Mean, Median and Mode and Weighted Approach
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1. Each of the three measures of central tendency?the mean, the median, and the mode?are more appropriate for certain populations than others.
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 dice 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?
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Solution Summary
The expert examines mean, median and modes for the weighted approach. Additional examples of populations for appropriate indication tendencies are determined.
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Question 1.
To think of populations most suited to the median, consider data that has a natural ordering and for which the population may be heavily skewed (ie. there are a few numbers with extreme values). The classic example is the population of household incomes. If there are a very few very high earners, then taking the mean of the population will give a value that may be higher than the earnings of nearly everyone. The median should give a sensible answer. As another population for the median consider the population of scores on a test.
The mean is the most appropriate average for populations of continuous data which is not heavily skewed - eg. the height and weight of people or animals in a population.
The mode is rarely used as it often does not provide a unique answer but may be appropriate for some forms of discrete data - especially if there is no obvious ordering for the data eg. favourite colours of people in a given population, or most common name for people in a population.
Question 2.
First of all to make the data clearer we'll sort it into ...
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