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

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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

2

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

## 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.