50. The bottom box plot is labeled Females. Minimum at 0, Q1 at 5, median at 6, Q3 at 10, maximum at 18 and a point at 43

Part A: Estimate the IQR for the males' data. (2 points)

Part B: Estimate the difference between the median values of each data set. (2 points)

Part C: Describe the distribution of the data and if the mean or median would be a better measure of center for each. (4 points)

Part D: Provide a possible reason for the outlier in the data set. (2 points)

A) IQR: 24

B)

C) Both distributions are asymmetrical (male: left-skewed, females: right-skewed), so the median is a better value for both

D) 43 females attended the same movie on the same day

Step-by-step explanation:

A)

The IQR (interquartile range) is the difference between the upper quartile (Q3) and the lower quartile (Q1):

where the upper quartile (Q3) is the value such that 25% of the values in the dataset are above Q3, while the other 75% of values are below Q3; instead, the lower quartile (Q1) is the value such that 25% of the values in the dataset are below Q1, while the other 75% of the values are above Q1.

For the males data in this problem, we have:

Upper quartile:

Lower quartile:

Therefore, the interquartile range for males data is:

B)

The median value (Q2) of a dataset is the "central value" of the dataset, such that half of the values of the dataset are above Q2 and the other half of the values are below Q2.

For the males data, the median is

For the females data, the median is

Therefore, the difference between the median values of each dataset is:

This means that the median for the males data is 14 points to the right of the median for the females data.

C)

For the males data, we have:

Minimum at 0

Q1 at 1

Q2 at 20

Q3 at 25

maximum at 50

As we can see, the distribution is very asymmetrical. In fact, the distance Q3-Q2 is much smaller than the distance Q2-Q1: since the two intervals contain the same number of points, this means that the distribution is much more "dense" on the left of Q2, so it has a large tail on the left (the distribution is said to be left-skewed)

The mean value is a good measure of the center of a distribution only if the distribution is symmetrical: this is not the case, therefore for the males data, the median is a better measure.

For the females data, we have

Minimum at 0

Q1 at 5

Q2 at 6

Q3 at 10

maximum at 18

a point at 43

For the female case, we notice that the interval Q3-Q2 is much larger than the interval Q2-Q1: this means that the distribution is very asymmetrical, but this time the interval on the right is much larger than the interval on the left, therefore the distribution has a large tail on the right (right-skewed distribution). Therefore, also in this case the median is a better measure over the mean.

D)

An outlier in a data set is a point that falls outside the interval (minimum-maximum).

In this problem, we see that the females dataset has an outlier, which is the point at

x = 43

Because the maximum of the females distribution is

max = 18

The meaning of an outlier is a point that has a behaviour much different from the rest of the distribution. In this specific case, the meaning of this outlier is that there has been a day where 43 females attended the same movie together. The explanation for this can be that the females were in a large group, so this explains why there were so many females attending the theater.