# CHMB16H3 Lecture Notes - Lecture 3: Gaussian Function, Triangular Distribution, Confidence Interval

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School
UTSC
Department
Chemistry
Course
CHMB16H3
Professor 1
CHMB16Fall2012 Lecture 3: Errors Chapter 3
o Precision = how close the results are to one another every time you do the experiment results
should be very near each other
o Accuracy = how close the result is to the actual (true) value
o In an experiment, you are aiming for similar results which are very close to the true value
high precision and high accuracy!
o Low accuracy and high precision is better than high accuracy and low precision because we
have statistics, we can calculate how far our measurements are from the true value (deviation
can be calculated) so the result is better than if you had high accuracy and low precision (high
deviation)
Mean ( ̅)
o Also known as average
o
o where xi = individual values of x making up the set of N replicate measurements
Median
o middle result of replicate data when arranged in ascending or descending order
o if data set has even number of replicates, the median can be evaluated as the mean of the middle
pair
Comparison of the Mean and Median
o ideally, the mean and the median should be identical
o when the data set contains an outlier, a large discrepancy occurs between the mean and the
median
o the outlier will have a significant effect on the mean, however no effect on the median
Precision
o describes the reproducibility of measurements
o standard deviation, varience or coefficient of varience
o how much an individual result differs from the mean
o the deviation from the mean, d, is the measure of how many any one result differs from the mean
o
Accuracy
o indicates the closeness of the measurement to the true accepted value
o the level of accuracy of a measurement is expressed by the error
o the absolute error, E, in the measurement x is given by E = xf x where xf is the true value
o usually error is expressed in relative terms, as relative error (Er)
o
Significant Figures
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o the number of significant figures is equal to the number of digits needed to represent a value in
scientific notation without the loss of accuracy
The significance of zeroes
o only significant when they appear in the middle of a number or at the end of a number on the right
side of the decimal point
o
Absolute and Relative Uncertainty
o Absolute uncertainty expresses the margin of uncertainty associated with a measurement (ex. The
estimated uncertainty associated with a calibrated buret)
o Relative uncertainty compares the size of the absolute uncertainty with the size of it associated
measurement
o
o
o the percent relative uncertainty changes as the magnitude of the measurement changes, but the
absolute uncertainty is always the same
Types of Error
o random error (indeterminate error) : causes data to scatter symmetrically about the mean value
and affects precision
o systemic error (determinate error) : causes the mean of the data to differ from the accepted/true
value and affects accuracy by biasing the data (negative bias if results are low with respect to the
true value and positive bias if results are greater than the true value)
o gross error : large in magnitude and rare, usually the result of human error leads to outliers
o there are statistical tests available for determination of outliers (ex. Q test)
Types of Systemic Error
o instrumental error faulty calibration, component failure, use of instrument outside of normal
operating parameters
o method error non-ideal chemical or physical behavior of the reagents or reactions on which the
analysis is based (ex. Slow or incompetent reactions, istability of reactants and products, side
product formation etc)
o personal error carelessness, inattention, being under the influence etc.
The significance of zeroes
Zeroes are significant only when they appear in the middle of a number or
at the end of a number on the right hand side of a decimal point.
0.0000572 (not significant zeroes!)
0.005702 (bold zero is significant, underlined ones are not!)
0.005720 (bold zero is significant, underlined ones are not!)
57200 (these zeroes are ambiguous! Because they could mean
5.72 x 104 or 5.720 x 104 or 5.7200 x 104
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## Document Summary

Mean ( : also known as average, where xi = individual values of x making up the set of n replicate measurements. Median: middle result of replicate data when arranged in ascending or descending order if data set has even number of replicates, the median can be evaluated as the mean of the middle pair. 1: the number of significant figures is equal to the number of digits needed to represent a value in scientific notation without the loss of accuracy. The significance of zeroes: only significant when they appear in the middle of a number or at the end of a number on the right side of the decimal point. Absolute and relative uncertainty: absolute uncertainty expresses the margin of uncertainty associated with a measurement (ex. Slow or incompetent reactions, istability of reactants and products, side product formation etc: personal error carelessness, inattention, being under the influence etc.