Target for todayBe able to:define uncertainty, accuracy, precision, error, magnitude of error.

calculate magnitude of error and % error.

determine the number of sig. figs. in a measurement.

re-write a measurement to a certain number of s.f.

use the correct number of s.f. when doing math.

Sig. Figs.

It’s 5007years old!(Exactly)

Basic Principles:

Every measurement has some ‘uncertainty’built in. The last digit only is uncertain.

“A chain is no stronger than its weakest link.”

A combination of measurements is nomore accurate than the least accuratemeasurement.

Accuracy – how close to the “true” valuea measurement (or the mean of a set ofmeasurements) is. Measured as a % ofthe true value.

Low accuracy meansthat you made amistake, or themeasurementscontain error.

Precision – how closely grouped a set ofmeasurements is one-to-another. Precisionis shown through the standard deviation ofthe measurements.

Good precision,poor accuracy

Low precision,good accuracy

Mistakes – something you do incorrectly:

Oops – I skipped step 5.

Oops – I measured out 25 mLs of waterbut wrote down 50 mLs.

Oops – I mis-read the thermometer!

Mistakes can be re-done or corrected, buterror is built in to the measuring equipment.

Error can only be eliminated by using betterequipment.

Error is defined as the inherent degree ofinaccuracy of a piece of equipment used tomake a measurement.

Estimating which equipment contributesthe most to the overall error:

A balance can be read to the nearest 0.01 gram.The last digit is uncertain.

If we weigh a test tube and it weighs 12.73 g, thebalance gives us a magnitude of error of0.01 g / 12.73 g x 100 = 0.08%

If we measure 100 mLs of water in a graduatedcylinder that’s marked every 1 mL, themagnitude of error is 1 mL / 100 mLs x 100 = 1%

Magnitude of error =

(readability / measurement) x 100

The readability of a piece of equipment is howit’s marked:

Ordinary lab balance: 0.01 gramsHigh-precision balance: 0.0001 gramsThermometer: 0.1oC

small graduated cylinder: 0.1 mLsmedium grad. cylinder: 1 mLlargest grad. cylinder: 10 mLs

beaker: 10 mLs, 25 mLs, 50 mLs (depends on

size of the beaker)

buret: 0.01 mLs

A medium graduated cylinder is used tomeasure out 20 mLs of water. What is themagnitude of error of this measurement?

(1 mL / 20 mLs) x 100 = 5%

A stopwatch is accurate to the nearest 0.01 s.If a receiver runs 40 yards in 6.51 seconds,what is the magnitude of error of the timemeasurement?

(0.01 x / 6.52 s) x 100 = 0.2%

After each lab, you will calculate how close tothe true value your experimental result was.

This is your % error. This single numberincorporates all the measurements errors intoone.

% error = │1 – (experimental result / true)│ x 100

A student does a lab to measure the densityof copper (Cu). She determines a density of9.04 g/cm3, but the true density is 8.94 g/cm3.How far off is she?

│ 1 – (9.04 / 8.94) │ x 100 = 1.12% error

(She earns an A!)

Formative Assessment time!

Take out half a sheet of paper and solve thisproblem:

A thermometer reads 58.3oC in water that isactually 60.0oC. What is the % error of themeasurement?

Target for todayBe able to:define uncertainty, accuracy, precision, error, magnitude of error.

calculate magnitude of error and % error.

determine the number of sig. figs. in a measurement.

re-write a measurement to a certain number of s.f.

use the correct number of s.f. when doing math.

Showing the proper level of accuracy or “confidence” in results.

Significant Figures

How many days are there in a century?• 7 days/week x 4 weeks/month

x 12 months/year x 100 yrs/century =

• 12 months/year x 30 days/month x 100 yrs/century =

• 7 days/week x 52 weeks/yr x 100 yrs/century =

• 365 days/year x 100 yrs/century =

• 365.24 days/year x 100 yrs/century =

33,600 days

36,000 days

36,400 days

36,500 days

36,524 days

Any non-0 digit is always significant, i.e. an actual part of the measurement.

12,345 g has 5 s.f. and 789 g has 3 s.f.

Zero’s trapped between non-0 digits are significant. 102 has 3 s.f.

Zero’s to the right of the decimal pt. after non-0 digits are significant.

128.0 mg has 4 s.f.

That final “0” tell us that the measurementwas made with an accuracy of 0.1 mg

A “0” by itself to the left of the decimal pt. is not significant (just a place holder)

0.23 g has 2 sf.0’s to the right of a decimal, but in front of non-0 digits are also just place holders (not significant - not a part of the measurement:

0.0023 g has 2 s.f. also

Without a decimal pt., zeros to the right are not significant.

1000 has 1 s.f.

With a decimal point, however, they become significant.

Write it as 1000. to have 4 s.f., or 1.000 x 103 for 4 s.f. also

Exact numbers arise from counting something (7 days/week) or from a definition (1 in = 2.54 cm exactly). These have an unlimited number of sig. figs.

Conversion factors that come from definitions also have unlimited s.f.

P Aresent bsent

How many s.f. are in 3002 mg?

0.00301 sec

1.204 x 103 kg

0.00007 Joules

0.0102050 moles

Rewrite…

13 grams to have 4 s.f.

0.04200 mol to have 5 s.f.

$52,238 to have 1 s.f.

0.0035229 sec to have 2 s.f.

Never change the basic value, only the s.f.

13.00 g

0.042000 mol

$50,000

0.0035 sec

Doing Math with Sig. Figs.

When a measurement with high accuracy (many s.f.) is combined with a low-accuracy measurement (few s.f.), the result can’t be any more accurate than the least-accurate measurement.

We shouldn’t try to make the result seem more accurate than it really is.

+ and - The result can’t have more digits to the right of the decimal than the measurement w/ the fewest number of digits:

100.2 98.65 10.35 -33.2877 12.655 +72123.2 137

x & /

The result can’t have more sig.figs. than the measurement with the fewest s.f.

28.300 x 12.887 x 0.12 = 44

Finding the Mean Average:

Add the results, keeping the correct sig. figs. for addition rules. Then divide by the number of results (unlimited sig. figs.)

Find the mean of 12.35, 12.04, 11.58

Sum is 35.97. Then divide by 3.

Result is 11.99