Measurement Reliability of Measurements

Much of what we know about the physical world has been obtained from measurements made in the lab

Quantitative Observations three parts to any measurement

Numerical value Unit of measurement An estimate of uncertainty

Measurements

All Measurements have some degree of error User error Instrument Error

Description of ErrorAccuracy and Precision

Uncertainty

Correct A measurement is accurate if it

correctly reflects the size of the thing being measured

Accuracy

"repeatable, reliable, getting the same measurement each time.“

Determined by the scale on the instrument

Precision

Precise and Accurate This pattern is both precise and

accurate. The darts are tightly clustered and their average position is the center of the bull's

This is a random-like pattern, neither precise nor accurate. The darts are not clustered together and are not near the bull's eye.

Neither Accurate nor Precise

PRECISION: – is a determination of the reproducibility of a measurement.

– tells you how closely several measurements agree with one another.

– precision is affected by random errors.

ACCURACY: – closeness of a measurement to a true, accepted value.

– is subject to systematic errors (errors which are off in the same direction, either too high or too low)

What went wrong? · The balance may not have been zeroed, · The pan of the balance may have been dirty? The instrument is damaged The skills of the user are bad

What Effects Accuracy & Precision

Numerical value must be recorded with the proper number of significant figures.

The number of significant figures depends on the scale of the instrument used and is equal to the known from the marked scale plus on estimated digit.

This last digit gives the uncertainty of the measurement and gives the precision of the instrument.

Scientist indicate the precision of a measurement with the use of significant figures

A system to communicate the precision of measurements Agreed Upon by all All known digits plus one estimated digit

Measurements and Significant Figures

The Metric Ruler Marked to the onesEstimate to the tenths placeLess precise 9.5 cm

Marked to the tenthsEstimate to the hundredths placeMore precise 9.51 cm

Error = (measured value – accepted value)

Percent Error = (measured value – accepted value) ÷ accepted value x 100 %

Calculations of Error

Two kinds of numbers are used in science:

· Exact or Defined: exact numbers; no uncertainty

· Measuredare subject to error; have uncertainty

Two kinds of numbers in science

Comparing Measurements

DecigramBalance

CentigramBalance

MilligramBalance

AnalyticalBalance

Mass Reading

3.1 g 3.12 g 3.121 g 3.1213 g

Sig. Figs 2 3 4 5

Less precise

More precise

Even more precise

Most precise

Non-Zero digits are significant.

256 36999 45

Any zeroes between two sig figs are significant. 205 1.0002 20.000005

Final zeroes to the right of the decimal point are significant. 1.0 25.0000 890000.00000 78.200

Placeholder zeroes are not significant. Convert to scientific notation to remove these placeholder zeroes. 2000 .0000002 .01 .010 .000500 1.5 x 10 2.50000 x 10 8.90 x 10

Counting numbers and defined constants have an infinite number of sig figs.

Rules for Recognizing Significant Figures

The answer to a calculation with measurements can be no more precise than the least precise number.

Significant Figures in Calculations

When you add and subtract with measurements your answer must have the same number of digits to the right of the decimal point as the value with the fewest digits to the right of the decimal point.

Example 28.00 cm + 23.538cm + 25.68cm = 77.218 cm

rounded to 2 places past the decimal 77.22 cm

Addition and Subtraction

When you multiply and divide with measurements your answer must have the same number of significant digits as the measurement with the fewest significant figures.

Example Calculate the volume of the rectangle that is 3.65

cm long, 3.20 cm high, and 2.05 cm wide. V = l x w x h V = 3.20cm x 2.05 cm x 3.65 cm = 23.944 cm3

rounded to 3 sig figs = 23.9 cm3

Multiplication and Division

If the rounded digit is < 5, the digit is dropped

If the rounded digit is > 5, the digit is increased

Example 1 7.7776 g rounded to 3 sig figs 7.78 g 124 g rounded to 2 sig figs 120g 14.4444 % rounded to 2 sig figs 14 % 0.02317 g rounded to 2 sig figs 0.023 g

Rounding

When performing multi step calculations, it is often better to carry the extra digits and round in the final step.

Calculate the volume of a cylinder with a diameter of 1.27 cm and a height of 6.14 cm

V = ∏d2h4

V = 7.7779598 cm3 round to 3 sig figs = 7.78 cm3

Example 2