Accuracy and Precision

of Measurement

O’Keefe - LBHS

Recording Measurements

• A measurement always includes a value

• A measurement always includes units

• A measurement always involves

uncertainty

– A measurement is the best estimate of a

quantity

– A measurement is useful if we can quantify

the uncertainty

Sources of Error in Measurement

• Potential errors create uncertainty

• Two sources of error in measurement

– Random Error

• Errors without a predictable pattern

• E.g., reading scale where actual value is between

marks and value is estimated

• Can be determined by repeated measurements

– Systematic Error

• Errors that consistently cause measurement value

to be too large or too small

• E.g., reading from the end of a meter stick instead

of from the zero mark

Uncertainty in Measurements• Scientists and engineers often use significant

digits to indicate the uncertainty of a

measurement

– A measurement is recorded such that all certain digits

are reported and one uncertain (estimated) digit is

reported

Uncertainty in Measurements

• Another (more definitive) method to indicate

uncertainty is to use plus/minus notation

– Example: 3.84 ± .05 cm

• 3.79 ≤ true value ≤ 3.89

• This means that we are certain the true

measurement lies between 1.19 cm and 1.29 cm

Uncertainty in Measurement

• In some cases the uncertainty from a

digital or analog instrument is greater than

indicated by the scale or reading display

– Resolution of the instrument is better than the

accuracy

• Example: Speedometers

How can we determine, with confidence,

how close a measurement is to the true

value?

Uncertainty in Measurement

• Uncertainty of single measurement− How close is this measurement to the true value?

− Uncertainty dependent on instrument and scale

• Uncertainty in repeated measurements− Random error

− Best estimate is the mean of the values

Accuracy and Precision

• Accuracy = the degree of closeness of

measurements of a quantity to the actual (or

accepted) value

• Precision (repeatability) = the degree to which

repeated measurements show the same result

High AccuracyLow Precision

Low AccuracyHigh Precision

High AccuracyHigh Precision

Accuracy and Precision

• Ideally, a measurement device is both accurate

and precise

• Accuracy is dependent on calibration to a

standard

– Correctness

– Poor accuracy results from procedural or equipment

flaws

– Poor accuracy is associated with systematic errors

• Precision is dependent on the capabilities of the

measuring device and its use

– Reproducibility

– Poor precision is associated with random error

Your Turn

Two students each measure the length of a

credit card four times. Student A measures

with a plastic ruler, and student B measures

with a precision measuring instrument called

a micrometer.

Student A Student B

85.1mm 85.701 mm

85.0 mm 85.698 mm

85.2 mm 85.699 mm

84.9 mm 85.701 mm

Your Turn

Plot Student A’s data on a number line

Student A Student B

85.1mm 85.301 mm

85.0 mm 85.298 mm

85.2 mm 85.299 mm

85.1 mm 85.301 mm

Plot Student B’s data on a number line

Your Turn

Student A’s data ranges from 85.0 mm to 85.2 mm

Student B’s data ranges from 85.298 mm to 85.301 mm

The accepted length of the credit card is 85.105 mm

85.1

05

Accepted

Value

Your Turn

Which student’s data is more accurate?

Which student’s data is more precise?

Student A

Student B

Quantifying Accuracy

Error = measured value – accepted values

Student A Student B

85.1mm 85.301 mm

85.0 mm 85.298 mm

85.2 mm 85.299 mm

85.1 mm 85.301 mm

Student A: xA = 85.10 mm

Student B: xB = 85.2998 mm

The accuracy of a measurement is related to

the error between the measurement value

and the accepted value

Quantifying Accuracy

Calculate the error of Student A’s measurements

Error A = mean of measured values – accepted value

Error A = 85.10 mm – 85.105 mm = − 0.005 mm

x A =

85.1

0

85.1

05

Error

- 0.005A

cce

pte

d

Va

lue

Quantifying Accuracy

x A =

85.1

0A

cce

pte

d

Va

lue

85.1

05

Error

- 0.005 Error

0.1948

x B=

85.2

998

Calculate the error of Student B’s measurements

Error B = mean of measured values – accepted value

Error B = 85.2998 mm – 85.105 mm = 0.1948 mm

Error

|0.1948|

= 0.1948

Quantifying Accuracy

x A =

85.1

0A

cce

pte

d

Va

lue

85.1

05

Error

- 0.005 Error

0.1948

x B=

85.2

998

Calculate the error of Student B’s measurements

Error B = mean of measured values – accepted value

Error B = 85.2998 mm – 85.105 mm = 0.1948 mm

Error

|- 0.005|

= 0.005

Error

|0.1948|

= 0.1948

Quantifying Accuracy

x A =

85.1

0A

cce

pte

d

Va

lue

85.1

05

Error

- 0.005 Error

0.1948

x B=

85.2

998

Calculate the error of Student B’s measurements

Error B = mean of measured values – accepted value

Error B = 85.2998 mm – 85.105 mm = 0.1948 mm

Error

|- 0.005|

= 0.005Student A

MORE ACCURATE

Quantifying Precision

Precision is related to the variation in

measurement data due to random errors that

produce differing values when a

measurement is repeated

Quantifying Precision

Student A Student B

85.1mm 85.301 mm

85.0 mm 85.298 mm

85.2 mm 85.299 mm

85.1 mm 85.301 mm

Student A:

sA= 0.08 mm

Student B:

sB = 0.0015 mm

The precision of a measurement device can

be related to the standard deviation of

repeated measurement data

Quantifying Precision

Use the empirical rule to express precision

• True value is within one standard deviation of the

mean with 68% confidence

• True value is within two standard deviations of the

mean with 95% confidence

Quantifying Precision

Express the precision indicated by Student A’s

data at the 68% confidence level

• True value is 85.10 ± 0.08 mm with 68%

confidence

85.10 − 0.08 mm ≤ true value ≤ 85.10 + 0.08 mm

Student A: xA= 85.10 mm

sA= 0.07 mm

85.02 mm ≤ true value ≤ 85.18 mm

with 68% confidence

Quantifying Precision

Express the precision indicated by Student A’s

data at the 95% confidence level

• True value is 85.10 ± 2(0.08) mm with 95%

confidence

85.10 − 0.16 mm ≤ true value ≤ 85.10 + 0.16 mm

Student A: xA= 85.10 mm

sA= 0.07 mm

84.94 mm ≤ true value ≤ 85.26 mm

with 95% confidence

The Statistics of Accuracy and Precision

A B

C D

High Accuracy

High Precision

Low Accuracy

Low Precision

High Accuracy

Low Precision

Low Accuracy

High Precision

Gauge Blocks (Gage Blocks)

• A block whose length is

precisely and accurately

known

Standard = basis of comparison

• Precision measuring devices are often

calibrated using gauge blocks

Calibrate = to check or adjust by

comparison to a standard