INTERPOLATION AND
EXTRAPOLATION
Tu/W 9/4,5
Honors Physics
INTERPOLATION AND
EXTRAPOLATION
A hypothesis can be used to predict the results of
observations or the existence of other phenomena.
There are two types of predictions that can be made.
One is called extrapolation;
the other is called interpolation.
EXTRAPOLATION
Extrapolated predictions are those that are
made outside of the known data points.
MEMORY TIP:
• “exit” = how to get outside;
• “extrapolate” = predicted outside the data points
Trends in the known data can often be used to make
accurate extrapolated predictions; however, this is not
always the case.
DANGER OF EXTRAPOLATION
A young
man's
parents
kept track
of his
height
through the
years, as
shown in
this graph.
DANGER OF EXTRAPOLATION
Extrapolation
shows that
Bryan will be
about 10 feet
tall when he's
30 years old.
What faulty
assumption
was made in
this
extrapolation?
INTERPOLATION
Interpolated predictions are those that are made between
known data points.
MEMORY TIP:
“inter”state highways go between states;
“inter”polations predict between data points
An
interpolation
of this data
would lead
one to the
prediction
that Bryan
was about 4.5
ft tall at the
age of 14.
Is this a
reasonable
prediction?
INTERPOLATION
PREDICTING USING BEST-FIT LINES/CURVES
AND THEIR EQUATIONS
Thanks to the genetic influence of my mother’s brothers (who
range from 6’2” to 6’7”), my oldest child is 6’4”. Here’s his height
data through the years:
Height (in)
100
y = 2.593x + 31.144
R² = 0.9567
80
60
40
20
0
0
5
10
15
Age (yrs)
20
25
Note that the equation of the best-fit line for this data is
given by
y = 2.593x + 31.144
Translating this into something meaningful
for height and age gives
Height, in = (2.593 in/yr)(Age,yrs) + 31.144 in
This means that if you need to find out the height
when he was 5 years 3 months old, you’d calculate
Height, in = (2.593inches/year)(5.25 yrs) + 31.144 in
= 44.75725 inches.
Does this seem reasonable? Why/why not?
How old was this child when he was 5’0” tall?
First, remember that the basic relationship is
Height, in = (2.593 in/yr)(Age,yrs) + 31.144 in
Let’s translate this into
H for height in inches and A for age in years:
H = 2.593A + 31.144
BUT
Remember that you’re asked to find the Age at a certain
Height, NOT the Height at a certain Age!
To do this, it’s time for ALGEBRA.
H = 2.593A + 31.144
To find the Age, we have to solve for A.
First, subtract 31.144 from both sides:
H - 31.144 = 2.593A + 31.144 – 31.144
leaving us with
H – 31.144 = 2.593A
Next, divide both sides by 2.593:
𝑯 − 𝟑𝟏. 𝟏𝟒𝟒 𝟐. 𝟓𝟗𝟑𝑨
=
𝟐. 𝟓𝟗𝟑
𝟐. 𝟓𝟗𝟑
which results in
𝑯 −𝟑𝟏.𝟏𝟒𝟒
=A
𝟐.𝟓𝟗𝟑
FINISHING UP
𝑯 −𝟑𝟏.𝟏𝟒𝟒
𝟐.𝟓𝟗𝟑
=A
So to find his age when he was 5’0” tall
(5.0 ft = 60.0 inches), substitute in 60.0 for H:
𝟔𝟎.𝟎 −𝟑𝟏.𝟏𝟒𝟒
𝟐.𝟓𝟗𝟑
=A
A = 11.1 years
Does this seem reasonable? Why/why not?
IS IT ACCURATE? IS IT PRECISE?
You can use information from your graph to
determine the accuracy and precision of your data.
Remember . . .
Accuracy is how close the experimental data is to
the accepted value;
Precision is how close your points are to each other;
in this case, how close all the points are to being on
that best-fit line.
Circumference (cm)
PRECISE: R2 ~ 1.00
ACCURATE: SLOPE ~ PI
This data is both
accurate and precise.
300
250
200
150
100
50
0
y = 3.14x - 0.0028
R² = 1
0
20
40
60
Diameter (cm)
80
100
Circumference (cm)
PRECISE: R2 ~ 1.00
NOT ACCURATE: SLOPE IS NOT PI
This data is inaccurate and precise.
350
300
250
200
150
100
50
0
y = 3.8501x - 0.0045
R² = 1
0
50
Diameter (cm)
100
NOT PRECISE: R2 IS NOT ~1.00
ACCURACY: SLOPE ~ PI
Circumference (cm)
This data is accurate and imprecise
250
200
150
100
50
0
y = 3.1396x - 3.9668
R² = 0.9403
0
20
40
Diameter (cm)
60
80
NOT PRECISE: R2 IS NOT ~1.00
NOT ACCURATE: SLOPE IS NOT ~ PI
Circumference (cm)
This data is inaccurate and imprecise.
120
y = 1.0578x + 23.204
R² = 0.765
100
80
60
40
20
0
0
20
40
60
Diameter (cm)
80
100
FOR NEXT TIME:
1. Make sure you have all questions answered and
your graph/data printed BEFORE you come to
class next time.
2. Download the Scientific Notation worksheet and
complete it on your own paper.
3. Practice Scientific Notation questions online. Be
ready for a quiz next class period!