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2.0 Measurement Errors

Learning Outcomes

At the end of this topic, you should be able to:

• Identify the causes of measurement errors

• Name various types of measurement errors

• Differentiate between systematic and random errors

• Suggest methods of reducing/eliminating errors

2.1 Causes of measurement errors

• Errors in measurement occur due to THREE main

factors:

1. Errors in instrument due to calibration error,

mechanical wear, aging, hysteresis, backlash,

improper handling and care etc.

2. Errors in measurement process due to improper

alignment, parallax, unclean surface, lack of skill

etc.

3. Errors due to environmental factors such as

temperature, relative humidity, vibration,

pressure etc.

Measurement

Errors

Random

Errors

Systematic

Errors

Controllable errors Uncontrollable errors

(Bias errors) (Precision errors)

2.1a Types of measurement errors

• Systematic errors are biases in the measurement that

lead to measured value being consistently too high or

too low compared to the actual value

25.862 mm

25.865 mm

25.861 mm

25.868 mm

25.866 mm

2.2 Difference between systematic error and

random error

25.860 mm

• Random errors are small differences in readings of

an instrument when the same quantity is measured a

number of times

25.861 mm

25.860 mm

25.859 mm

25.862 mm

25.858 mm

25.860 mm

25.860 mm

Xtrue = 18.00 mm

Xm = 20.08 mm

= 20.04 mm

= 19.96 mm

= 19.90 mm

= 20.03 mm

20.0820.0820.0820.08

Xavg = 20.00 mm

Systematic error =

How do you detect systematic error?

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xtruexmeasuredx

Systematic error

Total error

Frequency (no. of readings)

xm1

Random error

Random error < systematic error

Random errorSystematic error

Total error

xtrue

xmeasuredx xm1

Frequency (no. of readings)

Random error > systematic error

Causes of systematic error

• Calibration errors

• Certain consistently recurring human errors,

such as parallax error

• Errors caused by defective instrument

• Loading errors

Can be controlled

Causes of random (precision) error

• Errors caused by disturbances to the equipment

• Errors caused by fluctuating experimental

conditions

• Errors derived from insufficient measuring-system

sensitivity

Cannot be controlled!

2.3 Controllable errors

• Types of controllable errors :

- catastrophic error

- alignment error

- calibration error

- error due to ambient condition

- error due to elastic deformation

- parallax errors

• Catastrophic error

- large magnitude error caused by errors in

taking readings or in arithmetic

- in most cases, the readings vary

significantly from the other readings

- e.g :

20.2 mm, 20.1 mm, 19.9 mm,

20.1 mm, 23.0 mm, 20.0 mm

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• Alignment error (cosine error) - occurs when

measuring instrument is improperly aligned

relative to the workpiece. Example:

LD

Measured length = L

Actual length, D = Lcos θ

θ

D

θ

Dial gage

Workpiece

Figure below shows an arrangement for measuring

the dimension L on a block using a dial gage. If the dial

gage gives a reading of 5.212 mm, calculate the error

in the measurement. Given that angle θ = 5°.

Lθ

Dial gage

block

Activity 1 (3 minutes)

• Calibration error - caused by a difference between

the dimensions of standards, such as block gages,

angle gages and engraved scales, and the true

dimension

judgetool.com

Block gages

• Error due to ambient conditions - caused by

variation in ambient conditions compared to the

international standards:

temperature : 20°C

barometric pressure : 760 mmHg

humidity : 10 mmHg

Error ∆L in dimension L due to temperature rise is

given by:

20.0820.0820.0820.08∆L = L(∝1 ∆t1 - ∝2 ∆t2)

∆t1 = t1 - 20°C where

∆t2 = t2 - 20°C

∝1 = expansion coefficient of specimen

material

∝2 = expansion coefficient of instrument

material

t1°C

t2°C

LThe diameter of an aluminum rod was

measured using a micrometer in a laboratory

where the temperature is 28°C. The reading

shown by the micrometer is 52.725 mm. If the

coefficient of thermal expansion of aluminum is

26×10-6/°C determine the error in the

measurement and the true diameter of the rod.

Assume that the micrometer was calibrated at

20°C. Neglect the expansion of the micrometer.

Activity 2 (5 minutes)

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If, in Activity 2, the expansion of the

micrometer is taken into account what will be the

error in the measurement? Given that the

coefficient of thermal expansion of the micrometer

material is 10.5×10-6/°C.

Activity 3 (5 minutes) • Error due to elastic deformation - caused by

pressure at measuring probe of instrument that

causes small deformation at object surface and

the probe

Deformation

of probe

Deformation of

cylinder

Hollow

cylinder

Measuring probe

Pressure

���� Loading error

• Parallax error - occurs in instruments such as

dial gages where a gap exist between the pointer

and scale, and measurement is not taken normal

to the scale

Actual reading

Pointer

θ

Direction from

which reading

is taken

Reading taken

Normal

direction

Scale

Parallax error

2.4 Random errors

• Can be detected by repeating the measurement

using the same instrument in the same condition

• Readings will deviate slightly from one another

• Requires statistical treatment of measurement

data

• Mean and standard deviation:

- Arithmetic mean for a series of

measurement x1, x2, x3 ..... xn is defined as:

- Standard deviation is defined as:

n

xxxxx n++++=

.......321

( )1

2

−

−±=

n

xxσ

• The relationship between the standard deviation

of the measurement process and the standard

deviation caused by the manufacturing process will

determine the number of components that are

accepted or rejected by mistake at both ends of

the tolerance limits:

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Distribution of

measurement

process

Manufacturing tolerance

6σm

T = 6σp

Distribution of

manufactuing

process

20.10 20.11

20.09 19.90

19.91

19.89

20.00

Manufacturing tolerance = ±0.1 mm

Measurement accuracy = ±0.01 mm

Measurement accuracy must be

at least 10 times higher than

manufacturing tolerance

2.5 Compound errors

hrV 2π=

What is compound error?

Volume of cylinder,

h

r

What is the error in volume V if the error in

h is ∆h and error in radius r is ∆r ?

• Error in quantity M due to errors in the individual

measurements that determine M

- e.g. if M is a function of a, b and c, then error in

M is estimated by

where δa, δb, δc are the errors in the measurement of

a, b and c

cc

Mb

b

Ma

a

MM δ

∂

∂δ

∂

∂δ

∂

∂δ ++≈

2.5 Compound errors

The volume V of a cylinder is given by the

expression:

where r is the radius and h is the height of the

cylinder.

If r = 50 mm and h = 200 mm, and the errors

in the measurement of r and h are, respectively,

0.5 mm and -1.0 mm, estimate the error in the

volume V.

hrV 2π=

Activity 4 (5 minutes)

LEARNING POINTS

1. List the THREE main factors that cause errors

in measurement.

2. List the FIVE causes of systematic errors in

measurement.

3. List any FIVE types of controllable errors.

4. What is meant by ‘compound error’?