Algebra 1 Common Core Student Edition, Grade 8-9

C oncept Byte

Use W ith Lesson 5 -7

Using Residuals

Common Core State Standards

S-ID.B.6b Informally assess the f it o f a function by

plotting and analyzing residuals.

MP 7

In Lesson 5-7, you learned how to assess the line of best fit by using the correlation

coefficient r. In this activity you will learn how to determ ine w hether the best linear

function is a good fit for the data. As you learn different types of functions in future

chapters, residual plots will help you analyze the fit of other models.

A residual is th e difference betw een th e y-value of a d ata p o in t a n d the

corresponding y-value of a m odel for th e data set.

You can find a residual by calculating y — y , w here y represents th e y-value of the

data set an d y represents th e corresponding y-value predicted from th e m odel.

Activity

The linear function y = 2.3x + 33.4 models the data shown below.

Mean Heights of Boys Ages 5 to 13

Age (yr) (^678) 91011 12
......^13
Height (in.) 44.5 46.9 49.7 52.2 54.4 55.7 58.5 60.9 63.1

1. Com plete a table like th e one started below to calculate th e residuals.

[y hdataU line of best fit

point predit'lCU /

?> l 'it negative

[ l p o s i t i v e i/res id u al

j i f re.1 I !1^^ jal data

point

predicted

value dtt

X

Height (y) Predicted value y = 2.3x + 33.4 Residual y-y

5 44.5 2.3(5) + 33.4 = 44.9 44.5 - 44.9 = - 0 .4

I j. :

13

-U

■ ■

You can plot each of the points (x, y - y ) on a coordinate plane, a n d analyze the

residual plot to assess w hether the function is a good fit for th e data. For a good fit, the

points ap p ear to have a ra n d o m pattern.

y

i • •

• 
  


• X
  0 L
  Li i »> •
  9
  A

There is no apparent pattern in the

residual plot. This indicates th a t the

model function is a good fit fo r the data.

I •

• 4 A •

L

• 0*

(^0) • VI 3
The points in the residual p lo t form
a pattern. This indicates th a t the model
function is not a good fit for the data.
344 Concept Byt e Using Residuals