Quantitative Analysis Flashcards
Quantitative Analysis
Quantitative Chemist’s Toolkit
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xkcd.com/687
Part I: Collecting and Assessing Data
Dimensional analysis: just because things match, it doesn't mean they are correlated.
Measurement and Uncertainty Toolkit, Pt I
RULE ZERO: Every measurement has some degree of uncertainty.
Uncertainty: the inherent variability in data.
There is no perfect measurement.
Counting provides exact measurements, but it isn't efficient, so estimation becomes viable and counting becomes inaccurate.
The amount of uncertainty matters.
Example:
16.45 onscreen measurement: 0.01 digit of uncertainty
16.3 \, cm - there is no 0.01 unit established. Don't forget units!
Variability for class collective measurements:
Parallax: different viewing locations
Eyesight (Quality of measurement tool)
Positioning of ruler
Extrapolation
Accounting for Uncertainty Toolkit, Part I
Proper measurement technique is important to understanding the uncertainty of that measurement.
The last significant figure of a measurement is the digit of uncertainty.
How do we know the proper number of significant figures for a measurement?
When reading a scale from an object, keep one digit more than the markings; interpolate the value between the markings.
When reading a digital readout, record ALL digits.
On a print-out or from software on a screen… it's complicated.
Is it real? Or is it math?
Know the limits of the instrument.
Draw a line between 2 objects. The digit of uncertainty is the last digit recorded.
Reviewing “Significant Figures” Toolkit, Part I
WHAT ARE THE “RULES” FOR COUNTING SIGNIFICANT FIGURES?
Which digits are always significant?
Which are never significant?
Which are conditionally significant?
Why is scientific notation important relative to significant figures?
WHAT ARE THE “RULES” FOR SIGNIFICANT FIGURES AND CALCULATION?
How many do you keep when you MULTIPLY or DIVIDE numbers?
How many do you keep when you ADD or SUBTRACT numbers?
How many do you keep when working with LOGARITHMS?
A NOTE ON ROUNDING NUMBERS:
The most statistically valid way to round numbers is called the “round-half-to-even” method, which we will use in this class…
These rules are our means for maintaining reality in math
Examples:
0.01836 = 5 \, sig \, figs All nonzero digits are significant.
0.1370 = 4 \, sig \, figs Sandwiched zeros and trailing zero is significant.
Zeros at the start of a number (zeros to the left) are not significant. Ex: 0.002 = 1 \, sigfig
Trailing zeros are significant if the number has a decimal. Ex: 220. = 3 \, sigfigs, 220.0 = 4 \, sigfigs
Scientific notation shows the number of significant figures. Ex: 2 \times 10^3 = 1 \, sigfig or you can rewrite as 2.2 \times 10^2 \, to \, have \, 2 \, sigfigs
When multiplying or dividing, keep the smallest number of significant figures.
When adding or subtracting, keep the fewest number of decimal places
Logarithms:
Log(0.00526) = -2.27901425 So, -2.279 is the answer where the numbers represents significant figures location. Not part of significant figures (similar to + or -).
Rounding:
1-4 = round down
6-9 = round up
1/30/24 Warmup
2.6615 \, to \, 3 \, sig \, figs = 2.66
0.9159 \, to \, 2 \, sig \, figs = 0.92 (when should we pay attention to sig figs? ~Whenever you are doing a lab calculation). Also in this class
17.549 \, to \, 3 \, sig \, figs = 17.5
1.0710 = 5 \, sigfigs
0.000202 = 3 \, sigfigs
25.600 = 3 \, sigfigs
73.9 = 3 \, sigfigs
What is “Error”? Toolkit, Part I
Uncertainty and error are NOT the same thing!
Uncertainty is inherent in any data point as a consequence of limitations in our ability to make measurements.
Error is a deviation from an arbitrary reference point (or “true” value) that has been assigned to the data during the interpretation process
Why is the “true” part in quotation marks?
Error can be examined in two different ways:
Numerical (or mathematical) error: discrete, measurable error in data; based in numbers
Experimental error: problems in the process of making the measurement; based in actions
In lab work, an error is not necessarily a mistake! Similarly, a mistake may or may not create an error…
Reference from experience or expected
Calculation mistakes are not necessarily error.
Working with Data Sets: Summary Statistics Toolkit, Part I
Single measurements are generally not trustworthy. To understand actual uncertainty for a measurement, it must be repeated multiple times.
To summarize the collected data, summary statistics are used:
Standard deviation (s): expresses the “spread” of a set of measurements
Confidence interval (CI): defines the interval over which a measurement is expected to be repeatable.
Average (X) gives the "center" of the cluster of measurements Sample \, Average \, (X) = (X1 + X2 + … + Xn) / n = \sum{i=1}^{n} X_i / n
Where:
X_i = measured value
n = number of measurements
"average" (x) "mean" (µ) should not be used in labs
Limited (<50) number of samples
For a “population”; very large number (100+) of samples
Sig fig note: The sig figs for the final answer are based on the fewest sig figs of the starting measurements, NOT the decimal places from the summation.
Standard Deviation Toolkit, Part I
The standard deviation is the spread of the uncertainty inherent in a set of normally distributed measurements around a sample average.
The uncertainty in a normal (or Gaussian) distribution is distributed equally on either side of the central value.
For a population, this is expected to conform to this familiar bell-shaped curve.
Remember that µ represents the mean of the population. Similarly, the symbol σ represents the standard deviation of the population.
Standard Deviation (s): s = \sqrt{\frac{\sum (X_i - \overline{X})^2}{n-1}}
The “n-1” term is known as the “degrees of freedom” for the distribution.
A standard deviation can only describe the data set used to calculate it. It CANNOT be used for direct comparisons or predictions.
Sig fig note: Use as many sig figs as you want DURING the calculation, because the final calculated standard deviation only ever gets to have ONE sig fig!
X_i - \overline{X} can be calculated with 2 data points, but should have 3 points.
We can subscript 2 remaining digits after all sig figs have been written.
Confidence Intervals Toolkit, Part I
A confidence interval is a range of values within which there is a specified probability (1-α) of finding the expected value or population mean.
“α” is the probability that the “true” value is NOT in the interval given.
For large sample sets (>50 or so samples) where the population standard deviation is known, the standard normal distribution is used (or Z-score).
Confidence Interval (Population): CI = \overline{X} \,\, \pm \,\, Z \frac{s}{\sqrt{n}}
Where:
n = number of measurements
s = standard deviation
Z = Z-score for a given confidence/probability α
Desired Confidence % (1-α), Z-Score
0.1, 90%, 1.645
0.05, 95%, 1.960
0.01, 99%, 2.576
This interval is then combined with the centroid of the population and is expressed as:
*Note: This can be used to create a range in the format “(lowest, highest)” but this is NOT normally done as it is less descriptive in the range format.Example:
µ (+/-) Z(s/√n) units 0.25 (10.06, 95%) mL
Confidence Intervals for Smaller Data Sets Toolkit, Part I
For smaller sample sets (<50 samples) such as those commonly done in analytical laboratories, a different statistical distribution is needed, known as the Student’s T distribution.
Confidence Interval (Sample Set): CI = \overline{X} \,\, \pm \,\, t \frac{s}{\sqrt{n}}
Where:
T = Student’s T value for a given confidence and degrees of freedom (n-1) for the data set
Two-sided T values for a given α, d.f.: (A full table is in your text on p. 84)