Math in Our World Section 12.3 Measures of Average

Learning Objectives Compute the mean of a data set. Compute the median of a data set. Compute the mode of a data set. Compute the midrange of a data set. Compare the four measures of average.

Measures of Average In casual terms, average means the most typical case, or the center of the distribution. Measures of average are also called measures of central tendency, and include the mean, median, mode, and midrange.

Mean The Greek letter  (sigma) is used to represent the sum of a list of numbers. If we use the letter X to represent data values, then X means to find the sum of all values in a data set. The mean is the sum of the values in a data set divided by the number of values. If X1, X2, X3, . . . Xn are the data values, we use X to stand for the mean, and

EXAMPLE 1 Finding the Mean of a Data Set In 2003, there were 12 inmates on death row who were proven innocent and freed. In the 5 years after that, there were 6, 2, 1, 3, and 4. Find the mean number of death row inmates proven innocent for the 6 years from 2003 to 2008. SOLUTION

Mean for Grouped Data The procedure for finding the mean for grouped data uses the midpoints and the frequencies of the classes. This procedure will give only an approximate value for the mean, and it is used when the data set is very large or when the original raw data are unavailable but have been grouped by someone else. where f = frequency Xm = midpoint of each class n = f or sum of the frequencies

EXAMPLE 2 Finding the Mean for Grouped Data Find the mean for the price of GM stock for the first 74 days of trading in 2009. The frequency distribution is below.

EXAMPLE 2 Finding the Mean for Grouped Data SOLUTION We found the midpoint of each class in Example 4 of Section 12-2: the midpoints are 1.255, 1.755, 2.255, 2.755, 3.255, 3.755, and 4.255. Now we multiply each midpoint by the frequency: 1.255 31.59 31.57 44.08 39.06 30.04 21.275 198.87 Midpoint (Xm) Frequency (f) f Xm 1.255 1 1.755 18 2.255 14 2.755 16 3.255 12 3.755 8 4.255 5 Sums 74

EXAMPLE 2 Finding the Mean for Grouped Data SOLUTION Now we divide (f Xm) by the sum of the frequencies to get the mean: Midpoint (Xm) Frequency (f) f Xm 1.255 1 1.755 18 2.255 14 2.755 16 3.255 12 3.755 8 4.255 5 Sums 74 1.255 31.59 31.57 44.08 39.06 30.04 21.275 198.87 The mean closing price was $2.69.

Median According to payscale.com, the median salary for all federal government employees as of early 2009 was $60,680. This measure of average means that half of all government employees made less than $60,680, and half made more. Simply put, the median is the halfway point of a data set when it is arranged in order. When a data set is ordered, it is called a data array. The median will either be a specific value in the data set, or will fall between two values

Median Steps in Computing the Median of a Data Set Step 1 Arrange the data in order, from smallest to largest. Step 2 Select the middle value. If the number of data values is odd, the median is the value in the exact middle of the list. If the number of data values is even, the median is the mean of the two middle data values.

EXAMPLE 3 Finding the Median of a Data Set The weights of the five starting offensive linemen for the Pittsburgh Steelers in Super Bowl XLIII were 345, 340, 315, 285, and 317. Find the median weight. SOLUTION Step 1 Arrange the data in order. 285 315 317 340 345 Step 2 Select the middle value. The median weight was 317 pounds.

EXAMPLE 4 Finding the Median of a Data Set One of the authors of this book conducts a campus food drive at the end of each semester. The amount of food in pounds gathered over the last eight semesters is shown below. Find the median weight. 1,675 1,209 1,751 1,700 1,532 2,171 2,292 3,211

EXAMPLE 4 Finding the Median of a Data Set SOLUTION Step 1 Arrange the data in order. 1,209 1,532 1,675 1,700 1,751 2,171 2,292 3,211 Step 2 Find the middle value: in this case, it is in between 1,700 and 1,751, so we find the mean of those two values. The median weight is 1,725.5 pounds.

Mode The third measure of average is called the mode. The mode 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.

EXAMPLE 5 Finding the Mode of a Data Set These data represent the duration (in days) of U.S. space shuttle voyages for the years 2002–2008. Find the mode. 10 10 13 10 13 15 13 12 11 12 13 12 15 12 15 13 15 Source: Wikipedia

EXAMPLE 5 Finding the Mode of a Data Set SOLUTION If we construct a frequency distribution, it will be easy to find the mode—it’s simply the value with the greatest frequency. 10 10 13 10 13 15 13 12 11 12 13 12 15 12 15 13 15 The frequency distribution for the data is shown to the right, and the mode is 13. Days Frequency 10 3 11 1 12 4 13 5 15

EXAMPLE 6 Finding the Mode of a Data Set Six strains of bacteria were tested by the Centers for Disease Control 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 SOLUTION Since each value occurs only once, there is no mode.

EXAMPLE 7 Finding the Mode of a Data Set The number of wins in a 16-game season for the Cincinnati Bengals from 1997–2008 is listed below. Find the mode. 7 3 4 4 6 2 8 8 11 8 7 4 SOLUTION There are two numbers that occur three times: 4 and 8. No number occurs more than three times, so the data set has two modes, 4 and 8.

EXAMPLE 8 Finding the Mode for Categorical Data A survey of the junior class at Fiesta State University shows the following number of students majoring in each field. Find the mode. Business 1,425 Education 471 Liberal arts 878 General studies 95 Computer science 632 SOLUTION The values here are not the numbers, but the categories. The value that appears most often is business, so that is the mode.

Midrange The midrange is a very quick and rough estimate of the middle of a data set. It is found by adding the lowest and highest values and dividing by 2.

EXAMPLE 9 Finding the Midrange In Example 1, we saw that the number of death row inmates proven innocent for the years 2003–2008 was 12, 6, 2, 1, 3, and 4. Find the midrange. SOLUTION The lowest value is 1 and the highest is 12, so the midrange is

EXAMPLE 10 Comparing the Four Measures of Average The table below lists the number of golfers who finished the Masters tournament with a better score than Tiger Woods between 1997 and 2009. Find the mean, median, mode, and midrange for the data.

EXAMPLE 10 Comparing the Four Measures of Average SOLUTION It will be helpful to arrange the numbers in order: 0 0 0 0 1 1 2 4 5 7 14 17 21

Comparison of Measures of Average Strengths Weaknesses Mean Unique – there’s exactly one mean for any data set Factors in all values in the set Easy to understand Can be adversely affected by one or two unusually high or low values Can be time-consuming to calculate for large data sets Median Divides a data set neatly into two groups Not affected by one or two extreme values Can ignore the effects of large or small values even if they are important to consider

Comparison of Measures of Average Strengths Weaknesses Mode Very easy to find Describes the most typical case Can be used with categorical data like candidate preference, choice of major, etc. May not exist for a data set May not be unique Can be very different from mean and median if the most typical case happens to be near the low or high end of the range Midrange Very quick and easy to compute Provides a simple look at average Dramatically affected by extremely high or low values in the data set Ignores all but two values in the set