How do you find the mean of a set of numbers add up all the items and divide by?
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Mean, Median, Mode, and Range Three types of averages can be used to describe a data set. Averages
Math In Real Life Example  A marine biologist records the locations of deep sea jellies in relation to the ocean surface. Jellies are found at -2,278 feet, -1,875 feet, -3,210 feet, -2,755 feet, -2,407 feet, and -2,901 feet. What is the average location of a deep sea jelly? Find the Mean To find the mean of the 6 locations of the deep sea jellies in the problem above, divide the sum of the locations by 6.The mean location in relation to the ocean surface is -2571 ft. Guided Practice (Ask your tutor for help.)
2) Find the mean of the data. Finding Median, Mode, and Range
$7.20, $13.25, $14.94, $16.56, $18.74, $19.99, $19.99, $29.49 Median The data set has an even number of prices, so the median is the mean of the two middle values, $16.56 and $18.74. Mode The price that occurs most often is $19.99. This is the mode. Range
Range = $29.49-$7.20=$22.29 Choosing a Representative Average Groups A and B try a new ice cream flavor and rate it on a scale of 1 to 10 as shown. Which average best represents each group?
The mean, median, and mode are very close. So each average is a fair representation of the ratings as a group. The mean is higher than all but 3 ratings. The mode is equal to the lowest rating. So, mean and mode are not good choices. The median best represents the ratings. The mean of a set of numbers, sometimes simply called the average , is the sum of the data divided by the total number of data. Example 1 : Find the mean of the set {2,5,5,6,8,8,9,11} . There are 8 numbers in the set. Add them all, and then divide by 8 . 2 + 5 + 5 + 6 + 8 + 8 + 9 + 118=548=6.75 So, the mean is 6.75 . The Median of a Data SetThe median of a set of numbers is the middle number in the set (after the numbers have been arranged from least to greatest) -- or, if there are an even number of data, the median is the average of the middle two numbers. Example 1 : Find the median of the set {2,5,8,11,16,21,30} . There are 7 numbers in the set, and they are arranged in ascending order. The middle number (the 4 th one in the list) is 11 . So, the median is 11 . Example 2 : Find the median of the set {3,10,36,255,79,24,5,8} . First, arrange the numbers in ascending order. {3,5,8,10,24,36,79,255} There are 8 numbers in the set -- an even number. So, find the average of the middle two numbers, 10 and 24 . 10 + 242=342=17 So, the median is 17 . The Mode of a Data SetThe mode of a set of numbers is the number which occurs most often. Example 1 : Find the mode of the set {2,3,5,5,7,9,9,9,10,12} . 2 , 3 , 7 , 10 and 12 each occur once. 5 occurs twice and 9 occurs three times. So, 9 is the mode. Example 2 : Find the mode of the set {2,5,5,6,8,8,9,11} . In this case, there are two modes -- 5 and 8 both occur twice, whereas the other numbers only occur once. In this lesson supplement we will explore the the three most common summaries of a collection of numbers or data. These are called the mean, the median and the mode. The mean is a fancy word for the average, the median is the middle number, and the mode is the number that occurs most frequently. Finding the Mean It is a common occurrence to have several numbers, called data, that we want to make sense of. Most people are familiar with the word "average" and many know that to find the average, we add up all the numbers and divide by the total. Another word for the average is the mean. To find the mean, just all up all the numbers and divide by the total number of numbers. We can think of this as a three step process: Step-by-Step Process to Find the Mean Step 1: Add up all the numbers. The result is called the sum. Step 2: Count how many numbers there are. This number is called the sample size. We use the letter n for the sample size. Step 3: The mean is the sum divided by the sample size n. Now lets go to an example. Example 1 Over the past five weekends, Steve kept track of the total amount of money he spent. $20, $50, $70, $80, $30 Find the mean amount of money Steve spent over the past five weekends. Solution Step 1: Add the numbers to find their sum. 20 + 50 + 70 + 80 + 30 = 250 Step 2: How many numbers where there. n = 5 Step 3: Divide the sum of the numbers by the sample size 'n' 250 Therefore the mean of these numbers is 50. We conclude that the mean amount of money that Steve spent over the past 5 weekends was $50. Now try one by yourself. If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear. Exercise 1 The chart below shows the times in minutes that three runners had in their three trials for the one mile track race. Trial 1Trial 2Trial 3Trial 4Enrique 8779Monique 9667Sam 1012911What was Monique's mean time? Answer: Finding the Median Another way of describing the data is by looking at the middle number. When there are an odd number of values, we can just find the value so that there are the same number of values above as there are below this middle value. When there is an even number of values, there is an issue in there there is not one number that acts as a middle value. Instead, the two middle numbers such that there are the same number of values above as below these two middle numbers. As a compromise, we take the average of these two middle numbers. We call this result the median of the data. It might help to summarize this in a 3-step process. Step by Step Process for Finding the Median Step 1: Put the numbers in numerical order from smallest to largest. Step 2: If there is an odd number of numbers, locate the middle number so that there is an equal number of values to the left and to the right. If there is an even number of numbers locate the two middle numbers so that there is an equal number of values to the left and to the right of these two numbers. Step 3: If there is an odd number of numbers, this middle number is the median. If there is an even number of numbers add the two middles and divide by 2. The result will be the median. Example 2 Rosa measured the weight in pounds of seven packages bags of oranges that were purchased at her fruit stand. The weights are shown below. 18, 10, 13, 10, 17, 11, 9 Find the median weight Solution Step 1: First, put the numbers in numerical order from smallest to largest. 9, 10, 10, 11, 13, 17, 18 Step 2: Notice that there are 7 numbers. This is an odd number of values, so we locate the middle number. The middle number is 11. Notice that there are an equal number (3) of numbers to the left of 11 and to the right of 11. 9, 10, 10, 11, 13, 17, 18 Step 3: We are in the case that there are an odd number of values, so the median is this middle number. That is, the median is 11. Now try one by yourself. If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear. Exercise 2 Five of Chang-hee's homework scores for her math class have been graded. Her scores are shown below. Find her median homework score. Answer: Here is another example of finding the median. Example 3 There are six brothers and sisters in Lupe's family. Their ages are shown below 22, 18, 11, 14, 20, 11 Find the median age of the brothers and sisters in Lupe's family. Solution Step 1: First, put the numbers in numerical order from smallest to largest. 11, 11, 14, 18, 20, 22 Step 2: Notice that there are 6 numbers. This is an even number of values, so we locate the two middle numbers. The two middle numbers are 14 and 18. Notice that there are an equal number (2) of values to the left and to the right of these two middles. 11, 11, 14, 18, 20, 22 Step 3: We are in the case that there are an even number of values, so the median is this average of the two middle numbers. Add these two middle numbers and divide by 2. 14 + 18 So the median age of the brothers and sisters in Lupe's family is 16. Now try one by yourself. If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear. Exercise 3 Oscar priced eight desk lamps. The prices are shown below. $24.00, $48.00, $25.00, $30.00, $36.00, $52.00, $25.00, $90.00 What is the median price? Answer: Finding the Mode A third number that we use to describe data is called the mode of the data. The mode is the number or numbers that occur the most frequently. Here is a 2-step process for finding the mode. Two-Step Process for Finding the Median Step 1: Put the numbers in numerical order from smallest to largest. Step 2: Go down the list and determine if there is a number that appears in the list more than any other numbers. If the is a tie for the most frequently occurring number, then just state that both numbers are the mode. Example 4 The box below shows the number of customers that a restaurant has had during the past nine days 90, 50, 70, 80, 30, 60, 50, 30, 50 What is the mode of these data? Solution Step 1: First, put the numbers in size place, smallest to largest. 30, 30, 50, 50, 50, 60, 70, 80, 90 Step 2: Now go down the list and determine if there is a number that appears in the list more than any other number. 30, 30, 50, 50, 50, 60, 70, 80, 90 Here that number is 50. Therefore, 50 is the mode. Now try one by yourself. If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear. Exercise 4 The number of bookcases that each of 10 workers produced at a custom furniture factory is shown in the box below 4, 6, 2, 4, 6, 5, 2, 7, 6, 3 What is the mode of these data? Answer: Now for another example. Example 5 7, 9, 9, 4, 11, 7, 9, 12, 3, 7 What is the mode of these data? Solution Step 1: First, put the numbers in size place, smallest to largest. 3, 4, 7, 7, 7, 9, 9, 9, 11, 12 Step 2: Now go down the list and determine if there is a number that appears in the list more than any other number. 3, 4, 7, 7, 7, 9, 9, 9, 11, 12 Here we see that both the numbers 7 and 9 appear three times. Therefore, both 7 and 9 are both modes NOTE: There may be No mode, ONE mode, or multiple modes for a given sample of numbers. Now try one by yourself. If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear. Exercise 4 Sarah was studying how many pairs of Stellar Jays were living within each of 11 acres of land in the national forest. Her data are shown in the box below |
