Reliability of Measurements  

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

Uncertainty All Measurements have some degree of error User error Instrument Error Description of Error Accuracy and Precision

Accuracy Correct A measurement is accurate if it correctly reflects the size of the thing being measured

Precision "repeatable, reliable, getting the same measurement each time.“ Determined by the scale on the instrument

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

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

What Effects Accuracy & Precision 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

Measurements and Significant Figures 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

The Metric Ruler Marked to the ones Estimate to the tenths place Less precise 9.5 cm Marked to the tenths Estimate to the hundredths place More precise 9.51 cm

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

Two kinds of numbers in science Two kinds of numbers are used in science: · Exact or Defined: exact numbers; no uncertainty · Measured are subject to error; have uncertainty

Comparing Measurements Decigram Balance Centigram Milligram Analytical 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

Rules for Recognizing Significant Figures   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.

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

Addition and Subtraction 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

Multiplication and Division 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

Rounding 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

Example 2 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 = ∏d2h 4 V = 7.7779598 cm3 round to 3 sig figs = 7.78 cm3