Precision

1
Precision Accuracy

• Precision
• How close are the measurements to each other?
• Reproducibility
• Accuracy
• How close is a measurement to the true value?
• Could be affected by the following
• Systematic errors all higher or lower than
  actual value
• (lack of accuracy)
• Random errors some high and some low
• (lack of precision)

2
Significant Figures

• Presenting measurements and calculated results
  with the appropriate significant figures and
  units is an indication of the precision of
  values.

3
Rules for Sig Figs

• All nonzero digits are significant
• Trapped zeros are significant
• Trailing zeros are significant if there is a
  decimal point
• Leading zeroes are NOT significant

sig figs 1 3 4 1 2 3
300 300. 300.0 0.003 0.0030 0.00300
How would one write 300 with 2 sig figs?
4
Scientific Notation

• 3.0 x 10 2 has two sig figs
• Written as a number between 1-10 x a power of ten
• Unambiguously displays the precision of the value
  making it easier to make comparisons
• 300 3 x 102
• 300. 3 .00 x
  102
• 300.0 3.000 x
  102
• 0.003 3 x 10-3
• 0.0030 3.0 x 10-3
• 0.00300 3.00 x 10-3

5
Making Measurements - Thermometer

• The number of significant figures in your
  measurement depends on the measuring device.
  

87.5.1C
The bottom of the meniscus is between 87 and 88
C. This can be read to 1 digit more precision
than indicated by the calibration. The last
estimated digit can vary from person to person,
but each should record a value to the tenths
place. There are 3 sig figs and the last digit
is the uncertain digit. Generally, measurements
are uncertain by 1 in that last digit unless
otherwise indicated by your measuring device.
Usually, 1/10 of an increment.
6
Beaker vs. Graduated Cylinder

• Each contains the same amount of water.

7
Beaker
10. 1mL
8
Graduated Cylinder
10.05 .05 mL
9
The Analytical Balance
All digits should be recorded as given, precision
is to the 0.1 mg, the accuracy is determined by
the calibration.
10
Calculations Sig Figs

• Multiplication Division
• The total number of sig figs in the answer is
  equal to the same number of sig figs in the
  measurement used in the calculation with the
  smallest number of sig figs.
• Ex 5.1 cm x 2.01 cm 10.0701 cm2 10. cm2
  
• Round the final answer using the number to the
  right of the last sig fig.
• Avoid round off errors by keeping extra digits
  beyond the last sig fig when calculating
  intermediate values.

11
Calculations Sig Figs

• Addition Subtraction
• The final answer should be rounded to the
  right-most filled column (according to the value
  with the biggest uncertain digit the weakest
  link).
• Ex 6.5 cm
• 100.01 cm
• .044 cm
• 106.554 cm
• 106.6 cm

12
Scientific notation can make it easier..

• What is the sum of 4.5 x 10-6, 3.2 x10-5, and
  15.2 x 10-7?
• .45 x 10-5
• 3.2 x 10-5
• .152 x 10-5
• 3.802 x 10-5
• 3.8 x 10-5

13
SI Prefixes

• Prefix Symbol Meaning
  Power of 10
• Mega M 1,000,000 106
• Kilo k 1,000 103
• Deci d 0.1 10-1
• Centi c 0.01 10-2
• Milli m 0.001 10-3
• Micro µ 0.000001 10-6
• Nano n 0.000000001 10-9
• Femto f 0.000000000000001 10-15
• Atto a 0.000000000000000001
  10-18

14
Fundamental SI Units

• Physical Quantity Unit Abbreviation
• Mass kilogram kg
• Length meter m
• Time second s
• Temperature kelvin K

15
Dimensional Analysis

• Use conversion factors (definitions, ratios) to
  convert from one unit to another.
• Conversion factors are exact numbers that have no
  uncertainty.
• Ex. Convert 6.4 weeks to hours.
• 6.4 weeks x 7 days x 24 hours 1100 hrs
• 1 week 1 day

16
Group Problems

• Convert 47 hours to weeks.
• 47 hours x 1 day x 1 week 0.28
  weeks
• 24 hours 7 days
• The same conversions were used as in the previous
  example. The top equals the bottom.
• Round off answers at the end. Keep additional sig
  figs for intermediate answers.
• Calculate the sum of 2.5 3.5 4.5 5.5.
• 2.0
• 3.5
• 4.5
• 5.5
• 14.5
• The tread on a certain automobile tire wears
  0.00100 inches per 2,600 miles driven. If the car
  is driven 45 miles a day, how many months ( 1mo
  30 days) can a tire w/ 0.010 in of treat be used
  before it wears down and needs to be replaced?
• .010in x 2,600 mi x 1 day x 1 month 19.25
  19 months
• 0.00100in 45 mi 30 days
• In a displacement of water by gas experiment the
  initial volume of water in a burette is 45.50 mL
  and the final volume is 37.50 mL. What is the
  total volume of water displaced? In mL? in L?
• 45.50 (4sf)
• - 37.50 mL (4sf)
• 8.00 mL (3sf) 0.00800 L (still 3sf)

17
Statistical Analysis and Expression of Data

• Reading Lab Manual 29 - 40
• Today Some basics that will help you the entire
  year

The mean or average
? true value, ? measurements
? true value, finite of measurements
Uncertainty given by standard deviation
18
For finite of measurements Standard
deviation S
s S ?(xi-x)2/(n-1)1/2

(Calculators can calculate ? s quite easily!!!
Learn how to do this on your calculator.)
For small number of measurements s S is very
poor. Must use Student t value. s tS
where t is Student t
Usually use 95 Confidence Interval
So, 95 confident that if we make a measurement
of x it will be in the range
x t95s Uncertainty of a SINGLE MEASUREMENT
19
Usually interested in mean (average) and its
uncertainty
Standard Deviation of the mean
Then average and uncertainty is expressed as
x t95Sm
Often want to know how big uncertainty is
compared to the mean Relative Confidence
Interval (C.I.) (Sm / x )(t95)
Expression of experimental results
1. Statistical Uncertainties (S, Sm, t95S, Sm
(t95) / x) always expressed to 2
significant figures 2. Mean (Average) expressed
to most significant digit in Sm
(the std. dev.
of the mean)
20
Example Measure 3 masses 10.5763, 10.7397,
10.4932 grams Average 10.60307 grams Std. Dev.
S .125411 .13 grams
Sm .125411 / ?3 .072406 .072 grams
Then average ?
10.60 grams
t95 for 3 measurments
95 C.I. t95Sm 4.303 .072406 .311563
.31 grams of the mean
Relative 95 C.I.of the mean ?
95 C.I. / Average .311563/10.60307
.029384 .029 of the mean
Usually expressed at parts per thousands (ppt)
.029 1000 parts per thousand 29 ppt
Relative 95 C.I. of the mean
What if measure 10.5766, 10.5766, 10.5767 grams?
Ave. 10.57663 Sm .000033
Ave. ?
Ave. 10.5766, not 10.57663 because limited by
measurement to .0001 grams place
Now work problems.
21
Sally 5 times, average value of
15.71635 standard deviation of
0.02587. Janet 7 times, average
value of 15.68134 standard deviation of
0.03034. (different technique) Express the
averages and standard deviations to the correct
number of significant figures. Must use Sm.
Sally Sm 0.02587/?5 1.157 x 10-2 1.2 x
10-2 Janet Sm 0.03034/?7 1.147 x 10-2
1.1 x 10-2 Sally 15.72 S 0.026
Janet 15.68 S 0.030 Using the proper
statistical parameter, whose average value is
more precise? Must use Sm. Sm(Janet) lt Sm(Sally)
so Janets average value is more precise. 95
confidence intervals of the mean, relative 95
confidence intervals of the mean Sally 95 C.I.
t95Sm 2.776 (1.157 x 10-2) 3.21 x 10-2
3.2 x 10-2
Range 15.69 15.75 Relative
95 C.I. 3.21 x 10-2 / 15.71635 1000 ppt
2.04 2.0 ppt Janet 95 C.I. t95Sm
2.447(1.147 x 10-2) 2.81 x 10-2 2.8 x
10-2
Range 15.65 15.71 Relative
95 C.I. 2.81 x 10-2 / 15.68134 1000 ppt
1.79 1.8 ppt
22
Are the two averages in agreement at this
confidence level? Because the 95 C.I. for both
measurements overlap, the two averages are in
agreement If you owned a chemical company and
had to choose between Sallys and Janets
technique, whose technique would you choose and
why? Choose Sallys techniques because the
uncertainty in a single measurement based on S is
better than that using Janets technique.