Average

A Review of Averaging Formulas

Math

1

0

10

Intermediate Algebra Group Project

The Story:

Alyssa works for an online company that helps thousands of small business owners across the country to advertise themselves and find customers in need of their services.

After the job is completed, Alyssa’s company encourages the customers to review the business on a scale of 1 to

5

stars. This review is saved within Alyssa’s database so that future customers can see the average number of stars a business has received on the online service. That way the customers can find the best rated person for the job.

Alyssa found that many business owners had questions about how they could improve their rating. One common question was: “How many 5-star reviews in a row do I need to improve my rating from where I’m at to a

4

.5?” Or a 4.

2

? Or whatever threshold they were interested in? Rather than having to work this out from scratch in every scenario, it is more efficient to create a formula so the answer can be found quickly and efficiently.

We will start out with some simpler questions to become familiar with the situation and how averages work. Then we will work our way towards the full formula to help the business owners answer their question above. You will use skills you learned about solving formulas for a particular variable and working with rational expressions.

AVERAGES

We will be exploring how basic averages work in this section. This Average is also called the Mean, or Arithmetic Mean.

1. Write a formula which can find the average of two numbers x and y.

AVERAGE =

2. Use your formula to find the average of 16 and

3

4. Show your process.

AVERAGE =

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

4. Find the average of the six numbers 4, 6, 7,

12

, 14, and 17. Show your process.

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:

2

3

Number			Frequency
			1				2
			4
			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.

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.

Frequency

1

2

3

4

	Stars
21
	11
12
46
	5	60

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

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 __________

Sum of 2s __________

Sum of 3s ___36___

Sum of 4s __________

Sum of 5s __________

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.

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/amount of #

10(2.9) =(sum/10)

Sum=29

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 =29+4=33

11. What is the new average of all eight numbers? Show your process.

New Average = new sum/#of classes =33/11=3.00 New GPA

MANY MORE REVIEWS

12. Suppose a business has a current average review of 3.2 stars. This is from 30 customer reviews. Using a similar process to the previous three questions, find the new average review for the business after four new 5-star reviews. Show your process and round your answer to the nearest hundredth.

(3.2) (30)=96

New sum = 96+5(4)

96+20=116

New Average = 116/34

= 3.41 Star Rating.

PUTTING IT ALL TOGETHER

Suppose a business currently has an average rating of R stars. This average is calculated from N current reviews.

13. Find a formula for the new average rating A that the business will have after receiving X new reviews in a row, all of them 5-star reviews. Try to carefully follow the steps you used in Part 12.

A = 5x+RN/x+N

14. If the business wants to know how many 5-star reviews in a row are needed to achieve a desired new rating, the formula in the previous part must be solved for X. Solve your formula in the previous part for X, and show your process.

Let’s put our new formula into action! Kevin’s Tree Service is a small business that currently has ratings given in the table below:

Stars

Frequency

1

4

2

1

3

4

11

5

10

18

15. First calculate the current average review for Kevin’s Tree Service. Show your process. Leave your answer in the form of a fraction.

AVERAGE =

16. Now use your formula from Part 14 to calculate how many 5-star reviews in a row are needed for Kevin’s Tree Service to achieve an average review of 4.5.

REFLECTION

17. Did this project change the way you think about how math can be applied in the real world? Do you believe this kind of analysis could be important to business owners? Did you expect Kevin’s Tree Service to need that many 5-star reviews to reach its goal? Does what you learned here make you more likely to carefully consider ratings you give for services you receive? Write at least two paragraphs addressing the above questions. Refer to specific parts of this project where

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

20000001017220

20000001017220

20000001017220

11. What is the new average of all eight numbers? Show your process.
New Average = new sum/#of classes
New sum=102
Total number of lists=8
Average=
102
8
=12.75
MANY MORE REVIEWS
12. Suppose a business has a current average review of 3.2 stars. This is from 30 customer
reviews. Using a similar process to the previous three questions, find the new average
review for the business after four new 5-star reviews. Show your process and round your
answer to the nearest hundredth.
Sum= (3.2) (30) =96
New sum = 96+5(4)
96+20=116
New average=
𝑛𝑒𝑤 𝑠𝑢𝑚
𝑛𝑒𝑤 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑖𝑠𝑡𝑠
New Average = 116/34
= 3.41

20000001017220

20000001017220

20000001017220

PUTTING IT ALL TOGETHER
Suppose a business currently has an average rating of R stars. This average is calculated from N
current reviews.
13. Find a formula for the new average rating A that the business will have after receiving X
new reviews in a row, all of them 5-star reviews. Try to carefully follow the steps you
used in Part 12.
Average=R
Number of lists=N
Sum =NR
New number of lists=N+X
New sum=NR+5X
A=
𝑁𝑅 + 5𝑋
𝑁 + 𝑋

14. If the business wants to know how many 5-star reviews in a row are needed to achieve a
desired new rating, the formula in the previous part must be solved for X. Solve your
formula in the previous part for X, and show your process.
A=
𝑁𝑅 + 5𝑋
𝑁 + 𝑋
Making X the subject
A(N+X) =NR+5X
AN+AX=NR+5X
AX-5X=NR-AN
X(5-A) =N(A-R)
X=
𝑁(𝐴 ― 𝑅)
5 ― 𝐴

Let’s put our new formula into action! Kevin’s Tree Service is a small business that currently
has ratings given in the table below:
Stars Frequency
1 4
2 1
3 10
4 11
5 18
15. First calculate the current average review for Kevin’s Tree Service. Show your process.
Leave your answer in the form of a fraction.
AVERAGE =
∑𝑓𝑖𝑥𝑖
∑𝑓𝑖
=
(1 ∗ 4) + (2 ∗ 1) + (3 ∗ 10) + (4 ∗ 11) + (5 ∗ 18)
4 + 1 + 10 + 11 + 18
=
170
44
=85
22
16. Now use your formula from Part 14 to calculate how many 5-star reviews in a row are
needed for Kevin’s Tree Service to achieve an average review of 4.5.
4.5=
𝑁𝑅 + 5𝑋
𝑁 + 𝑋
4.5=
44 ∗
170
44 + 5𝑋
44 + 𝑋
=170 + 5𝑋
44 + 𝑋
9
2
340+10x=396+9x

X=56
REFLECTION
17. Did this project change the way you think about how math can be applied in the real
world? Do you believe this kind of analysis could be important to business owners? Did
you expect Kevin’s Tree Service to need that many 5-star reviews to reach its goal? Does
what you learned here make you more likely to carefully consider ratings you give for
services you receive? Write at least two paragraphs addressing the above questions.
Refer to specific parts of this project where
This project really changed my thinking on how one can find an average of a large data
given the numbers with their respective frequencies. This analysis is important in
business owners since can be used for a large sample of data with various frequencies.
For the case of Kevin’s Tree Service, the star reviews were many compared to the
previous study where only 4 reviews were needed.
From this topic on average, ratings should be considered carefully since some many give
a large output which is unexpected from the usual rating. This will ensure there is a high
accuracy.

Becky Connelly
20000001017220

Becky Connelly
20000001017220

Becky Connelly
20000001017220