Qualitative and Quantitative Statistics

1. For each type of measure, give two additional examples of populations where it would be the most appropriate indication of central tendency.

2. Find the mean, median, and mode of the following data set:

5 15 9 22 67 42 2 72 81 53 6 70 41 9 42 23

3. Sometimes, we can take a weighted approach to calculating the mean. Take our example of high temperatures in July. Suppose it was 98°F on 7 days, 96°F on 14 days, 88°F on 1 day, 100°F on 6 days and 102°F on 3 days. Rather than adding up 31 numbers, we can find the mean by doing the following:

Mean = ( 1 x 88 + 14 x 96 + 7 x 98 + 6 x 100 + 3 x 102) / 31

...where 1, 14, 7, 6, and 3 are the weights or frequency of a particular temperature's occurrence. Then we divide by the total of number of occurrences.

Suppose we are tracking the number of home runs hit by the Boston Red Sox during the month of August:

Number of Games HRs Hit each Day
2 3
5 2
6 1
7 0

Using the weighted approach, calculate the average number of home runs per game hit by the Sox.

4. When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over.

a. Why will some numbers come up more frequently than others?

b. Each die has six sides numbered from 1 to 6. How many possible ways can a number be rolled? In other words, we can roll (2,3) or (3,2) or (6,1) and so on. What are the total (x,y) outcomes that can occur?
c.How might you then estimate the percentage of the time a particular number will come up if the dice are rolled over and over?

d. Once these percentages have been calculated, how might the mean value of the all the numbers thrown be determined?

e. There is a very famous distribution that describes the frequency of the number of times a number comes up in a series of dice rolls. What is its name?

f. What is the definition of what is called the distribution of a data population. Also, find the statistic that measures the width of dispersion ("looseness" or "tightness") of the population data about its mean. Give an example of the type of situation where this statistic might be critical to making good decisions about the population under study.