Statistical Analysis

1
Statistical Analysis of Experimental Density Data
Purpose of the Experiment

• Determine the density of a material (in this
  experiment glass beads).
• Estimate the random error in your density
  measurement based on your experimental
  measurements of mass and volume.

2
Error Analysis
Note A density measurement involves the
experimental determination of the mass and volume
of a material, both of which introduce an error
in the final result. Thus the contribution of
each error must be determined as a part of any
measurement.
3
Two Types of Errors
Systematic or Determinate Errors Shifts in the
measured values from the true values which
reduces the accuracy of a result. They have a
definite source that usually can be identified by
the observer. They can often be eliminated by
simply changing the experimental procedure.
(An example of a systematic error is misreading
a buret). Random or Indeterminate Errors
Shifts in the measured values from the true
values which influences the precision or scatter
of the result. They have an indeterminate source
and are usually not by the observer. (Examples
of random errors might be the imprecision among
multiple readings or an unnoticed change in
temperature, pressure, or humidity or a
fluctuation in voltage during a measurement).
4

Accuracy versus Precision
The following targets illustrate the difference
between accuracy and precision.
AccurateThe average is accurate butnot precise.
PreciseThe average is precise but not accurate.
Accurate Precise The average is both accurate
and precise.
5
Absolute Error
The absolute error is the difference between the
measured value of an observable and its true
value, Xt.
Relative Error
The relative error is the difference between the
measured value of an observable and its true
value, normalized to the true value.
6
The Average or Mean Value (xbar)
The average or mean of a set of numbers, Xi, is
found by adding the numbers and dividing by the
number of values, N. Thus the average of 3, 5,
7, 3, and 5 23 / 5 or 4.6.
Population Mean
7
The Standard Deviation
The standard deviation, a measure of the spread
of N values, Xi, about the average value, ?, a
measure of precision, is given by,
Estimate of the Standard Deviation

If the number of values, N, is small, i.e., if N
lt 10, an estimate of the standard deviation, s,
is given by,
8
Determination of the Density of Glass Beads
You will measure the mass and volume of the
glass beads several times. Thus you will know
their average mass, m, and an estimate its
standard deviation, sm, and their average volume,
V, and an estimate of its standard deviation, sV.
The density, d, of the beads is given by,
and an estimate of the standard deviation in the
density, sd, is given by,
Note It is meaningless to report a density
without units (e.g. g/cm3 ). The appropriate
units MUST always be given.
9

Standard Deviation in the Density
Note the corresponding standard deviation in
the density, ?d, is given by,
But where does this equation come from?
d f (mass, volume) As density is a function of
both mass and volume, clearly the error in both
mass and volume contribute to the error in the
density.
10
But how should they be combined?
In calculus you will eventually study error
propagation and learn that the error in a
quantity that is a function of several variables
is determined by the partial derivative of the
function with respect to each variable times the
error in that variable. Then, the terms are
combined through the square root of the sum of
the squares,
11
For the density, the partial derivatives are
and the weighted contribution to the error is
given by
and combined by taking the square root of the sum
of the squares,
yielding
or
12
Confidence Limit
where t is Students t-factor.
At the 90 confidence limit, 90 times out of 100
the true value will be within 1.64? of the
experimental results.
The confidence limit defines an interval about
the average that most likely contains ?.
Students t-factors are given in tables for
different probabilities.
13
Significant Figures
All non-zero digits are significant, for example,
123 has three significant figures. Zeros between
non-zero digits are significant, for example,
12.507 has five significant figures. Zeros to
the left of the first non-zero digit are not
significant, for example, 1.02 has three
significant figures, 0.12 has two significant
figures, and 0.012 also has two significant
figures. If a number ends in zeros to the right
of the decimal point, those zeros are
significant, for example, 2.0 has two significant
figures and 2.00 has three significant figures.
Throughout Chemistry 2, the proper number of
significant figures must be used in all
laboratory reports and on all examinations.
Failure to do so will result in the loss of
credit.
14
Calculations Using Significant Figures
Significant figures in additions and subtractions
Decimal places are overwriting the significant
figure rule. The answer should have the same
number of decimal places as the quantity with the
least decimal places. For Example,
3.7 m 9.40 m 13.1 m 2.35 L 1.2 L 3.6
L 3.67 kg 12.498 kg 16.17 kg
Significant figures in multiplications and
divisions
The product or quotient should have the same
number of significant digits as the quantity with
least significant figures. For Example,
(0.023 m) x (3.40 m) 0.078 m2 56.90 s / 2.45 s
23.2
15
Electronic Top-Loading Balance
Determine the mass using an electronic balance.
It should be left on at all times. When you are
through using it, leave it on. If the components
of an electronic balance are cold when you start,
they will drift while warming up, causing
changes in your measurements.
16
Determine Weight Using A Balance
Never place items to be weighed directly on the
balance. They have shown a weighing boat here.
You will be using a beaker for this experiment.
17
To Tare The Balance
Taring sets the balance to Zero. In order to
avoid drift, make sure the balance has been set
to zero before starting the experiment. Press
the Tare button. Wait for the balance to read
0.000 g.
18
To Weigh The Glassbeads
Weigh the Beaker. Record the weight on the
datasheet (in pen). Add the glassbeads to the
beaker. Record the weight on the datasheet.
Repeat.
19
What Can Go Wrong?
Fingerprints! The balance measures the net
downward force. For example, if the weight of
the sample is 1.0 mg and the beaker weighs 100
grams, then the accuracy of the balance must be 1
part in 100,000. Oily fingerprints do have a
measurable weight on this level. The most
accurate procedures therefore use tongs or lab
tissues to handle objects that must be accurately
weighed.
20
Types of Glassware
zero
zero
Graduated Cylinder
Buret
21
Graduated Cylinder
To determine the volume contained in a graduated
cylinder, read the bottom of the meniscus at eye
level.
Determine the volume using all certain digits
plus one uncertain digit. Certain digits are
determined from the calibration marks on the
cylinder. The uncertain digit, the last digit of
the volume, is then estimated.
22
Determination of Certain Digits
10 mL graduated cylinder volume is 6.62 mL
100 mL graduated cylinder volume is 52.7 mL
23
Viewing the Meniscus
Correct viewing the meniscusat eye level.
Incorrect viewing the meniscusfrom an angle.
24
Viewing the Meniscus
Viewing the meniscus from an angle can lead to
false readings of the volume.
25
Filling A Buret
To fill a buret, close the stopcock at the bottom
use a funnel. You may need to lift the funnel
slightly to allow the solution to flow freely
into the buret.
26
Air Bubbles in the Buret Tip
Check the tip of the buret for an air bubble.
To remove an air bubble, tap the side of the
buret tip while solution is flowing.
27
Reading the Buret
A buret reading card with a black rectangle may
help you to obtain a more accurate volume
reading.
28
Reading the Buret
Hold a buret reading card behind the buret. Move
the card until you can easily see the
meniscus. Read the buret from top to bottom.
This buret reads 11.34mL
29
Transferring Glass Beads
Do not transfer beads with hands. This is
because oil from your hands can affect the mass.
Always use forceps.
30
Calculations
Calculations should be done in lab. All data
entered on datasheets must be in pen.
31
Computer Simulation
Purpose of the Experiment
This experiment allows the measurements of the
volumes and masses of randomly-selected groups of
glass beads. These beads 1. vary
considerably in size 2. are not perfectly
spherical 3. are limited to a range of
diameters, but vary appreciably in weight and
volume Directions for the computer simulation
are located on your lab bench.