1

Chapter 2: Measurement Errors

Gross Errors or Human Errors– Resulting from carelessness, e.g.

misreading, incorrectly recording

2

Systematic Errors– Instrumental Errors

• Friction

• Zero positioning

– Environment Errors• Temperature

• Humidity

• Pressure

– Observational Error Random Errors

3

Absolute Errors and Relative Errors

ValueMeasuredX

ValueTrueXwhere

XXeErrorAbsolute

m

t

mt

:

:

100%X

XX%ErrorErrorRelative

t

mt

4

Accuracy, Precision, Resolution, and Significant Figures– Accuracy (A) and Precision

• The measurement accuracy of 1% defines how close the measurement is to the actual measured quality.

• The precision is not the same as the accuracy of measurement, but they are related.

n

xx

x

xx1Precision n

n

nn

%Error1Accuracy

5

a) If the measured quantity increases or decreases by 1 mV, the reading becomes 8.936 V or 8.934 V respectively. Therefore, the voltage is measured with a precision of 1 mV.

b) The pointer position can be read to within one-fourth of the smallest scale division. Since the smallest scale division represents 0.2 V, one-fourth of the scale division is 50 mV.

– Resolution• The measurement precision of an

instrument defines the smallest change in measured quantity that can be observed. This smallest observable change is the resolution of the instrument.

– Significant Figures• The number of significant figures

indicate the precision of measurement.

6

Example 2.1: An analog voltmeter is used to measure voltage of 50V across a resistor. The reading value is 49 V. Find

a) Absolute Error

b) Relative Error

c) Accuracy

d) Percent Accuracy

Solution

%98%2%100%d)

98.0%21%1c)

%2%10050

4950

%100%b)

14950a)

Acc

ErrorA

V

VV

X

XXError

VVVXXe

t

mt

mt

7

Example 2.2: An experiment conducted to measure 10 values of voltages and the result is shown in the table below. Calculate the accuracy of the 4th experiment.

Solution

No. (V) No. (V)

1 98 6 103

2 102 7 98

3 101 8 106

4 97 9 107

5 100 10 99

%..

.

x

xxPrecision

.

x...xx

n

xx

n

nn

n

9695901101

11019711

110110

9910710698103100971011029810

1021

8

Class of Instrument– Class of instrument is the number

that indicates relative error.– Absolute Error

– Relative Error

rangeClass

e(range) 100

valuemeasuredx,%X

e%Error

valuetruex,%X

e%Error

mm

range

tt

range

100

100

9

Example 2.3 A class 1.0 Voltmeter with range of 100V, 250V, and 1,000V is used to measure voltage source with 90V. Calculate range of voltage and its relative errors

Solution

%11.11%10090V

10V%Error

V010,1V990V,101,000V

V10V000,1100

1ec)

%77.2%10090V

2.5V%Error

V5.252V5.247V,5.2250V

V5.2V250100

1eb)

%11.1%10090V

1V%Error

101V99V1V,100V

V1V100100

1ea)

1,000V

250V

100V

10

Measurement Error Combinations– When a quantity is calculated

from measurements made on two (or more) instruments, it must be assumed that the errors due to instrument inaccuracy combine is the worst possible way.

– Sum of Quantities• Where a quantity is determined as

the sum of two measurements, the total error is the sum of the absolute errors in each measurement.

2121

2211

ΔVΔVVVE

ΔVVΔVVE

giving

11

– Difference of Quantities• The error of the difference of two

measurements are again additive

– Product of Quantities• When a calculated quantity is the

product of two or more quantities, the percentage error is the sum of the percentage errors in each quantity

212

2211

ΔVΔVVV

ΔVVΔVVE

1

EIΔIEΔEIP

,

ΔEΔIEIΔIEΔEI

ΔIIΔEE

EIP

smallveryisΔEΔIsince

12

%10

E

E

E

I

IEI

EIIEerrorpercentage

inerror%Iinerror%Pinerror%

%100

%100

Quotient of Quantities

Quantity Raised to a Power

Example 2.4 An 820Ω resistance with an accuracy of carries a current of 10 mA. The current was measured by an analog ammeter on a 25mA range with an accuracy of of full scale. Calculate the power dissipated in the resistor, and determine the accuracy of the result.

Iinerror%Einerror%I

Einerror%

Ainerror%BAinerror% B

%2

13

Solution

mW

mARIP

82

82010 22

%5%10010

5.0

5.0

25%2

%10

mA

mA

mA

mAofIinerror

Rinerror

%20%10%10

%%%

%10%52%2

2

RinerrorIinerrorPinerror

Iinerror

Basics in Statistical Analysis Arithmetic Mean Value

• Minimizing the effects of random errors

n

xxxxx n

...321

14

15

– Deviation• Difference between any one

measured value and the arithmetic mean of a series of measurements

• May be positive or negative, and the algebraic sum of the deviations is always zero

• The average deviation (D) may be calculated as the average of the absolute values of the deviations.

xxd nn

n

d...dddD n

321

16

– Standard Deviation and Probable of Error

• Variance: the mean-squared value of the deviations

• Standard deviation or root mean squared (rms)

• For the case of a large number of measurements in which only random errors are present, it can be shown that the probable error in any one measurement is 0.6745 times the standard deviation:

n

d...ddσorSD

2n

22

21

67450.ErrorProbable

n

d...dd n22

22

12

17

Example 2.5 The accuracy of five digital voltmeters are checked by using each of them to measure a standard 1.0000V from a calibration instrument. The voltmeter readings are as follows: V1 = 1.001 V, V2 = 1.002, V3 = 0.999, V4 = 0.998, and V5 = 1.000. Calculate the average measured voltage and the average deviation.

Solution

V

dddD

Vd

Vd

Vd

VVVd

VVVd

V

VVVVVV

av

av

av

0012.05

0002.0001.0002.0001.05

...

0000.1000.1

002.0000.1998.0

001.0000.1999.0

002.0000.1002.1

001.0000.1001.1

000.15

000.1998.0999.0002.1001.15

521

5

4

3

22

11

54321