3.2-Measures of Center

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1 3.2-Measures of Center Characteristics of Center: Measures of center, including mean, median, and mode are tools for analyzing data which reflect the value at the center or middle of a set of data. We will learn to not only determine the value of each measure of center, but also interpret those values. Arithmetic Mean: Referred to as just the mean, it is the measure of center obtained by adding the values and dividing the total by the number of values. What most people call an average. Notation: S denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of data values in a sample. N represents the number of data values in a population. x is pronounced x-bar and denotes the mean of a set of sample values. µ is pronounced mu and denotes the mean of all values in a population. Formula for the mean of a sample: Formula for the mean of a population: Example: The amount of time (in hours) that Sam studied for an exam on each of the last five days is given below. Find the mean study time. Round your answer to one more decimal place than is present in the original data values Solution: If we assume that Sam studied on more than just these 5 days, then these values are a sample of Sam s daily study times and therefore we choose the formula for a sample. x n Advantages and Disadvantages of the Mean: One advantage of the mean is that it is relatively reliable, so that when samples are selected from the same population, sample means tend to be more consistent than other measures of center. Another advantage is that it takes every data value into account. The mean is not a resistant measure of center. This means that it is very sensitive to changes caused by extreme values.

2 Median: The median of a data set is the measure of center that is the middle value when the original data values are arranged in order of increasing or decreasing magnitude. The median is often written using the notation x ~ which is pronounced x-tilde. To find the median, first sort the values, then follow one of the following procedures: 1. If the number of data values is odd, the median is the number located in the exact middle of the list. 2. If the number of data values is even, the median is found by computing the mean of the two middle numbers. Example: The temperatures (in degrees Fahrenheit) in 7 different cities on New Year's Day are listed below. Find the median temperature Solution: Because there are an odd number of values, simply arrange the values in order and the median is the middle value The median temperature is 59 F. Example: The distances (in miles) driven in the past week by each of a company's sales representatives are listed below. Find the median distance driven Solution: Because there are an even number of data values, arrange the values in order and then locate the two values closest to the middle. Determine the mean of these two values x n The median number of miles driven is 222. Advantages and Disadvantages of the Median: An advantage to the median is that it is easy to determine quickly. The median is a resistant measure of center because it does not change by large amounts due to the presence of extreme values. Example: Given the two data sets below, notice how the mean is skewed to the low side and yet the median is unaffected by the extreme low value in the second data set ~ ~ 85

3 Mode: The mode of a data set is the value that occurs with the greatest frequency. A data set can have one mode, more than one mode, or no mode. Bimodal: Multimodal: No Mode: Two data values occur with the same greatest frequency More than two data values occur with the same greatest frequency No data value is repeated Advantages and Disadvantages of the Mode: The mode is not used much with numerical data. However, the mode is the only measure of center that can be used with the nominal level of measurement (data that is categorical, such as names or labels.) Example: Find the mode of the data set below: Solution: The values of 56 and 92 both occur twice and no other value occurs more than that. Therefore, the data set is bimodal. Midrange: The midrange of a data set is the measure of center that is the value midway between the maximum and minimum values in the original data set. It is found by adding the maximum data value and the minimum data value then dividing by two. Midrange Maximum.. data.. value + Minimum.. data.. value 2 Example: A meteorologist records the number of clear days in a given year in each of 21 different U.S. cities. The results are shown below. Find the midrange Solution: The maximum data value is 169 and the minimum data value is 52. Therefore we have: Midrange The midrange of the data set is Advantages and Disadvantages of the Midrange: Because the midrange uses only the maximum and minimum vales of a data set, it is too sensitive to these extremes, so the midrange is rarely used. The midrange however is very easy to compute and helps to reinforce that there are several different ways to define the center of a data set. It is sometimes incorrectly used for the median.

4 Mean from a Frequency Table: When working with data summarized in a frequency table, we do not know the exact values falling within a class. To make calculations possible, we assume that all sample values in each class are equal to the class midpoint and use the following formula. ( f x) f where f is the frequency of each value in a particular class. Example: A company had 80 employees whose salaries are summarized in the frequency distribution below. Find the mean salary. Solution: Using the formula, we have: 17(7500.5) + 20(12,500.5) + 12(17,500.5) + 14(22,500.5) + 17(27,500.5) 80 17, Weighted Mean: When data values are assigned different weights, we can compute a weighted mean using the following formula: ( w x) w where w is the weight of each value in the data set. Example: A student earned grades of C, A, B, and A. Those courses had these corresponding numbers of credit hours: 4, 5, 1, and 5. The grading system assigns quality points to letter grades as follows: A4, B3, C2, D1 and F0. Compute the grade point average (GPA) and round the result to two decimal places. Solution: We will use the number of credit hours as the weights, therefore C4, A5, B1, A5. ( w x) 4(2) + 5(4) + 1(3) + 5(4) w 15 15

5 The student has a GPA of 3.4. Compare this to the mean if the GPA was not weighted. x x n The weighted mean is slightly higher because an A is weighted more than the B or C. Example: Michael gets test grades of 73, 77, 82, and 86. He gets a 93 on her final exam. Find the weighted mean if the tests each count for 15% and the final exam counts for 40% of the final grade. Round to one decimal place. Solution: The percentages act as weights. Note that because all the percentages must add up to one, the denominator will simply be one. 0.15(73) (77) (82) (86) + 0.4(93) ( w x) w Skewness: A comparison of the mean, median and mode can reveal information about the skewness of a set of data. A distribution of data is skewed if it is not symmetric and extends more to one side then the other. Contrast this to a symmetric distribution where the left side is a mirror image of the right side.

6 Best Measure of Center: Example: Professor Strong s statistic class took an exam which had a median of 85%, a mode of 86% and a mean of 75%. Give a reason why the mean is so much lower than the median and mode. Which of the three measures of center is the most useful for understanding how the class as a whole did on the exam? Solution: One student scored a 15% on the exam. Because the mean is sensitive to extreme values it will be lowered by this score. The median and mode will not be affected by it at all. The median provides a more accurate picture of the class as a whole. The fact that the mean is much lower suggests that at least one student needs extra help.