INSTRUCTIONAL GOALS:

1) REVIEWING THE SI AND METRIC BASE UNITS FOR MASS, VOL., LENGTH

2) REVIEWING THE METRIC AND SI UNITS AND PREFIXES

3) REVIEWING SOME COMMON METRIC EQUALITIES

4) DEFINING ERROR, ACCURACY AND PRECISION

5) DETERMINING ACCURACY AND PRECISION OF MEASUREMENTS

I. Units of Measurement

A) SI(International System) and Metric Base Units

Type of Measurement Metric SI

· length meter(m) meter(m)

· volume liter(L,or dm3) cubic meter(m3)

· mass gram(g) kilogram(kg)

· time second(s) second(s)

· temperature celsius(oC) Kelvin(K)

(B) SI and Metric prefixes

Prefix Symbol Meaning Multiplier Multiplier

(numerical) (exponential)

greater than 1

tera- T trillion 1,000,000,000,000 10^12

giga- G billion 1,000,000,000 10^9

mega- M million 1,000,000 10^6

kilo- k thousand 1,000 10^3

hecto- h hundred 100 10^2

deka- da ten 10 10^1

no prefix used one 1 10^0

less than 1

deci- d tenth 0.1 10^-1

centi- c hundredth 0.01 10^-2

milli- m thousandth 0.001 10^-3

micro- u millionth 0.000001 10^-6

nano- n billionth 0.000000001 10^-9

pico- p trillionth 0.000000000001 10^-12

In general, the relationship between units is that 10^x of the smaller units = 1 of the larger unit, where x = (larger exponent of ten - smaller exponent of ten) Example: 1 millisecond = _______ microseconds

10^(-3 - (-6)) = 10^3 microseconds = 1000 us

Example: 1 kilometer = _______ nanometers

10^(+3 - (-9)) nm = 10^12 nm = 1000000000000 nm

1) Length

· 1 m = 100 cm

· 1 m = 1000 mm

· 1 cm = 10 mm

2) Volume

· 1 L = 1 dm3

· 1 L = 1000 mL

· 1 L = 1000 cm3

· 1 mL = 1 cm3

3) Mass

· 1 kg = 1000 g

· 1 g = 1000 mg

· 1 mg = 1000 ug

II. Experimental Error: Accuracy and Precision

(A) Three types of errors in measurement:

1) gross, careless error - due to mistake by person measuring

· spilling of liquid before it is weighed or volume read

· reading the weight or volume incorrectly

This type is not likely to be repeated in similar determinations.

2) indeterminate(random) error - due to limitations of

· observation - difficult to read scale because it is to small to see

· equipment - not sensitive enough

Repeating the measurement reduces the effect of random error

3) determinate(systematic) error - affect each individual measurement in

exactly the same way, due to

· using impure materials as standards of purity

· using improperly calibrated volumetric glassware to measure volume

· using the wrong procedure to make your measurements

(B) Recognizing errors

1) careless and random errors are recognized by deviations of separate

measurements from each other. This is precision.

2) systematic error is recognized when the experimental results are

compared with the generally accepted value for that measurement.

This is accuracy.

1) nearness of an observed value to the correct or accepted value.

2) expressed in terms of percent error or percent difference:

%error = l observed - accepted l x 100 %

accepted

accepted values are found in handbooks, such as Handbook of Chemistry and Physics.

Example: density of Aluminum measured to be 2.81 g/cm3, accepted value is 2.70 g/cm3. % error = l 2.81 - 2.70 l x 100% = 4.07 % error

2.70

(D) Precision

1) how closely the measurements in a series agree with each other

2) 3 ways to express the degree of precision. They are the average deviation, relative average deviation, and standard deviation. Suppose an experiment was performed to determine the percent by mass of water in a crystal. To indicate the precision, the experiment was repeated 4 times with the following results:

SAMPLE % WATER

A 44.02

B 44.11

C 43.98

D 44.09

3) Average deviation or a.d. is calculated as follows:

a) calculate mean(average) of the trials:

mean = 44.05

b) calculate the deviations of each experimental value from the mean. Deviation = l experimental - mean l

SAMPLE %WATER DEVIATION FROM MEAN

A 44.02 .03

B 44.11 .06

C 43.98 .07

D 44.09 .04

c) a.d. = sum of deviations = .20/4 = .05

number of measurements

The mean is then expressed followed by + average deviation. The

first non-zero digit of the a.d. decides what decimal place the mean is rounded to:

% WATER = 44.05 + .05

4) Relative average deviation or r.a.d. is calculated as follows:

a) calculate the average deviation(see above)

b) r.a.d. = average deviation x 100%

mean

EXAMPLE: r.a.d. for %H20 = .05/44.05 x 100% = 0.1%

5) Standard deviation or s.d. is calculated as follows:

a) s.d. = d1^2 + d2^2 + d3^2 + .... + dn^2

n - 1

where d = deviation from mean for each measurement

and n = number of measurements

Using the data above,

SAMPLE %H20 DEV. FROM MEAN DEV. SQUARED

A 44.02 .03 .0009

B 44.11 .06 .0036

C 43.98 .07 .0049

D 44.09 .04 .0016

MEAN 44.05 .05

s.d. = (.0009 + .0036 + .0049 + .0016) = .06

4 - 1

The mean is then expressed followed by + s.d. The first non-zero digit of

the s.d. decides what decimal place the mean is rounded to:

% WATER = 44.05 + .06

(E) Rejecting Measurements

1) If a set of measurements contains gross error, it is usually obvious and should be excluded from any accuracy or precision considerations.

2) If an indeterminant error is suspected in one of the measurements, use the Q-test or Rejection Quotient to judge whether to throw the measurement out.

Q = (difference between deviant result and nearest neighbor)

(range of all the measurements)

Compare the value of Q to the standard Q values. If calculated Q> std.Q, then the measurement can be rejected with 90% certainty. If calculated Q < std. Q, keep it along with the others.

Std. Q Table

NUMBER OF MEASUREMENTS Q-90

3 0.94

4 0.76

5 0.64

6 0.56

7 0.51

8 0.47

9 0.44

10 0.41

Example: test the measurement 43.98 above:

Q = 44.02 - 43.98 = 0.31

44.11 - 43.98

Since Q calculated < Q std., 43.98 is precise enough to keep