Scientific MeasurementScientific MeasurementMeasurements and their UncertaintyMeasurements and their Uncertainty

Dr. YagerDr. YagerChapter 3.1Chapter 3.1

ObjectivesObjectives

ConvertConvert measurement to scientific notation measurement to scientific notation

DistinguishDistinguish between accuracy, precision, and between accuracy, precision, and error of a measurementerror of a measurement

DetermineDetermine the number of significant figures in the number of significant figures in a measurement and in a calculated answer.a measurement and in a calculated answer.

A A measurementmeasurement is a quantity that has both is a quantity that has both a a numbernumber and a and a unit.unit.

Measurements are fundamental to the Measurements are fundamental to the experimental sciences. It is important to be experimental sciences. It is important to be able to make measurements and to decide able to make measurements and to decide whether a measurement is correct.whether a measurement is correct.

In In scientific notationscientific notation, , a given number is a given number is written as the product written as the product of two numbers: a of two numbers: a coefficient and 10 coefficient and 10 raised to a power. raised to a power.

The number of stars in The number of stars in a galaxy is an example a galaxy is an example of an estimate that of an estimate that should be expressed in should be expressed in scientific notation.scientific notation.

Accuracy and PrecisionAccuracy and Precision

AccuracyAccuracy is a measure of how close a is a measure of how close a measurement comes to the actual or true measurement comes to the actual or true value of whatever is measured. value of whatever is measured.

PrecisionPrecision is a measure of how close a series is a measure of how close a series of measurements are to one another.of measurements are to one another.

Key ConceptsKey Concepts

To evaluate the accuracy of a measurement, To evaluate the accuracy of a measurement, the measured value must be compared to the measured value must be compared to the correct value. the correct value.

To evaluate the precision of a measurement, To evaluate the precision of a measurement, you must compare the values of two or more you must compare the values of two or more repeated measurements.repeated measurements.

Determining ErrorsDetermining Errors The The accepted valueaccepted value is the correct value based is the correct value based

on reliable references. on reliable references.

The The experimental valueexperimental value is the value is the value measured in the lab. measured in the lab.

The difference between the experimental value The difference between the experimental value and the accepted value is called the and the accepted value is called the errorerror..

The The percent errorpercent error is the absolute value of the is the absolute value of the error divided by the accepted value, multiplied error divided by the accepted value, multiplied by 100%.by 100%.

Just because a measuring device works, you Just because a measuring device works, you cannot assume it is accurate. The scale below cannot assume it is accurate. The scale below has not been properly zeroed, so the reading has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate.obtained for the person’s weight is inaccurate.

Suppose you estimate a weight that is Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The first two digits (2 and 4) are known. The last digit (6) is an estimate and The last digit (6) is an estimate and involves some uncertainty. All three digits involves some uncertainty. All three digits convey useful information, however, and convey useful information, however, and are called significant figures. are called significant figures.

The The significant figuressignificant figures in a measurement in a measurement include all of the digits that are known, include all of the digits that are known, plus a last digit that is estimated.plus a last digit that is estimated.

Significant Figures in Significant Figures in MeasurementsMeasurements

Measurements must always be reported to Measurements must always be reported to the correct number of significant figures the correct number of significant figures because calculated answers often depend because calculated answers often depend on the number of significant figures in the on the number of significant figures in the values used in the calculation.values used in the calculation.

Rule 1Rule 1

Every nonzero digit in a reported measurement is assumed to be significant.

24.7 , 0.743, 714

all have three significant figures

Rule 2Rule 2

Zeros between nonzero digits are significant.Zeros between nonzero digits are significant.

7004, 40.79, 1.5037004, 40.79, 1.503

all have four significant figuresall have four significant figures

Rule 3Rule 3

Look for a decimal point. If there is no decimal, Look for a decimal point. If there is no decimal, then you are done. then you are done.

Otherwise:Otherwise: Look for zeros at the end of the number after the Look for zeros at the end of the number after the

decimal point - they are significant.decimal point - they are significant.

0.00710, 42.0, 9.000.00710, 42.0, 9.00all have three significant figures all have three significant figures

Note:Note:

Left most zeros in front of nonzero digits are Left most zeros in front of nonzero digits are notnot significant. significant.

0.0071, 0.42, 0.0000990.0071, 0.42, 0.000099

all have two significant figuresall have two significant figures

Zeros to the right of nonzero digits but left of Zeros to the right of nonzero digits but left of implied decimal point are implied decimal point are notnot significant. significant.

300, 7000, 1,000,000300, 7000, 1,000,000

all have one significant figureall have one significant figure

Scientific NotationScientific Notation

Scientific notation is a means of expressing Scientific notation is a means of expressing significant figures:significant figures:

300 = 3 x 10300 = 3 x 1022

7,000.0 = 7.0000 x 107,000.0 = 7.0000 x 1033

0.000456 = 4.56 x 100.000456 = 4.56 x 10-4-4

The first part of scientific notation holds all The first part of scientific notation holds all the significant figures.the significant figures.

Rule 4Rule 4

There are two special situations with an There are two special situations with an unlimitedunlimited (infinite)(infinite) numbernumber of significant figures: of significant figures:

A. A. CountingCounting

i.e. 23 peoplei.e. 23 people

B. B. Exactly defined quantitiesExactly defined quantities

i.e. 60 min = 1 hri.e. 60 min = 1 hr

Significant Figures in Significant Figures in CalculationsCalculations

In general, a calculated answer cannot be more In general, a calculated answer cannot be more precise than the least precise measurement precise than the least precise measurement from which it was calculated.from which it was calculated.

The calculated value must be The calculated value must be roundedrounded to to make it consistent with the measurements from make it consistent with the measurements from which it was calculated.which it was calculated.

RoundingRounding

To round a number, you must first decide how To round a number, you must first decide how many significant figures your answer should many significant figures your answer should have. The answer depends on the given have. The answer depends on the given measurements and on the mathematical measurements and on the mathematical process used to arrive at the answer.process used to arrive at the answer.

Addition and SubtractionAddition and Subtraction

The answer to an addition or subtraction calculation The answer to an addition or subtraction calculation should be rounded to the same number of decimal should be rounded to the same number of decimal places (places (not digitsnot digits) as the measurement with the ) as the measurement with the leastleast number of decimal places. number of decimal places.

Multiplication and DivisionMultiplication and Division

In calculations involving multiplication and In calculations involving multiplication and division, you need to round the answer to the division, you need to round the answer to the same number of significant figures as the same number of significant figures as the measurement with the least number of measurement with the least number of significant figures.significant figures.

The position of the decimal point has nothing to The position of the decimal point has nothing to do with the rounding process when multiplying do with the rounding process when multiplying and dividing measurements.and dividing measurements.

1.1. In which of the following expressions is In which of the following expressions is the number on the left NOT equal to the the number on the left NOT equal to the number on the right?number on the right?

a)a) 0.00456 0.00456 1010–8–8 = 4.56 = 4.56 10 10–11–11

b)b) 454 454 10 10–8–8 = 4.54 = 4.54 10 10–6–6

c)c) 842.6 842.6 10 1044 = 8.426 = 8.426 10 1066

d)d) 0.00452 0.00452 10 1066 = 4.52 = 4.52 10 1099

1.1. In which of the following expressions is In which of the following expressions is the number on the left NOT equal to the the number on the left NOT equal to the number on the right?number on the right?

a)a) 0.00456 0.00456 1010–8–8 = 4.56 = 4.56 10 10–11–11

b)b) 454 454 10 10–8–8 = 4.54 = 4.54 10 10–6–6

c)c) 842.6 842.6 10 1044 = 8.426 = 8.426 10 1066

d)d) 0.00452 0.00452 10 1066 = 4.52 = 4.52 10 1099

2.2. Which set of measurements of a 2.00 g Which set of measurements of a 2.00 g standard is the most precise?standard is the most precise?

a)a) 2.00 g, 2.01 g, 1.98 g2.00 g, 2.01 g, 1.98 g

b)b) 2.10 g, 2.00 g, 2.20 g2.10 g, 2.00 g, 2.20 g

c)c) 2.02 g, 2.03 g, 2.04 g2.02 g, 2.03 g, 2.04 g

d)d) 1.50 g, 2.00 g, 2.50 g1.50 g, 2.00 g, 2.50 g

2.2. Which set of measurements of a 2.00 g Which set of measurements of a 2.00 g standard is the most precise?standard is the most precise?

a)a) 2.00 g, 2.01 g, 1.98 g2.00 g, 2.01 g, 1.98 g

b)b) 2.10 g, 2.00 g, 2.20 g2.10 g, 2.00 g, 2.20 g

c)c) 2.02 g, 2.03 g, 2.04 g2.02 g, 2.03 g, 2.04 g

d)d) 1.50 g, 2.00 g, 2.50 g1.50 g, 2.00 g, 2.50 g

3. A student reports the volume of a liquid 3. A student reports the volume of a liquid as 0.0130 L. How many significant as 0.0130 L. How many significant figures are in this measurement?figures are in this measurement?

a)a) 22

b)b) 33

c)c) 44

d)d) 55

3. A student reports the volume of a liquid 3. A student reports the volume of a liquid as 0.0130 L. How many significant as 0.0130 L. How many significant figures are in this measurement?figures are in this measurement?

a)a) 22

b)b) 33

c)c) 44

d)d) 55