3 excel trendline

8/2/2019 3 Excel Trendline

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Trendline Analysis

• Experimental measurements are never

perfect which results in data “scatter”

• From an inspection of a plot of the data, it

is apparent that there is a clear “trend” in

the dependent variable y ( x ) with respect to

the independent parameter x .

• A “trendline” analysis is used to find the

best “fit” of a function or best choice of a

set of coefficients for a function to match

the trend indicated by the data.

• For the example shown, the functional

form is a second order polynomial,

• MS Excel automatically determines that

the polynomial best represents or fits the

data when the coefficients are

cbxax y 2

8810

9416

9440

.

.

.

c

b

a

y = 0.944x2 - 6.941x - 10.88

R² = 0.965

-200

0

200

400

0 10 20 30

y ( x ) - ( d e p e

n d e n t p a r a m e t e r )

x - (independent parameter)

Trendline Analysis

exper imental data

2nd order polynomial



y = 0.944x2 - 6.941x - 10.88

R² = 0.965

-200

0

200

400

0 10 20 30

y ( x ) - ( d e p e

n d e n t p a r a m e t e r )

x - (independent parameter)

Trendline Analysis

exper imental data

2nd order polynomial

Trendline Analysis

The trendline analysis uses a “least squares”

curve fitting procedure.

• the deviation of the i th data point is the

difference between the value of thedependent variable y i at x i and the curve

fitting function f() evaluated at x i .

• the coefficients a, b, and c in f ( x ) are

chosen to minimized the total sum of the

deviations squared,

iii x f yd

n

iid S

1

2



inlet

exit

motor

The function of a water pump is to

cause an increase in the fluid pressureas it passes through the pump.

The pressure increase depends on the

flow rate and the pump speed. In a

pump test, these parameters must be

measured.

inlet

pressure

transducer

exit

inP

out V computerized

DACQ

A pressure transducer is used to

“transform” the pressure differential

across the pump into a proportional

voltage. The voltage is measured and

recorded by the computerized data

acquisition system (DACQ).

A calibration is need to establish the

relationship between the pressure

input to the transducer and the output

voltage.

Transducer Calibration



Transducer Calibration

• apply known pressure differentials to

the transducer and measure the

corresponding transducer output

voltage

(known pressures might be determined

by using another previously calibrated

pressure measurement system)• Apply a Trendline Analysis to find the

best fit of a linear curve fit equation to

the data

fitcurvefromfound

offsetzero

ysensitivit

0

0

0

V a

V

a

V PaV

,

• Use the calibrated pressure transducer

to determine unknown pressures by

measuring the transducer output

voltage and applying the calibration

relation

01

V V aP



• Type in the data values shown

• Create the table headings and

format the table as shown

• Plot the data on a Scatter Plot

(refer to the previous tutorial as

necessary)

• Format the plot as shown:

axis range,

tick mark intervals,

tick mark type,

number format

horizontal & vertical gridlines

data point marker style



Add a Trendline to the plot by:

• right-click on a data point

marker and select Add

Trendline from the popup

window• select a Linear curve fit

• set the Trendline Name for

the plot legend

• display the curvefit equation

on the plot

•display the r-squared valueon the plot (indicates how

well the curve fit matches

the data - the closer it is to

one, the better the fit)



Change the format for the Trendline

Label by:

• right-clicking on the label

• select Format Trendline Label• select the Number option group

and choose the Scientific format

with 3 decimal places

• select the Fill option group and

choose a Solid Fill with the Color

set to White

• select the Border Color option

group and choose a Solid Line with

the Color set to Black



We have established the relationship

between the pressure applied to the

transducer and it’s corresponding voltageoutput.

In a normal experimental application, an

unknown pressure will be applied to the

transducer and the calibration equation

will be used to find the pressure from themeasured voltage.

offsetzeroV0.1195

ysensitivitPa

V00220

0

0

V

a

V PaV

.

01

V V a

P



• Start a new worksheet Sheet2

•

Create a data table: type in the measured

transducer output voltages

enter the coefficients

determined from the trendline

analysis

enter the formula to calculatethe pressures corresponding to

each voltage

complete the table formatting

as shown

•

Create a plot of the pressures withthe formatting shown



0

2000

4000

6000

8000

0 20 40 60

y ( x ) = 3 * x ^ 2

x

Power Law y = a*x^b, b>1

Power Law Trendlines

• Many processes of interest to engineers

follow a power-law relationship,

• The plot above is for a power-law

relationship with an exponent greater

than 1.

b xa y

0

20

40

60

80

0 200 400 600

y ( x ) = 3 * x ^ 0 . 5

x

Power Law y = a*x^b, b<1

•

The plot above illustrates the typicaltrend for a power-law variation with an

exponent less than 1.



10

100

1000

10000

1 10 100

y ( x ) = 3 * x ^ 2

x


1

10

100

10 100 1,000

y ( x ) = 3

* x ^ 0 .

5

x

Power Law y = a*x^b, b<1

Log-Log Plots

•

Data that follows a power-law is oftenshown on a log-log plot. This is the same

data that was presented on the previous

slide.

• Note that the data follows a linear trend on

the log-log plot.

•

On log-log plots, the distance along an axisis proportional to the log of the parameter.



10

100

1000

10000

1 10 100

y ( x ) = 3 * x ^ 2

x


Log-Log Plots• On log-log plots, the distance along an axis is

proportional to the log of the parameter.

• Taking the log of the power-law relation,

note that log y is linear with respect log x.

• Since the difference between log(100) and

log(10) is the same as the difference

between log(1000) and log(100), it follows

that the distance between the corresponding

tick marks is the same.

• The minor grid lines from 1-10 are 2, 3, 4,etc.and from 10-100 are 20, 30, 40, etc.

• Note that the coordinates for the first 3

points are (5,75), (10,300), and (15,675).

a xb y

xa yb

logloglog

loglog



To find the trendline for data that

follows a power-law relationship:

•

key-in the data and format the tableas shown

• create the plot with the formatting

features shown

• right-click on one of the data points

and select Add Trendline from thepopup menu

• set the Trend/Regression Type to

Power

• set the Trendline Name to “power-

law curve fit”

• display the equation and R-squared

value on the plot



• Format the trendline label as shown

• right-click on the x-axis and select

format axis

• turn-on the auto-scaling

• select the Logarithmic Scale

• repeat these selections for the y-axis



• select Major & Minor Gridlines for

the vertical and horizontal axis



The final forma of the plot



Exponential Trendlines

• Many processes of interest to engineers

follow an exponential relationship,

• The plot above is for an exponential

relationship with a positive exponent .

• The process is described as exponential

growth.

bxea y

• The plot above illustrates the typical

trend for an exponential process with anegative exponent.

• The process is described as exponential

decay.

0

100

200

300

0 5 10 15 20

y = 8 * e x p ( 0 . 2

* x )

x

Exponential, y = a*exp(b*x), b>0

0

2

4

6

8

0 5 10 15 20

y = 8 * e x p ( - 0 . 2

* x )

x

Exponential, y = a*exp(b*x), b<0



Semi-Log Plots

•

Data that follows an exponential variation isoften shown on a semi-log plot. This is the

same data that was presented on the

previous slide.

• Note that the data follows a linear trend on

the semi-log plot.

•

On semi-log plots, the distance along the x-axis is proportional to the parameter and

distance along the y-axis is proportional to

the log of the parameter

1

10

100

1000

0 5 10 15 20

y = 8 * e x p ( 0 . 2

* x )

x


0

1

10

0 5 10 15 20

y = 8 * e x p ( - 0 . 2

* x )

x

Exponential, y = a*exp(b*x), b<0



Log-Log Plots• On semi-log plots, the distance along the y-

axis is proportional to the log of the

parameter and along the x-axis, the distanceis proportional to the parameter.

• Taking the log of the exponential relation,

note that log y is linear with respect x.

a xeb yaebx y

ea y bx

loglogloglogloglog

loglog

1

10

100

1000

0 5 10 15 20

y = 8 * e x p ( 0 . 2

* x )

x




On Sheet4 of your workbook:

•

Create and format the datatable as shown.

• Create and format the plot

as shown – you will choose

the Exponential Trendline

Type.

3 excel trendline

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