Chapter 1(5)Measurement andError

Measurement Measurement andandErrorError

PGCC CHM 103 SinexPGCC CHM 103 Sinex

Some terminologySome terminology

True value – standard or reference of known value or a theoretical value

Accuracy – closeness to the true value

Precision – reproducibility or agreement with each other for multiple trials


Types of ErrorsTypes of Errors

• Determinate (or Systematic)– Sometimes called bias due to error in one

direction- high or low– Known cause

• Operator• Calibration of glassware, sensor, or

instrument

– When determined can be corrected– May be of a constant or proportional

nature


Types of Errors continuedTypes of Errors continued

• Indeterminate (or Random)– Cannot be determined (no control over)– Random nature causes both high and

low values which will average out– Multiple trials help to minimize


Accuracy and PrecisionAccuracy and Precision

Nature of accuracy and precision

Both accurate and precise

Precise only Neither accurate nor precise

Target shooters comments

Great shooting!

Gun barrel must be bent!

Can’t hit the broad side of a barn!

The center ofthe target isthe true value.


Standard Deviation of the…Standard Deviation of the…• Population

Actual variation in the population

• Sample – part of population

Estimatesthe variation

in the population

May not be representative sample


• THE SAMPLE STANDARD DEVIATION allows for more variation in the sample compared to the population, since sample is only part of population. Dividing by n-1 increases the estimate of the population variation. This attempts to eliminate the possibility of bias.

populat ion

deviat ionsn

( ) 2

sample

deviat ionsn

( ) 2

1

Population

Sample

n

xxxdeviation

i


Measurements- The UncertaintyMeasurements- The Uncertainty

Example:True value of thickness of a book is 5cm. Student A uses meter ruler and measures the thickness to be 4.9cm with an uncertainty of 0.1cm.Student B , with Vernier caliper, found it to be 4.85cm with an uncertainty of 0.01cm.

We may say,Student A has more accurate value, but less precise.Student B got a more precise value, but less accurate (due to the faulty caliper. Un-calibrate !)However, after sending the caliper to be calibrated, student B performs the measurement again and found the thickness is 4.98cm. So, now he has more accurate and more precise value.

Note: We always report a measurement in a way that would includes the uncertainty carried by the instrument.

For instance:

cm)01.098.4(


Combining uncertainties Combining uncertainties ++ and and --

• Adding or subtracting quantities then sum all individual absolute uncertainties

• eg 2.1 ± 0.1 + 2.0 ± 0.2 = 4.1 ± 0.3

• eg 2.1 ± 0.1 - 2.0 ± 0.2 = 0.1 ± 0.3

• this method overestimates the final uncertainty


Combining uncertainties Combining uncertainties xx and and //

• When Dividing or multiplying quantities, then

sum all of the individual relative uncertainties• eg (2.5 ± 0.1) x (5.0 ± 0.1)

= (2.5 ± 4%) x (5.1 ± 2%) =12.5 ± 6% (or 0.75 or 0.7)

• eg (21 ± 6%) / (5.0 ± 4%) –= 4.12 ± 10% or 4.2 ± 0.42 or 4.2 ± 0.4

• also overestimates final uncertainty


Measurements- The Significant figures, What is ?Measurements- The Significant figures, What is ?In general,

The number of Significant figures of a numerical quantity is the number of reliably known digits it contains.For measured quantity, it is defined as all of the digits that can be read directly from the instrument used in making the measurement plus one uncertain digit that is obtained by estimating the fraction of the smallest division of the instrument’s scale.

Note: Exact quantities are considered as having unlimited number of significant figures. We need to be concerned with significant figures only when dealing with measurements that have required some estimation.

For example,

Reading of the thickness of a book is 5.0cm or 50mm from meter ruler (with 2 sf)5.00cm or 50.0 mm from vernier caliper. (with 3 sf)


Significant Figures

The rules of significant figures:

1. Any figures that is non-zero, are considered as a significant figure.

2. Zeros at the beginning of a number are not significant

Example: 0.254 ----------------- 3 s.f

3. Zeros within a number are significant.

Example: 104.6 m ---------------- 4 s.f

4. Zeros at the end of a number after the decimal point are significant.

Example: 27050.0 ------------------- 6 s.f


Significant Figures …

5. Zeros at the end of a whole number without a decimal point may or may not be significant.

It depends on how that particular number was obtained, using what kind of instrument, and the uncertainty involved.

Example: 500m ------------------- could be 1 or 3 sf.

Convert the unit:

500m = 0.5km (would you say it has 1 sf ? )

500m = 50 000cm (would you say it has 1 or 5 sf ? )

How to solve this problem ?


Significant figures – Addition and Subtraction processes

The rule:

The final result of an addition and/or subtraction should have the same number of significant figures as the quantity with the least number of decimal places used in the calculation.

Example:

23.1 + 45 + 0.68 + 100 = 169

Example:

23.5 + 0.567 + 0.85 = 24.9


Significant figures – Multiplication and division processes

The rule:

The final result of an multiplication and/or division should have the same number of significant figures as the quantity with the least number of significant figures used in the calculation.

Example:

0.586 x 3.4 = 1.9924

= 2.0

Example:

13.90 / 0.580 = 23.9655 = 24.0


Estimating the slopeEstimating the slope

1. Simple conservative method

Don’t forget the uncertainty on the slope


Simple conservative method Simple conservative method

Plot error bars on the graph

Draw maximum (mmax

) and minimum (m

min)

slopes.

m = (mmax

+ mmin)/2

m = (mmax

- mmin)/2

C = (Cmak-Cmin)/2

Chapter 1(5)Measurement andError

Technology

Transcript of Chapter 1(5)Measurement andError