LESSON 1: Measures of Central Tendency

 Measures of Central Tendency

The Arithmetic Mean 

        Statistics involves the collection, organization, summarization, presentation, and interpretation of data. The branch of statistics that involves the collection, organization, summarization, and presentation of data is called descriptive statistics. The branch that interprets and draws conclusions from the data is called inferential statistics. One of the most basic statistical concepts involves finding measures of central tendency of a set of numerical data. We will consider three types of averages, known as the arithmetic mean, the median, and the mode. Each of these averages is a measure of central tendency for the numerical data.

        In statistics it is often necessary to find the sum of a set of numbers. The traditional symbol used to indicate a summation is the Greek letter sigma, . Thus the notation x, called summation notation, denotes the sum of all the numbers in a given set. We can define the mean using summation notation.

        Statisticians often collect data from small portions of a large group in order to determine information about the group. In such situations the entire group under consideration is known as the population, and any subset of the population is called a sample. It is traditional to denote the mean of a sample by (which is read as “x bar”) and to denote the mean of a population by the Greek letter  (lowercase mu).

The Median 

        Another type of average is the median. Essentially, the median is the middle number or the mean of the two middle numbers in a list of numbers that have been arranged in numerical order from smallest to largest or largest to smallest. 

      Any list of numbers that is arranged in numerical order from smallest to largest or largest to smallest is a ranked list.

Solution: 

 a. The list 4, 8, 1, 14, 9, 21, 12 contains 7 numbers. The median of a list with an odd number of entries is found by ranking the numbers and finding the middle number. Ranking the numbers from smallest to largest gives

 1, 4, 8, 9, 12, 14, 21 

 The middle number is 9. Thus 9 is the median. 

b. The list 46, 23, 92, 89, 77, 108 contains 6 numbers. The median of a list of data with an even number of entriesis found by ranking the numbers and computing themean of the two middle numbers. Ranking the numbers from smallest to largest gives

23, 46, 77, 89, 92, 108

The two middle numbers are 77 and 89. The mean of 77 and 89 is 83. Thus 83 is the median of the data.

The Mode

        A third type of average is the mode. The mode of a list of numbers is the number that occurs most frequently. Some lists of numbers do not have a mode. For instance, in the list 1, 6, 8, 10, 32, 15, 49, each number occurs exactly once. Because no number occurs more often than the other numbers, there is no mode. 

        A list of numerical data can have more than one mode. For instance, in the list 4, 2, 6, 2, 7, 9, 2, 4, 9, 8, 9, 7, the number 2 occur three times and the number 9 occurs three times. Each of the other numbers occurs less than three times. Thus 2 and 9 are both modes for the data.

Solution:

a. In the list 18, 15, 21, 16, 15, 14, 15, 21, the number 15 occurs more often than the other numbers. Thus 15 is the mode. 

b. Each number in the list 2, 5, 8, 9, 11, 4, 7, 23 occurs only once. Because no number occurs more often than the others, there is no mode.

        The mean, the median, and the mode are all averages; however, they are generally not equal. The mean of a set of data is the most sensitive of the averages. A change in any of the numbers changes the mean, and the mean can be changed drastically by changing an extreme value. In contrast, the median and the mode of a set of data are usually not changed by changing the extreme value.

        When the data set has one or more extreme values that are very different from the majority of data values, the mean will not necessarily be a good indicator of an average value. In the following example, we compare the mean, median, and mode for the salaries of 5 employees of a small company.

        The median is the middle number, ₱36,000. Because the ₱20,000 salary occurs the most, the mode is ₱20,000. The data contain one extreme value that is much larger than the other values. This extreme value makes the mean considerably larger than the median. Most of the employees of this company would probably agree that the median of ₱36,000 better represents the average of the salaries than does either the mean or the mode. 

The Weighted Mean

        A value called the weighted mean is often used when some data values are more important than others. For instance, many professors determine a student’s course grade from the student’s tests and the final examination. Consider the situation in which a professor counts the final examination score as 2 test scores. To find the weighted mean of the student’s scores, the professor first assigns a weight to each score. In this case the professor could assign each of the test scores a weight of 1 and the final exam score a weight of 2.

Solution: The B is worth 3 points, with a weight of 4; the A is worth 4 points with a weight of 3; the D is worth 1 point, with a weight of 3; and the C is worth 2 points, with a weight of 4. The sum of all the weights is 4 + 3 + 3 + 4, or 14.

Dillon’s GPA for the fall semester is 2.5. 

Solution: The numbers in the right-hand column of table 4.4 are the frequencies f for the numbers in the first column. The sum of all the frequencies is 40.

The mean number of laptop computers per household for the homes in the subdivision is 1.975 .