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Calculate for the mean, median and mode of each set of ungrouped data. For mode, determine if the mode is unimodal, bimodal or trimodal.
1. 8, 9, 11, 13, 14
2. 114, 117, 200, 119
3. 89, 92, 83, 88, 89
4. 12, 31, 29, 6, 10, 8, 12, 27, 12, 16
5. 52, 52, 52, 50, 55, 50, 57, 49, 60, 49, 62
Answers & Comments
Answer:
Presentation
The Mean, Median, and Mode
Mean, median, and mode are three basic ways to look at the value of a set of numbers. You will start by learning about the mean.
The mean, often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. This is also referred to as the arithmetic mean. The mean is the balance point of a distribution.
Mean = sum of the values
the number of values
For instance, take a look at the following example. Use the formula to calculate the mean number of hours that Stephen worked each month based on the example below.
Example
Problem
Stephen has been working on programing and updating a Web site for his company for the past 15 months. The following numbers represent the number of hours Stephen has worked on this Web site for each of the past 7 months:
24, 25, 31, 50, 53, 66, 78
What is the mean (average) number of hours that Stephen worked on this Web site each month?
Step 1: Add the numbers to determine the total number of hours he worked.
24 + 25 + 33 + 50 + 53 + 66 + 78 = 329
Step 2: Divide the total by the number of months.
329 over 7 equals 47
Answer
The calculations for the mean of a sample and the total population are done in the same way. However, the mean of a population is constant, while the mean of a sample varies from sample to sample.
Example
Problem
Mark operates Technology Titans, a Web site service that employs 8 people. Find the mean age of his workers if the ages of the employees are as follows:
55, 63, 34, 59, 29, 46, 51, 41
Step 1: Add the numbers to determine the total age of the workers.
55 + 63 + 34 + 59 + 29 + 46 + 51 + 41 = 378
Step 2: Divide the total by the number of months.
378 over 8 equals 47.25
Answer
Look at another approach. If you were to take a sample of 3 employees from the group of 8 and calculate the mean age for these 3 workers, would the results change?
Use the ages 55, 29, and 46 for one sample of 3, and the ages 34, 41, and 59 for another sample of 3:
mean equals sum of 55 plus 29 plus 46 over 3 mean equals sum of 34 plus 41 plus 59 over 3
mean equals 130 over 3 mean equals 134 over 3
mean equals 43.33 mean equals 44.66
The mean age of the first group of 3 employees is 43.33 years.
The mean age of the second group of 3 employees is 44.66 years.
The mean age for a sample of a population depends upon the values that are included in the sample. From this example, you can see that the mean of a population and that of a sample from the population are not necessarily the same.
In addition to calculating the mean for a given set of data values, you can apply your understanding of the mean to determine other information that may be asked for in everyday problems.
Example
Problem
Two weeks before Mark opened Technology Titans, he launched his company Web site. During those 14 days, Mark had an average of 24.5 hits on his Web site per day. In the first two days that Technology Titans was open for business, the Web site received 42 and 53 hits respectively. Determine the new average for hits on the Web site.
Step 1: Multiply the given average by 14 to determine the total number of hits on Mark's Web site.
24.5 x 14 = 343
Step 2: Add the hits for the first two days his business was open.
343 + 42 + 53 = 438
Step 3: Divide this new total by 16 to determine the new average.
mean equals to 438 over 16 equals 27.375
Answer
All values for the means you have calculated so far have been for ungrouped, or listed, data. A mean can also be determined for data that is grouped, or placed in intervals. Unlike listed data, the individual values for grouped data are not available, and you are not able to calculate their sum. To calculate the mean of grouped data, the first step is to determine the midpoint of each interval or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean.
The following example will show how the mean value for grouped data can be calculated.
Example
Problem
In Tim's office, there are 25 employees. Each employee travels to work every morning in his or her own car. The distribution of the driving times (in minutes) from home to work for the employees is shown in the table below.
Driving Times (minutes)
0 to less than 10
10 to less than 20
20 to less than 30
30 to less than 40
40 to less than 50
Number of Employees
3
10
6
4
2