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Assessment of
Analytical Data
V. Santhanam
Department of Chemistry
SCSVMV
MPC101 : UNIT – IV - RESEARCH METHODOLOGY
Errors in chemical analysis – classification of
errors – determination of accuracy of methods –
improving accuracy of analysis – significant figures –
mean, standard deviation.
comparison of results : “ t ” test, “ F ” test and “
χ2 ” – rejection of results – presentation of data.
Sampling – introduction – definitions – theory of
sampling – techniques of sampling – statistical
criteria of good sampling and required size.
Nature of Quantitative Analysis
• Modern analytical chemistry is concerned with the
– Detection
– identification
– measurement of the chemical composition
using existing instrumental techniques, and the
development or application of new techniques and
instruments.
• It is a quantitative science - desired result is
always numeric.
• Quantitative results are obtained using devices or
instruments that allow us to determine the
concentration of a chemical in a sample from an
observable signal.
• There is always some variation in that signal / value
measured over time due to noise and / or drift
within the instrument.
• We also need to calibrate the response as a
function of analyte concentration in order to
obtain meaningful quantitative data.
• As a result, there is always an error, a deviation
from the true value, inherent in that
measurement.
• One of the uses of statistics in analytical
chemistry is therefore to provide an estimate of
the likely value of that error; in other words, to
establish the uncertainty associated with the
measurement.
When using an analytical method we make
three separate evaluations of experimental
error.
1. Before beginning an analysis we evaluate potential
sources of errors to ensure that they will not
adversely effect our results.
2. During the analysis we monitor our measurements
to ensure that errors remain acceptable.
3. At the end of the analysis we evaluate the quality
of the measurements and results, comparing them
to our original design criteria.
ERROR
• Any measurement made with a measuring
device is approximate.
• If you measure the same object two different
times, the two measurements may not be
exactly the same.
• The difference between two measurements is
called a variation in the measurements. Another
word for this variation - or uncertainty in
measurement - is "error."
• This "error" is not the same as a "mistake."
• It does not mean that you got the wrong
answer. The error in measurement is a
mathematical way to show the uncertainty in
the measurement.
• It is the difference between the result of the
measurement and the true value of what you
were measuring.
Error = True value ~ Measured value
Absolute and Relative Errors
• Error in measurement may be represented by
the actual amount of error, or by a ratio
comparing the error to the size of the
measurement (Relative error)
• The absolute error of the measurement shows
how large the error actually is, while the relative
error of the measurement shows how large the
error is in relation to the correct value.
• Absolute errors do not always give an indication of
how important the error may be.
If you are measuring a football field and the
absolute error is 1 cm, the error is virtually irrelevant.
• But, if you are measuring a small machine part (<
3cm), an absolute error of 1 cm is very significant.
While both situations show an absolute error of 1 cm.,
the relevance of the error is very different. For this
reason, it is more useful to express error as a relative
error.
Types of Errors
Depending on the origin of errors they can be
classified in to
1. Gross errors
2. Systematic / Determinate errors
3. Random / Indeterminate errors
Gross Errors
• This category of errors includes all the human
mistakes while reading, recording and the
readings.
• Mistakes in calculating the errors also come
under this.
Reasons for Gross Errors
– Incompetency of the observer
• Observer may not be knowing all the technical details of the
measurement.
– Carelessness
• The observer may not be measuring the value with full attention.
For example the students recording a value in lab while chatting
with their batch mates.
• Spilling of solutions
• Contaminations
• Arithmetic errors
– Transposition
• Data transposition occurs to all many times. Checking and
rechecking before recording can avoid this error
Ex Recording 1.1125 as 1.1215
How to reduce gross errors
• Proper care should be taken in reading,
recording the data. Also calculation of error
should be done accurately.
• By increasing the number of experimenters
we can reduce the gross errors.
• If each experimenter takes different reading
at different points, then by taking average of
more readings we can reduce the gross
errors.
Systematic / Determinate Errors
• The constant error that occurs no matter how
many times the measurement is done and
averages are taken.
• This error has a definite value and sign (+/-).
• Systematic errors cause a bias in
measurements.
• Bias is a positive or negative deviation of all
the measured values from the true value.
Bias
0
5
10
15
20
25
30
0 2 4 6 8 10 12
TRUE
Pos.Bias
Neg.Bias
Causes of systematic errors
• Instrument errors
• Method errors
• Personal errors
Instrument errors
• Failure to calibrate
• Degradation of parts in the
instrument
• Power fluctuations
• Variation in temperature
• Can be corrected by
calibration or proper
instrumentation
maintenance..
Method errors
• The method used is having an inherent problem
– Due to non-ideal physical or chemical behavior
– Completeness and speed of reaction
– Interfering side reactions
– Sampling problems
• Can be corrected with proper method
development.
• Ex: We take concentration of solutions instead
of activities in many calculations.
• End point – Equivalence point
Personal errors
• This type of error occurs where
measurements
– require judgment
– result from prejudice
– color acuity problems
• Can be minimized or eliminated with proper
training and experience.
….Personal errors
Where is the end point?
Here
Or here?
Minimization of systematic errors
• Calibration of instruments and apparatus.
• Performing duplicate runs simultaneously.
• Performing a blank run.
• Performing control runs ( with standards).
• Measurements by independent methods.
• Method of standard addition.
• Method of internal standard.
RANDOM ERRORS
Random Errors
Random Errors
• Any repeated measurement gives slightly
differing values, even when done with utmost
care and under similar conditions.
• This error stems from unpredictable inaccuracies
in each step of the measurement.
• It is not having a fixed value or sign – known as
indeterminate error.
• They occur with statistical distribution and
treated by statistical methods.
Error Handling
Definition of Some Statistical Terms
• True value – The actual value to be got
• Measured value – the value got in a trail
• Error = True value ± measured value
• Relative error = Error / True value
• % error = Relative error X 100
Measures of Central Tendency
• Mean – its types
– Mean or Arithmetic mean is the average of a set
of data.
– Population Mean / Limiting Mean – is the
arithmetic average of a set of data where the
number of data (population) is nearing ∞
Sample mean / Arithmetic mean
• Simple average
• When you know the number of sample is
finite.
• Example arithmetic mean of 1,2,3,4,5 is 3
1+2+3+4+5 = 15
15/5 = 3
Population Mean
The formula to find the population mean is
μ = (Σ * X)/ N
where:
Σ means “the sum of.”
X = all the individual items in the group.
N = the number of items in the group.
Example
• All 57 residents in a nursing home were
surveyed to see how many times a day they
eat meals. 1 meal (2 people), 2 meals (7
people),3 meals (28 people),4 meals (12
people),5 meals (8 people).What is the
population mean for the number of meals
eaten per day?
Solution
(1*2)+(2*7)+3*28)+(4*12)+(5*8)=188
Divide your answer to Step 1 with the number of
items in your data set. There are 57 people,
so:
188 / 57 = 3.29824561404
That’s an average of 3.3 meals per person, per
day.
The population mean is 3.3.
• Figuring out the population mean should feel
familiar. You’re just taking an average, using the
same formula you probably learned in basic
math (just with different notation).
• However, care must be taken to ensure that you
are calculating the mean for a population (the
whole group) and not a sample (part of the
group).
• The symbols for the two are different:
Population mean symbol = μ
Sample mean symbol = x̄
Median
• The median, is the middle value when we order our
data from the smallest to the largest value.
• When the data set includes an odd number of
entries, the median is the middle value.
• For an even number of entries, the median is the
average of the n/2 and the (n/2) + 1 values, where n
is the size of the data set.
Median
• Determine the median value of the following
data 1,3,1,8,5,4,3,9,5 ( n is odd)
1, 1, 3, 3, 4, 5, 5, 8, 9
• If n is even 1,3,1,8,5,4,3,9,5,7
1 1 3 3 4 5 5 7 8 9
Median = 4+5 / 2 = 4.5
Standard deviation & Variance
• Standard Deviation
– The Standard Deviation is a measure of how spread out
numbers are.
– A quantity expressing by how much the members of a
group differ from the mean value for the group
– Its symbol is σ (the greek letter sigma)
– The formula is easy: it is the square root of
the Variance. So now you ask, "What is the Variance?"
• Variance
– The Variance is defined as:
– The average of the squared differences from the Mean.
Distribution of Errors
• Random errors are distributed normally i.e. it
follows normal distribution.
Where x – value measured
µ - Mean
σ - Standard deviation
Normal distribution curve
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normal distribution curve
•The Normal Distribution has:
mean = median = mode
•symmetric about the center
•50% of values less than the mean
and 50% greater than the mean
Effect of Different Errors on The Measurement
Precision and Accuracy
• Assume a titration experiment is done for 10 times.
The actual concentration is 0.1122 N
• Student-1 is getting – Accurate and Precise
•
• Student – 2 is getting – Precise but not accurate
0.1122 0.1121 0.1122 0.1122 0.1121 0.1122 0.1122 0.1122 0.1122 0.1122
0.1126 0.1125 0.1127 0.1126 0.1126 0.1126 0.1126 0.1125 0.1126 0.1126
Thank
You

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Assessment of Analytical Data Errors and Statistics

  • 1. Assessment of Analytical Data V. Santhanam Department of Chemistry SCSVMV
  • 2. MPC101 : UNIT – IV - RESEARCH METHODOLOGY Errors in chemical analysis – classification of errors – determination of accuracy of methods – improving accuracy of analysis – significant figures – mean, standard deviation. comparison of results : “ t ” test, “ F ” test and “ χ2 ” – rejection of results – presentation of data. Sampling – introduction – definitions – theory of sampling – techniques of sampling – statistical criteria of good sampling and required size.
  • 3. Nature of Quantitative Analysis • Modern analytical chemistry is concerned with the – Detection – identification – measurement of the chemical composition using existing instrumental techniques, and the development or application of new techniques and instruments. • It is a quantitative science - desired result is always numeric.
  • 4. • Quantitative results are obtained using devices or instruments that allow us to determine the concentration of a chemical in a sample from an observable signal. • There is always some variation in that signal / value measured over time due to noise and / or drift within the instrument.
  • 5. • We also need to calibrate the response as a function of analyte concentration in order to obtain meaningful quantitative data. • As a result, there is always an error, a deviation from the true value, inherent in that measurement. • One of the uses of statistics in analytical chemistry is therefore to provide an estimate of the likely value of that error; in other words, to establish the uncertainty associated with the measurement.
  • 6. When using an analytical method we make three separate evaluations of experimental error. 1. Before beginning an analysis we evaluate potential sources of errors to ensure that they will not adversely effect our results. 2. During the analysis we monitor our measurements to ensure that errors remain acceptable. 3. At the end of the analysis we evaluate the quality of the measurements and results, comparing them to our original design criteria.
  • 7. ERROR • Any measurement made with a measuring device is approximate. • If you measure the same object two different times, the two measurements may not be exactly the same. • The difference between two measurements is called a variation in the measurements. Another word for this variation - or uncertainty in measurement - is "error." • This "error" is not the same as a "mistake."
  • 8. • It does not mean that you got the wrong answer. The error in measurement is a mathematical way to show the uncertainty in the measurement. • It is the difference between the result of the measurement and the true value of what you were measuring. Error = True value ~ Measured value
  • 9. Absolute and Relative Errors • Error in measurement may be represented by the actual amount of error, or by a ratio comparing the error to the size of the measurement (Relative error) • The absolute error of the measurement shows how large the error actually is, while the relative error of the measurement shows how large the error is in relation to the correct value.
  • 10. • Absolute errors do not always give an indication of how important the error may be. If you are measuring a football field and the absolute error is 1 cm, the error is virtually irrelevant. • But, if you are measuring a small machine part (< 3cm), an absolute error of 1 cm is very significant. While both situations show an absolute error of 1 cm., the relevance of the error is very different. For this reason, it is more useful to express error as a relative error.
  • 11. Types of Errors Depending on the origin of errors they can be classified in to 1. Gross errors 2. Systematic / Determinate errors 3. Random / Indeterminate errors
  • 12. Gross Errors • This category of errors includes all the human mistakes while reading, recording and the readings. • Mistakes in calculating the errors also come under this.
  • 13. Reasons for Gross Errors – Incompetency of the observer • Observer may not be knowing all the technical details of the measurement. – Carelessness • The observer may not be measuring the value with full attention. For example the students recording a value in lab while chatting with their batch mates. • Spilling of solutions • Contaminations • Arithmetic errors – Transposition • Data transposition occurs to all many times. Checking and rechecking before recording can avoid this error Ex Recording 1.1125 as 1.1215
  • 14. How to reduce gross errors • Proper care should be taken in reading, recording the data. Also calculation of error should be done accurately. • By increasing the number of experimenters we can reduce the gross errors. • If each experimenter takes different reading at different points, then by taking average of more readings we can reduce the gross errors.
  • 15. Systematic / Determinate Errors • The constant error that occurs no matter how many times the measurement is done and averages are taken. • This error has a definite value and sign (+/-). • Systematic errors cause a bias in measurements. • Bias is a positive or negative deviation of all the measured values from the true value.
  • 16. Bias 0 5 10 15 20 25 30 0 2 4 6 8 10 12 TRUE Pos.Bias Neg.Bias
  • 17. Causes of systematic errors • Instrument errors • Method errors • Personal errors
  • 18. Instrument errors • Failure to calibrate • Degradation of parts in the instrument • Power fluctuations • Variation in temperature • Can be corrected by calibration or proper instrumentation maintenance..
  • 19. Method errors • The method used is having an inherent problem – Due to non-ideal physical or chemical behavior – Completeness and speed of reaction – Interfering side reactions – Sampling problems • Can be corrected with proper method development. • Ex: We take concentration of solutions instead of activities in many calculations. • End point – Equivalence point
  • 20. Personal errors • This type of error occurs where measurements – require judgment – result from prejudice – color acuity problems • Can be minimized or eliminated with proper training and experience.
  • 21. ….Personal errors Where is the end point? Here Or here?
  • 22. Minimization of systematic errors • Calibration of instruments and apparatus. • Performing duplicate runs simultaneously. • Performing a blank run. • Performing control runs ( with standards). • Measurements by independent methods. • Method of standard addition. • Method of internal standard.
  • 25. Random Errors • Any repeated measurement gives slightly differing values, even when done with utmost care and under similar conditions. • This error stems from unpredictable inaccuracies in each step of the measurement. • It is not having a fixed value or sign – known as indeterminate error. • They occur with statistical distribution and treated by statistical methods.
  • 27. Definition of Some Statistical Terms • True value – The actual value to be got • Measured value – the value got in a trail • Error = True value ± measured value • Relative error = Error / True value • % error = Relative error X 100
  • 28. Measures of Central Tendency • Mean – its types – Mean or Arithmetic mean is the average of a set of data. – Population Mean / Limiting Mean – is the arithmetic average of a set of data where the number of data (population) is nearing ∞
  • 29. Sample mean / Arithmetic mean • Simple average • When you know the number of sample is finite. • Example arithmetic mean of 1,2,3,4,5 is 3 1+2+3+4+5 = 15 15/5 = 3
  • 30. Population Mean The formula to find the population mean is μ = (Σ * X)/ N where: Σ means “the sum of.” X = all the individual items in the group. N = the number of items in the group.
  • 31. Example • All 57 residents in a nursing home were surveyed to see how many times a day they eat meals. 1 meal (2 people), 2 meals (7 people),3 meals (28 people),4 meals (12 people),5 meals (8 people).What is the population mean for the number of meals eaten per day?
  • 32. Solution (1*2)+(2*7)+3*28)+(4*12)+(5*8)=188 Divide your answer to Step 1 with the number of items in your data set. There are 57 people, so: 188 / 57 = 3.29824561404 That’s an average of 3.3 meals per person, per day. The population mean is 3.3.
  • 33. • Figuring out the population mean should feel familiar. You’re just taking an average, using the same formula you probably learned in basic math (just with different notation). • However, care must be taken to ensure that you are calculating the mean for a population (the whole group) and not a sample (part of the group). • The symbols for the two are different: Population mean symbol = μ Sample mean symbol = x̄
  • 34. Median • The median, is the middle value when we order our data from the smallest to the largest value. • When the data set includes an odd number of entries, the median is the middle value. • For an even number of entries, the median is the average of the n/2 and the (n/2) + 1 values, where n is the size of the data set.
  • 35. Median • Determine the median value of the following data 1,3,1,8,5,4,3,9,5 ( n is odd) 1, 1, 3, 3, 4, 5, 5, 8, 9 • If n is even 1,3,1,8,5,4,3,9,5,7 1 1 3 3 4 5 5 7 8 9 Median = 4+5 / 2 = 4.5
  • 36. Standard deviation & Variance • Standard Deviation – The Standard Deviation is a measure of how spread out numbers are. – A quantity expressing by how much the members of a group differ from the mean value for the group – Its symbol is σ (the greek letter sigma) – The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?" • Variance – The Variance is defined as: – The average of the squared differences from the Mean.
  • 37. Distribution of Errors • Random errors are distributed normally i.e. it follows normal distribution. Where x – value measured µ - Mean σ - Standard deviation
  • 39. Normal distribution curve •The Normal Distribution has: mean = median = mode •symmetric about the center •50% of values less than the mean and 50% greater than the mean
  • 40. Effect of Different Errors on The Measurement
  • 41.
  • 42. Precision and Accuracy • Assume a titration experiment is done for 10 times. The actual concentration is 0.1122 N • Student-1 is getting – Accurate and Precise • • Student – 2 is getting – Precise but not accurate 0.1122 0.1121 0.1122 0.1122 0.1121 0.1122 0.1122 0.1122 0.1122 0.1122 0.1126 0.1125 0.1127 0.1126 0.1126 0.1126 0.1126 0.1125 0.1126 0.1126