Accuracy and Precision

SIGNIFICANT FIGURES (Sig Figs)

Uncertainties in Scientific Measurement

All measurements are subject to error.

Types of Errors:1)  Systematic error: arise because to some extent, measuring

instruments have built-in, or inherent, errors.

2) Random errors: they arise from intrinsic limitation in the sensitivity of the instrument and inability of observer to read a scientific instrument and give results that may be either too high or too low.

In talking about the degree of uncertainty in a measurement, we use the words accuracy and precision.

Accuracy and precisionAccuracy and precision

Precision: reproducibility of a measurement

Accuracy: nearness to the actual value **based on the accuracy of the measuring device**

Example: My 8 year old cousin sucks at bowling.

She puts the ball in the gutter every time.

Her aim may not be accurate but she has great precision!

Example: How many students are in the front row today?

Swami Willets predicts there will be 8 students.

How close I am reflects the accuracy of my measurement (or here, prediction) and the accuracy of the measuring device.

Accuracy and precision: the target exampleAccuracy and precision: the target example

Precise, but not accurate

Accuracy and precision: the target exampleAccuracy and precision: the target example

Accurate, not precise

Accuracy and precision: the target exampleAccuracy and precision: the target example

Neither accurate, nor precise

Accuracy and precision: the target exampleAccuracy and precision: the target example

Accurate and precise

Beware of Parallax – the apparent shift in position when viewed at a different angle.

Significant FiguresThe term Significant figures refers to digits that were

measured.

When rounding calculated numbers, we pay attention to significant figures so we do not overstate the accuracy of our answers.

Significant Figures

The rules for determining significant figures (sig. fig.).

1) Zeros in the middle of a numbers are significant figures. E.g. 4023 mL

has 4 significant figures.

2) Zeros at the beginning of a number are not significant; they act only to locate the decimal point. E.g. 0.00206L

has 3 significant figures. The zeros to the left of 2 are not sig. fig.

3) Zeros at the end of a number and after decimal point are always significant. E. g. 2.200 g

has 4 sig. fig.

4) Zeros at the end of number and before the decimal point may or may not be significant figures

5) A useful rule of thumb to use for determining whether or not zeros are significant figures is that zeros are not sig. fig. if the zeros disappear

when scientific notation is used. E.g. 0.0197 = 1.97 x10-2 the zeros are not sig. fig.; 0.01090 = 1.090 x10-2

the first two zeros are not significant and the second two are.

0. 0 0 4 0 0 4 5 0 0

Not significantzero for cosmetic purpose

Not significant: zeros used only to locate the decimal point

Significant:all nonzerointegers

Significant: zeros at the end of a number and after decimal point

Significant: all zeros between nonzero numbers

The number 0.004004500 has 7 sig. Fig.

Examples 32445 = 5 sig figs

0.23435 = 5 sig figs

2348.23 = 6 sig figs

0.0023 = 2 sig figgs

0.02300 = 4 sig figs

1.009 = 4 sig figs

230,004 = 6 sig figs

100 = 1 sig fig

1.00 x 102 = 3 sig figs

100. = 3 sig figs

EXACT NUMBERS These are non-measured numbers (8 atoms), numbers in formulas (2r2),

and numbers from definitions (1 in=2.54 cm). Exact numbers have an

infinite number of sig figs.

10 molecules = infinite sig figs

4/3r3 = infinite sig figs

= infinite sig figs

3) Addition and Subtraction:

The result of addition or subtraction must be expressed with the same digits beyond the decimal point as the quantity carries the smallest number of such digits.

E.g. 3.18

+ 0.01315

3.19

When use scientific notion, All numbers are converted to the same power of 10, and the digit terms are added or subtracted.

E.g. (4.215 x 10-2) + (3.2 x 10-4) = (4.215 x 10-2) + (0.032 x 10-2) = 4.247 x 10-2

2) Exact numbers can be considered to have an unlimited number of sig. fig. There are two situations when a quantity appearing in a calculation may be exact.

By definition such as 1 in = 2.54 cm

Counting such as six faces on a cube or two hydrogen atoms in a water molecule.

Significant Figures in Numerical Calculations

Addition Subtraction

Significant Figures in Numerical Calculations

1) Multiplication or DivisionThe result of multiplication or division, may contain only as many sig. fig. as the least precisely known quantity in the calculation.

E.g. 14.79cm x 12.11cm x 5.05cm = 904cm (4 sig.fig) (4 sig. fig) (3 sig. fig) (3 sig fig.)

If use scientific notion,

E.g. (3.4 x 106)(4.2 x 103)

= (3.4)(4.2) x 10(6+3) = 14.28 x 109 = 1.4 x 1010

(to 2 significant figures)the digit terms are multiplied in the normal way and the exponents are added. The end result is changed so that there is only one nonzero digit to the left of the decimal.

Multiplication

Division

2 sigfigs3 sigfigs 2 sigfigs

1 sigfig4 sigfigs 1 sigfig

Rounding Numbers

1) If the first digit you remove is less than 5, round down by drop it and all following digits.

E.g. 5.664525 5.66 when rounded to three sig. fig.

2) If the first digit you remove is greater than 5 or 5 followed by nonzero, round up by adding 1 to the digit on the left.

E.g. 5.664525 5.7 when rounded to two sig. fig.

*In a series of calculations, carry the extra digits through to the* final result, THEN round

3.05 x 5.555 x 3.0 = 50.82825 = 51NOT

3.1 x 5.6 x 3.0 = 52.08 = 52

3) What happens if there is a 5 or 5 following by zeros? There is an arbitrary rule “round 5 to even”:

If the number before the 5 is odd, round up. If the number before the 5 is even, let it be. The justification for this is that in the course of a series of

many calculations, any rounding errors will be averaged out.

E.g. Round the following number to 2 significant figures:

(a) 2.35 x 102 (b) 2.45 x 102

(Answer: 2.4 x 102)

Rounding Numbers