Average

A R

eview of A

veraging Form ulas

M ath 1010 Interm

ediate A lgebra G

roup Project

T he Story:

A lyssa w

orks for an online com pany that helps thousands of sm

all business ow ners across the

country to advertise them selves and find custom

ers in need of their services.

A fter the job is com

pleted, A lyssa’s com

pany encourages the custom ers to review

the business on a scale of 1 to 5 stars. T

his review is saved w

ithin A lyssa’s database so that future custom

ers can see the average num

ber of stars a business has received on the online service. T hat w

ay the custom

ers can find the best rated person for the job.

A lyssa found that m

any business ow ners had questions about how

they could im prove their

rating. O ne com

m on question w

as: “H ow

m any 5-star review

s in a row do I need to im

prove m

y rating from w

here I’m at to a 4.5?” O

r a 4.2? O r w

hatever threshold they w ere interested in?

R ather than having to w

ork this out from scratch in every scenario, it is m

ore efficient to create a form

ula so the answ er can be found quickly and efficiently.

W e w

ill start out w ith som

e sim pler questions to becom

e fam iliar w

ith the situation and how

averages w ork. T

hen w e w

ill w ork our w

ay tow ards the full form

ula to help the business ow ners

answ er their question above. Y

ou w ill use skills you learned about solving form

ulas for a particular variable and w

orking w ith rational expressions.

A V

E R

A G

E S

W e w

ill be exploring how basic averages w

ork in this section. T his A

verage is also called the M

ean, or A rithm

etic M ean.

1. W

rite a form ula w

hich can find the average of tw o num

bers x and y.

A V

E R

A G

E = 𝑥

+ 𝑦

2

2. U

se your form ula to find the average of 16 and 34. Show

your process.

A V

E R

A G

E =

= =25

16 +

34 2

502

3. Explain in words: How does the process of finding the average change if there are 6 numbers to average?

The average will entail summing all the 6 numbers and dividing the total by 6.

4. Find the average of the six numbers 4, 6, 7, 12, 14, and 17. Show your process.

Average= = = =10 ∑𝑛

𝑖 = 1𝑥𝑖 𝑛

4 + 6 + 7 + 12 + 14 + 17 6

60 6

FREQUENCY TABLES Often very large data sets are averaged. The data sets include many numbers which are the same. A Frequency Table describes how many of each number are included in the set. Examine the following table as an example:

Number Frequency 1 2 2 4 3 3

The table explains the data set 1, 1, 2, 2, 2, 2, 3, 3, 3.

5. Find the average of the nine numbers above. Show your process. Round your answer to the nearest hundredth.

Average= = = =2.11 ∑𝑛

𝑖 = 1𝑥𝑖 𝑛

1 + 1 + 2 + 2 + 2 + 2 + 3 + 3 + 3 9

19 9

When the data set becomes too large, it becomes too tedious to add all of the numbers one by one. We need a different strategy. Examine the next frequency table, which shows the number of reviews a particular business owner has received of each number of stars.

Stars Frequency 1 21 2 11 3 12 4 46 5 60

6. Determine how many total reviews this business has received. Show your process.

Total =sum of the frequency

=21+11+12+46+60

=150

7. Rather than add up all of those numbers one at a time, we will add them in groups. As an example, if we add up all twelve of the 3s, we should use multiplication (since it represents adding the same number repeatedly). All of the 3s sum to 3 times 12 which is 36. In the following blanks, similarly write the sum of all numbers of each type.

Sum of 1s ____21______

Sum of 2s _____22_____

Sum of 3s ___36___

Sum of 4s ____184______

Sum of 5s ____300______

8. Use your results from the previous two parts to find the average number of stars the business has received. Show your process. Round your answer to the nearest hundredth.

Average= = = 3.7521 + 22 + 36 + 184 + 300 150

563 150

ONE MORE NUMBER To answer our final question, we will need to know what happens to an average if we add one more number.

9. Suppose a list of seven numbers has an average of 12. What is the SUM of those seven numbers? Show your process.

Average =sum/numbers in the list

Average=12

Numbers in the list=7

Sum=average*number in the list

=12*7

=84

10. Suppose the number 18 is added to the previous list of seven numbers to make eight numbers. What is the new SUM of all eight numbers?

New sum=sum+ new GPA points

New sum =84+18

=102