Data Description Measures of central tendency Measures of average are called measures of central tendency and include the mean, median, mode, and midrange. Measures taken by using all the data values in the populations are called parameters. Measures obtained by using the data values of samples are called statistics. A statistic is a characteristic or measure obtained by using the data values from a sample. A parameter is a characteristic or measure obtained by using all the data values for a specific population 1-The Mean: - The mean, also known as the arithmetic average, is found by adding the values of the data and dividing by the total number of values. For example, the mean of 3, 2, 6, 5, and 4 is found by adding 3 + 2 + 6 + 5 + 4 = 20 and dividing by 5; hence, the mean of the data is 20 5 = 4. The values of the data are represented by X's. In this data set, X 1 = 3, X 2 = 2, X 3 = 6, X 4 = 5, and X 5 = 4. To show a sum of the total X values, the symbol (the capital letter sigma) is used, and X means to find the sum of the X values in the data set. The mean is the sum of the values divided by the total number of values. The symbol X represents the sample mean. Where n represents the total number of values in the sample. For a population, the Greek letter µ (mu) is used for the mean. Where N represents the total number of values in the population. In statistics, Greek letters are used to denote parameters and Roman letters are used to denote statistics. Example -1- The ages in weeks of six kittens at an animal shelter are 3, 8, 5, 12, 14, and 12. Find the mean. 1
Example -2- The fat contents in grams for one serving of 11 brands of packaged foods, as determined by the U.S. Department of Agriculture, are given as follows. Find the mean. 6.5, 6.5, 9.5, 8.0, 14.0, 8.5, 3.0, 7.5, 16.5, 7.0, 8.0 Example -3- The scores for 25 students on a 5-point quiz are shown below. Find the mean. 2
Example -4- A random sample of the life expectancy of residents for 25 countries in Asia was selected, and the following frequency distribution was obtained. Find the mean. 3
2-The Median: - is the halfway point in a data set. Before one can find this point, the data must be arranged in order. When the data set is ordered, it is called a data array. The median either will be a specific value in the data set or will fall between two values. The median is the midpoint of the data array. The symbol for the median is MD. Steps in Computing the Median of a Data Array STEP 1 Arrange the data in order. STEP 2 Select the middle point. 4
Example -5- The weights (in pounds) of seven army recruits are 180, 201, 220, 191, 219, 209, and 186. Find the median. Each of these examples had an odd number of values in the data set; hence, the median was an actual data value. When there is an even number of values in the data set, the median will fall between two given values. Example -6- The number of tornadoes that have occurred in the united states in the last 8 years follows. Find the median. 684,764,656,702,856,1133,1132,1303 Example -7- Six customers purchased the following number of magazines: 1, 7, 3, 2, 3, 4. Find the median. 5
Example -8- Holmes Appliance recorded the number of videocassette recorders (VCRs) sold per month over a two-year period. Find the median. Example -9- Find median using relative frequencies for the distribution (shown here) of the miles 20 randomly selected runners ran during a given week? 6
STEP 1 STEP 2 STEP 3 STEP 4 7
3-The Mode: - The third measure of average is called the mode. The mode is the value that occurs most often in the data set. It is sometimes said to be the most typical case. The value that occurs most often in a data set is called the mode. A data set can have more than one mode or no mode at all. These situations will be shown in some of the examples that follow. Example -10- The following data represent the duration (in days) of U.S. space shuttle voyages for the years 1992-94. Find the mode? 8, 9, 9, 14, 8, 8, 10, 7, 6, 9, 7, 8, 10, 14, 11, 8, 14, 11 It is helpful to arrange the data in order, although it is not necessary. 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 14, 14, 14 Since 8-day voyages occurred five times a frequency larger than any other number the mode for the data set is 8. 8
Example -11- Six strains of bacteria were tested to see how long they could remain alive outside their normal environment. The time, in minutes, is recorded below. Find the mode. 2, 3, 5, 7, 8, 10 Since each value occurs only once, there is no mode. Example -12- Find the modal class for the frequency distribution of miles 20 runners ran in one week? The modal class is 20.5-25.5, since it has the largest frequency. Sometimes the midpoint of the class is used rather than the boundaries; hence, the mode could also be given as 23 miles per week. Example -13- A small company consists of the owner, the manager, the salesperson, and two technicians, all of whose annual salaries are listed here. (Assume that this is the entire population.). Find the mean, median, and mode? 9
4-The Midrange: - The midrange is a rough estimate of the middle. It is found by adding the lowest and highest values in the data set and dividing by 2. It is a very rough estimate of the average and can be affected by one extremely high or low value. The midrange is defined as the sum of the lowest and highest values in the data set divided by 2. The symbol MR is used for the midrange. Example -14- Last winter, the city of Brownsville, Minnesota, reported the following number of water-line breaks per month. Find the midrange? 2, 3, 6, 8, 4, 1 4-The Weighted mean: - Find the weighted mean of a variable X by multiplying each value by its corresponding weight and dividing the sum of the products by the sum of the weights Example -15- A student received an A in English Composition I (3 credits), a C in Introduction to Psychology (3 credits), a B in Biology I (4 credits), and a D in Physical Education (2 credits). Assuming A = 4 grade points, B = 3 grade points, C = 2 grade points, D = 1 grade point, and F = 0 grade points, find the student's grade point average. 10
Exercises For exercises 1 to 3 find mean, the median, the mode and midrange? 1- Twelve members of the high school cross-country team were asked how many minutes each ran during practice sessions. Their answers are recorded here? 32,28,35,37,43,51,61,39,48,51,53,49 2- The manager of a sports shop recorded the number of baseball caps he sold during the week. The data are shown here? 132, 121, 119, 116, 130, 121, 131 3- For 108 randomly selected college students, the following IQ frequency distribution was obtained? 4- The following numbers of books were read by each of the 28 students in a literature class. Find mean, the median, the mode class? 5- Find the weighted mean price of three models of automobiles sold. The number and price of each model sold are shown in the following list? 11