Experimental ErrorsExperimental Errorsand Uncertaintiesand Uncertainties

Errors and UncertaintiesErrors and Uncertainties

Errors can be divided into Errors can be divided into 2 main classes 2 main classes Random errors Random errors Systematic errorsSystematic errors

MistakesMistakes Mistakes on the part of an individual such Mistakes on the part of an individual such

as as misreading scales misreading scales poor arithmetic and computational skills poor arithmetic and computational skills wrongly transferring raw data to the final wrongly transferring raw data to the final

report report using the wrong theory and equations using the wrong theory and equations

These are sources of error but are not These are sources of error but are not considered as an experimental errorconsidered as an experimental error

Systematic ErrorsSystematic Errors Cause a random set of Cause a random set of

measurements to be spread about a measurements to be spread about a value rather than being spread about value rather than being spread about the accepted value the accepted value

It is a system or instrument errorIt is a system or instrument error

Systematic Errors result Systematic Errors result fromfrom

Badly made instruments Badly made instruments Poorly calibrated instruments Poorly calibrated instruments An instrument having a zero error, a form An instrument having a zero error, a form

of calibration of calibration Poorly timed actions Poorly timed actions Instrument parallax error Instrument parallax error Note that systematic errors are not Note that systematic errors are not

reduced by multiple readingsreduced by multiple readings

Systematic ErrorsSystematic Errors Two types of systematic error can occur Two types of systematic error can occur

with instruments having a linear response: with instruments having a linear response: OffsetOffset or or zero setting errorzero setting error in which in which

the instrument does not read zero when the instrument does not read zero when the quantity to be measured is zero. the quantity to be measured is zero.

MultiplierMultiplier or or scale factor errorscale factor error in in which the instrument consistently reads which the instrument consistently reads changes in the quantity to be measured changes in the quantity to be measured greater or less than the actual changes. greater or less than the actual changes.

Systematic ErrorsSystematic ErrorsExamples of systematic errors caused by the Examples of systematic errors caused by the

wrong use of instruments are: wrong use of instruments are: errors in measurements of temperature errors in measurements of temperature

due to poor thermal contact between the due to poor thermal contact between the thermometer and the substance whose thermometer and the substance whose temperature is to be found, temperature is to be found,

errors in measurements of solar radiation errors in measurements of solar radiation because trees or buildings shade the because trees or buildings shade the radiometer. radiometer.

Random ErrorsRandom Errors Are due to variations in performance Are due to variations in performance

of the instrument and the operator of the instrument and the operator Even when systematic errors have Even when systematic errors have

been allowed for, there exists error.been allowed for, there exists error.

Random Errors result fromRandom Errors result from Vibrations and air convection Vibrations and air convection Misreading Misreading Variation in thickness of surface being Variation in thickness of surface being

measured measured Using less sensitive instrument when a Using less sensitive instrument when a

more sensitive instrument is available more sensitive instrument is available Human parallax errorHuman parallax error

Reducing Random ErrorsReducing Random ErrorsRandom errors can be reduced by Random errors can be reduced by taking multiple readings, andtaking multiple readings, and eliminating obviously erroneous result or eliminating obviously erroneous result or by averaging the range of results.by averaging the range of results.

Example of a random errorExample of a random error You measure the mass of a ring You measure the mass of a ring

three times using the same balance three times using the same balance and get slightly different values: and get slightly different values: 17.46 g, 17.42 g, 17.44 g17.46 g, 17.42 g, 17.44 g

Practice Problem Practice Problem Which of the following procedures would lead Which of the following procedures would lead

to systematic errors, and which would to systematic errors, and which would produce random errors? produce random errors?

(a) Using a 1-quart milk carton to measure 1-(a) Using a 1-quart milk carton to measure 1-liter samples of milk. liter samples of milk.

(b) Using a balance that is sensitive to ±0.1 (b) Using a balance that is sensitive to ±0.1 gram to obtain 250 milligrams of vitamin C. gram to obtain 250 milligrams of vitamin C.

(c) Using a 100-milliliter graduated cylinder (c) Using a 100-milliliter graduated cylinder to measure 2.5 milliliters of solution. to measure 2.5 milliliters of solution.

AnswerAnswer Procedure (a) would result in a systematic Procedure (a) would result in a systematic

error. The volume would always be too small error. The volume would always be too small because a quart is slightly smaller than a because a quart is slightly smaller than a liter. liter.

Procedures (b) and (c) would lead to random Procedures (b) and (c) would lead to random errors because the equipment used to make errors because the equipment used to make the measurements is not sensitive enough. the measurements is not sensitive enough.

PrecisionPrecision Precision is how close the measured Precision is how close the measured

values are values are to each otherto each other. . A precise experiment has a low A precise experiment has a low

random errorrandom error

AccuracyAccuracy Accuracy is how close a measured value is Accuracy is how close a measured value is

to the to the actual (true) valueactual (true) value. . An accurate experiment has a low An accurate experiment has a low

systematic error.systematic error. Accuracy tells us something about the Accuracy tells us something about the

quality or correctness of the result.quality or correctness of the result.

Examples of Precision Examples of Precision and Accuracy:and Accuracy:

Low AccuracyLow AccuracyHigh PrecisionHigh Precision

High AccuracyHigh AccuracyLow PrecisionLow Precision

High AccuracyHigh AccuracyHigh PrecisionHigh Precision So, if you are So, if you are

playing soccer and playing soccer and you always hit the you always hit the left goal post left goal post instead of scoring, instead of scoring, then you are then you are notnot accurate, but you accurate, but you areare precise! precise!

Limit of Reading and Limit of Reading and UncertaintyUncertainty

The The Limit of ReadingLimit of Reading of a measurement of a measurement is equal to the smallest graduation of the is equal to the smallest graduation of the scale of an instrument scale of an instrument

The The Degree of Uncertainty Degree of Uncertainty of a of a measurement is equal to half the limit of measurement is equal to half the limit of reading reading

e.g. If the limit of reading is 0.1cm then e.g. If the limit of reading is 0.1cm then the uncertainty range is the uncertainty range is 0.05cm 0.05cm

This is the absolute uncertainty or This is the absolute uncertainty or absolute error.absolute error.

ExampleExample The measurement is taken below using a The measurement is taken below using a

ruler. The limit of reading is 0.05 cm and ruler. The limit of reading is 0.05 cm and the uncertainty is ½ of the limit of reading, the uncertainty is ½ of the limit of reading, or ± 0.025 cm. Since uncertainties are or ± 0.025 cm. Since uncertainties are only given to only given to oneone significant figure, the significant figure, the length is stated as 0.44 ± 0.02 cm. length is stated as 0.44 ± 0.02 cm.