Math Module 4.1 Using Averages

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Averages What are averages? Surely you’ve heard someone say they’re “about average” at performing some kind of skill. This means that they’re not excellent, but not terrible; they’re right in the middle. In math, the definition of an average is actually very similar: A number that describes a central or common value in a set of numbers A good example would be the average score that a certain class achieved on an exam. This number would be very close to the middle of the various scores that the students received. There are probably a few scores above and below the average, and some right around it. This person’s score was probably above the average!

But did you know there are actually 3 common types of averages?! If you’ve used averages before, you were probably using the mean. This is the most common average used in daily life. In fact, most people just call the mean “the average.” But did you know there are actually 3 common types of averages?! They are: MEDIAN MODE MEAN AND

Add up all the numbers in the set MEAN Like we said before, the mean is definitely the most commonly used type of average. This is because the mean gives you the most accurate idea of what the range of numbers is like, and where most of the numbers fall in that range. Oh! Well the range just means the difference between the greatest and lowest numbers in the set. For example, the set of numbers from 0 to 10 have a range of 10! So a bigger range basically means a wider spread of numbers. Now just what is a range? So how do we find the mean? Actually, there are two steps: WHAT? Add up all the numbers in the set 2) Then divide the answer by the number of values

This is the mean score for the class! Let’s look at an example, and then this may make a little more sense… 12 students in a math class took a 10 point quiz. Here are the scores for everyone in the class: 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10 First of all, what is the range? 10 – 2 = 8 Well the greatest number is 10, and the lowest number is 2. So our range is… But it looks like most of the students scored a 6 or more, right? So what do you think the mean is? Let’s find out… STEP 1 STEP 2 Add all the numbers together Divide the answer by the number of values 2 + 3 + 3 + 4 + 5 + 6 + 6 + 7 + 8 + 9 + 9 + 10 72 = 6 = This is the mean score for the class! 72 There were 12 students who each got a quiz score. That means the number of values in the set is 12 12

Let’s look at one more example… 10 patients were in a medical study to test a new weight loss drug for a period of 3 months. The total amount of weight in pounds that each person lost was recorded: 2, 4, 6, 7, 10, 11, 15, 16, 17, 18 Okay, in this problem what is the range? 18 – 2 = 16 This time the greatest number is 18, and the lowest number is 2. So our range is… But it looks like most of the patients lost 10 pounds or more, right? So what do you think the mean is? STEP 1 STEP 2 Add all the numbers together Divide the answer by the number of values 2 + 4 + 6 + 7 + 10 + 11 + 15 + 16 + 17 + 18 106 = = 106 10.6 lbs There were 10 patients in the study. That means the number of values in the set is 10 10

Okay, ready to try it? Calculate the mean in the following sets of numbers. Write your answers on your Lesson Answer Sheet, then click to check your work. A college wants to find the mean age of students in a particular nursing class. There are 14 students in the class with the following ages: 18, 19, 21, 24, 26, 27, 28, 28, 29, 32, 36, 37, 42, 53 La Plata county wants to calculate the mean monthly rainfall for the area. The total rainfall for the county in inches was recorded every month for 1 year: 0.7, 1.0, 1.1, 1.2, 1.8, 1.9, 2, 2.3, 2.4, 3, 3.4, 3.8 A dietician wants to calculate a patients mean calorie intake per day. The patient’s total calorie intake was recorded daily for one week: 1900, 1950, 2200, 2300, 2350, 2400, 2650

Check your answers. Cross out any wrong answers and correct them on your answer sheet. Step 1 1) 18+19+21+24+26+27+28+28+29+32+36+37+42+53 = 420 Step 2 420 ÷ 14 = 30 years old Step 1 2) 0.7+1.0+1.1+1.2+1.8+1.9+2+2.3+2.4+3+3.4+3.8 = 24.65 Step 2 24.6 ÷ 12 = 2.05 inches Step 1 3) 1900+1950+2200+2300+2350+2400+2650 = 15,750 Step 2 15,750 ÷ 7 = 2,250 calories per day

Put the numbers in order from lowest to highest median The median is a very simple average that really just means the middle value in a ordered list of numbers; and by ordered we simply mean they are in order from lowest to highest. The median is useful because it tells us EXACTLY what the middle number of the set is… usually half of the numbers are below that value and half of the number are above that value So how do we find the median? Actually this is much easier than finding the mean, but there are still two steps: Put the numbers in order from lowest to highest 2) Then cross off one number on each end over and over until you’re left with the middle value

Put the numbers in order from lowest to highest Let’s see this in action with an example… At a particular company, the weekly salaries of 5 employees are: $750, $245, $550, $310, $400 STEP 1 Put the numbers in order from lowest to highest 245, 310, 400, 550, 750 STEP 2 Cross off one number on each end over and over until you’re left with the middle value The mode is $400 245, 310, 400, 550, 750

Easy, right? Well, there’s one more thing about medians that we need to talk about. When there is an odd number of values in the set, you will get the median easily. BUT, where there is an even number of values, you will actually end up with two numbers in the middle: Brittany needs to buy six ingredients at the store to make dinner. Here are the prices of the ingredients: $4.60, $1.25, $6.75, $3.40, $5, $2 We can start out with the same two steps as before… STEP 1 STEP 2 Put the numbers in order from lowest to highest Cross off one number on each end over and over until you’re left with the middle value 1.25, 2, 3.40, 4.60, 5, 6.75 1.25, 2, 3.40, 4.60, 5, 6.75 But now we need to add one more step… STEP 3 (3.4 + 4.6) Mean step 1 = The median is 4.0 Find the number that is exactly half way between the remaining values. If this isn’t obvious, you can easily find it by calculating the mean of the two numbers. 2 Mean step 2

Okay, ready to try it? Calculate the median in the following sets of numbers. Write your answers on your Lesson Answer Sheet, then click to check your work. 13 college biology students just took their midterm exam. Here are the scores for everyone in the class: 37, 96, 18, 74, 87, 68, 85, 73, 75, 45, 79, 83, 92 A man tracked his weight once a week for 2 months. His weights were: 190, 193, 195, 192, 196, 198, 200, 195 A nurse took the heart rate for a patient 10 times during a fitness test. The patient’s pulse rates were: 105, 118, 132, 138, 137, 142, 150, 148, 154, 157

Take the mean of the two middle numbers Check your answers. Cross out any wrong answers and correct them on your answer sheet. Step 1 1) 18, 37, 45, 68, 73, 74, 75, 79, 83, 85, 87, 92, 96 Step 2 18, 37, 45, 68, 73, 74, 75, 79, 83, 85, 87, 92, 96 Step 1 2) 190, 192, 193, 195, 195, 196, 198, 200 Step 2 *In this problem, we’re left with two numbers in the middle, but they’re the same number! So the median is just 195. 190, 192, 193, 195, 195, 196, 198, 200 Step 1 3) 105, 118, 132, 137, 138, 142, 148, 150, 154, 157 Step 2 105, 118, 132, 137, 138, 142, 148, 150, 154, 157 Step 3 (138 + 142) = 140 Take the mean of the two middle numbers *You may have even figured this out without having to take the average, just by seeing that the two numbers were 4 apart, and so 2 more than 138 and 2 less than 142 would be the middle between the two. 2

Mode So how do we find the mode? BUT WAIT! The mode is probably the simplest average of all! The mode is just the most common number in a list of values. The mode is useful because it tells us which values tend to happen the most times in the set, even if that number is not near the middle. So how do we find the mode? This is super easy, and it’s only one step… Just look for the number that appears the most times in the set. That number is the mode! What if two different numbers appear the same number of times? BUT WAIT! That’s actually one unique thing about the mode… The mode is the ONLY average that can have more than 1 number for the answer!!!

Let’s look at a couple examples… The cost of renting a hotel room varies by the day of the week: Mon Tue Wed Thurs Fri Sat Sun $55 $55 $55 $60 $70 $70 $60 So what is the mode? Yes, the mode here is $55 because there are more of them than any other number in the set. Okay, how about this one… The county soccer league is having a tournament with a different number of games every day this week: Mon Tue Wed Thurs Fri Sat Sun 2 3 3 3 5 5 5 So what is the mode here? Did you say 3 and 5? Yes! They are both the most common numbers because there are 3 of each of them.

Alright, your turn again Alright, your turn again. Find the mode in the following sets of numbers. Write your answers on your Lesson Answer Sheet, then click to check your work. A company is trying to figure out the most common age of their entry-level workers. There are currently 12 entry-levels at the company, and their ages are: 21, 25, 32, 26, 25, 27, 27, 32, 32, 28, 25, 32 A woman tracked her daily calorie intake for 1 week to find the most common amount of calories she eats. Her totals were: 1700, 1900, 2050, 1700, 1750, 1900, 1850 A runner is training for a marathon and wants to figure out his most common total time. He records his total time during 5 separate practice runs: 2 hrs 37 min, 2 hrs 36 min, 2 hrs 36 min, 2 hrs 39 min, 2 hrs 37 min, 2 hrs 36 min, 2 hrs 38 min, 2 hrs 35 min, 2 hrs 35 min, 2 hrs 37 min

Check your answers. Cross out any wrong answers and correct them on your answer sheet. 1) 21, 25, 32, 26, 25, 27, 27, 32, 32, 28, 25, 32 There are four “32s” in the set, making it the mode. 2) 1700, 1900, 2050, 1700, 1750, 1900, 1850 In this set, there are actually two modes! There are two “1700s” AND two “1900s.” So both of these numbers are the mode! 3) 2 hrs 37 min, 2 hrs 36 min, 2 hrs 36 min, 2 hrs 39 min, 2 hrs 37 min, 2 hrs 36 min, 2 hrs 38 min, 2 hrs 35 min, 2 hrs 35 min, 2 hrs 37 min Here’s a set with two modes again! There are three“2 hrs 37 min” AND three “2 hrs 36 min.” So both of these numbers are the mode!

Practice problems Ready to put it all together? Click on the link below to take a practice quiz using all three types of averages! (You can use the bottom and back sides of your Lesson Answer Sheet for scratch paper) http://www.mathopolis.com/questions/q.php?id=1876&site=1&ref=/data/central-measures.html&qs=1876_1877_1878_1879_1880_1881_3050_3051_3052 Don’t be afraid to ask your coach if you need a little bit of help. The last few questions can be a little tricky!

Review and Practice Turn in your first answer sheet to your coach. Then complete the following worksheets (you should have a print out of them): Calculating Range and Mean Finding Median and Mode Once you complete each worksheet, ask your coach for the answer key and correct your work. Don’t worry, you’ll only be graded for completion on this part. Finally, click here to review some flash cards with all of the terms you learned in this lesson (you can also play games with the terms!): https://quizlet.com/101456269/flashcards

CONGRATULATIONS! You now have a good introduction to using averages! Ready to take the quiz?