MEASURE of CENTRAL TENDENCY of UNGROUPED DATA

Mean The mean for ungrouped data is obtained by dividing the sum of all values by the number of values in the data set. Thus, Mean for population data: Mean for sample data:

Example 1 : Given the 2002 total payrolls of five Major League Baseball (MLB) teams. Find the mean of the 2002 payrolls of these five MLB teams. MLB Team 2002 Total Payroll (millions of dollars) Anaheim Angels Atlanta Braves New York Yankees St. Louis Cardinals Tampa Bay Devil Rays 62 93 126 75 34

Solution: Thus, the mean 2002 payroll of these five MLB teams was $78 million.

Example 2: The following are the ages of all eight employees of a small company: 53 32 61 27 39 44 49 57 Find the mean age of these employees.

Solution: Thus, the mean age of all eight employees of this company is 45.25 years, or 45 years and 3 months.

Median The median is the value of the middle term in a data set that has been ranked in increasing order.

Median The calculation of the median consists of the following two steps: Rank the data set in increasing order Find the middle term in a data set with n values. The value of this term is the median.

Value of Median for Ungrouped Data

Example 1: 10 5 19 8 3 Find the median. The following data give the height gained (in centimeter) by a sample of five members of a health club at the end of two months of membership: 10 5 19 8 3 Find the median.

Solution: First, we rank the given data in increasing order as follows: 3 5 8 10 19 There are five observations in the data set. Consequently, n = 5 and

Therefore, the median is the value of the third term in the ranked data. 3 5 8 10 19 The median height gained for this sample of five members of this health club is 8 cm. Median

Mode The mode is the value that occurs with the highest frequency in a data set.

Example: The following data give the grades of 8 groups in a class. 77 69 74 81 71 68 74 73 Find the mode.

Solution: In this data set, 74 occurs twice and each of the remaining values occurs only once. Because 74 occurs with the highest frequency, it is the mode. Therefore, Mode = 74

Mode A data set may have none or many modes, whereas it will have only one mean and only one median. The data set with only one mode is called unimodal. The data set with two modes is called bimodal. The data set with more than two modes is called multimodal.

Example: Last year’s incomes of five randomly selected families were $36,150, $95,750, $54,985, $77,490, and $23,740. Find the mode. Solution: Because each value in this data set occurs only once, this data set contains no mode.

Example: The prices of the same brand of television set at eight stores are found to be $495, $486, $503, $495, $470, $505, $470 and $499. Find the mode.

Solution: In this data set, each of the two values $495 and $470 occurs twice and each of the remaining values occurs only once. Therefore, this data set has two modes: $495 and $470, bimodal .