The Statistical Mean

Definition

Mean: The average of a set of N numbers. Below are the fundamental equations used for finding the mean.

Example

• Data = [ 5 2 4 3 8 6 ]

• Mean = ( 5 + 2 + 4 + 3 + 8 + 6 )/ 6 = 4.67

Note: Since all the observations were included in the calculation of this mean, it is referred to as the population mean.

Population Mean vs. Sample Mean

The mean of a set can be measured in two ways, either as a population mean or as a sample mean.

• If all possible observations for the data are included, then is used to represent the mean.
• is the population mean
• If a portion of observations are extracted from the whole, is used to represent the mean.
• is the sample mean.

The sample mean is used more often than population mean in engineering applications. It is used because it might be impossible to attain all the data, or there might be impossible to process all the data, or there is a specific period of time when the testing must be done. In these situations a good generalization of data results can be obtained from the sample mean.

Advantages and Disadvantages of the Sample Mean

The mean is a popular Statistical Measure because:

• It is familiar to most people.
• It reflects the inclusion of every item in the data set.
• It always exists.
• It is unique.
• It is easily used with other statistical measurements.

The mean is the center of gravity of the data, and for the most part, is better to use with other statistical measurements.

The major disadvantage to the use of the statistical mean is that it can be affected by extreme values in the data set and therefore be biased.

Exercises

1. Find the mean
   data = [ 1 2 3 4 5 6 7 ]

Mean:

2. A reporter travels deep into the woods to do a story on a group of monkeys. The reporter, realizing that these monkeys are very paranoid, decides to keep a careful watch on each monkey as it is brought into his presence. He notices that the monkeys jump more rapidly as he approaches them in leveling degrees and took note of the number of times that each monkey jumped when he approached them. These are the results of his count:

        Monkey          Number of Jumps

        Bubba                   6
        Waldo                   3
        Edna                    2
        Fred                    7
        Buford                  9
        Elizabeth               16
        Harvey                  4
        Nelson                  8
        Pete                    1
        Barbara                 6
        Herbert                 5
        Johnny                  10
        Red                     11
        Catherine               3
        Willard                 15

• What is the mean number of jumps by the monkeys whose names start with the letter "B" ?

Mean:

• Is the mean computed above
  a population mean?
  a sample mean?

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