Terminology (3.1-3, 4.1, 5.1-2)
Terminology (3.1-3, 4.1, 5.1-2)
(Q3.1-3, 6, 9-13)
Range: difference between min and max values Resolution: ability to discriminate between different values (ie. cm, mm, decimal place) Error vs Uncertainty Measurement Error (E): difference between measured (Xm) and true (Xtrue) values • true value should be known • can use conventional or accepted true value (i.e. = 3.14) Uncertainty (U): our estimate of the likely error in Xm • more realistic to use uncertainty (to it more times) Absolute: Ex = Xm – Xture Relative: Ex’ =
–
=
or
± Ux
or
U’x =
Mean: -Ex when Xm is an average value Standard of Reference for calibration: • determines quality (usefulness) of measurement • determines quality (usefulness) of results Error types Systematic: - all results either too high / low - result in bias - affects accuracy (closeness to true value) are errors that always have the same magnitude and sign, resulting in a bias of the measured values from the true value. An example would be a ruler missing the first 1 mm of its length – it will consistently give lengths that are 1 mm too short. Systematic errors affect the accuracy of the final result. Random:
- results fall either side of average - affects precision (spread) of results - express relative to mean or median will have different magnitudes and signs, and result in a spread or dispersion of the measured values from the true value. An example would be any electronic measuring device – random electrical noise within its electronic components will cause the reading to fluctuate, even if the signal it is measuring is completely constant. Random errors affect the precision of the final result; they may also affect accuracy if the number of replicates used is too small.
Gross error: irredeemable error resulting in an outlier are errors that are so serious (i.e. large in magnitude) that they cannot be attributed to either systematic or random errors associated with the sample, instrument, or procedure. An example would be writing down a value of 100 when the reading was actually 1.00. If included in calculations, gross errors will tend to affect both accuracy and precision. Precision: describes the reproducibility of a result. If you measure a quantity several times, and the values agree closely with one another, your measurement is precise. If the values vary widely, the measurement is not precise. Accuracy: describes how close a measured value is to the “true” value Replicates What are they? Repeated measurement of the same sample / same parameter / identical conditions Central Value / Central Tendency • arithmetic mean • geometric mean • mean • mode ( ) Residuals: comparison between successive individual values within a set of measurements, show the quality of the data set Sample mean:
∑
Sample standard deviation:
√
Degrees of freedom: (d.o.f., v)
(
Relative standard deviation: Coefficient of Variation:
∑ (
)
)
k = parameter
During experiment: Result of extensive inter-laboratory trials Best attainable %RSD (CV) as function of analyte concentration Places limit on ability to quantify analytes
Analytical measurement: Error, Uncertainty, Canfidence (4.1-6) Population mean, standard deviation: difference between min and max values Precision of mean value: random error in the mean decrease and n increase Measurement uncertainty: the range within the value of a measured quantity is reasonably expected to lie with a stated level of confidence (probability)
Question 1. Systematic or random error a) A 25mL transfer pipette consistently delivers 25.032 + 0.009mL 25.032 is Systematic error + 0.009 is Random error b) A 10mL buret consistently delivers 1.98 + 0.01mL when drained from exactly 0 to exactly 2mL and consistently delivers 2.03 + 0.02mL when drained from 2 to 4mL 1.98, 2.03 are Systematic error + 0.01 and 0.02 are Random error c) A 10mL buret delivered 1.9839g of water when drained from exactly 0.00 to 2.00 mL. the next time I delivered water from the 0.00 to the 2.00mL. the delivery mass was 1.9900g Random error d) 4 consecutive 20.0uL injections of a solution into a chromatograph were made and the area of a particular peak was 4383, 4410, 4401, 4390 unit Random error CHM 217 TUT1
Questions
1. The analysis of a calcite sample yielded CaO percentage of 55.95 56.04 56.08 61.08. Mean = 57.03 standard deviation = 2.26. The last value is appeared suspicious should it be retained or rejected? 2. A standard home CO detector has a standard deviation of 0.21 ppm CO. A modification had been made to the existing detector. Is this modification significantly more precise then the original? Detector StdCO Modified CO
Standard Deviation 0.21 0.12
N 20 21
3. A new method was developed to determine sulfur content in kerosenes. A known sample containing 0.123% s was tested against this method. Resultant % S are as follows: 0.112, 0.118, 0.115, 0.119. Do the data indicate that there is a bias in the method at 95% confidence level? Mean = 0.116%S
Standard Deviation = 0.0032%S
F(19,11) = 2.54
4. Two barrels of wine analyzed for alcohol content to determine if they were from the same source (by our definition same source would have similar means) Do our data indicate a difference between the wines? *Spooled = 0.070 (n=10) Barrel 1 2
Mean%EtOH 12.61 12.53
N 6 4
(Q3.1-3, 6, 9-13)
Range: difference between min and max values Resolution: ability to discriminate between different values (ie. cm, mm, decimal place) Error vs Uncertainty Measurement Error (E): difference between measured (Xm) and true (Xtrue) values • true value should be known • can use conventional or accepted true value (i.e. = 3.14) Uncertainty (U): our estimate of the likely error in Xm • more realistic to use uncertainty (to it more times) Absolute: Ex = Xm – Xture Relative: Ex’ =
–
=
or
± Ux
or
U’x =
Mean: -Ex when Xm is an average value Standard of Reference for calibration: • determines quality (usefulness) of measurement • determines quality (usefulness) of results Error types Systematic: - all results either too high / low - result in bias - affects accuracy (closeness to true value) are errors that always have the same magnitude and sign, resulting in a bias of the measured values from the true value. An example would be a ruler missing the first 1 mm of its length – it will consistently give lengths that are 1 mm too short. Systematic errors affect the accuracy of the final result. Random:
- results fall either side of average - affects precision (spread) of results - express relative to mean or median will have different magnitudes and signs, and result in a spread or dispersion of the measured values from the true value. An example would be any electronic measuring device – random electrical noise within its electronic components will cause the reading to fluctuate, even if the signal it is measuring is completely constant. Random errors affect the precision of the final result; they may also affect accuracy if the number of replicates used is too small.
Gross error: irredeemable error resulting in an outlier are errors that are so serious (i.e. large in magnitude) that they cannot be attributed to either systematic or random errors associated with the sample, instrument, or procedure. An example would be writing down a value of 100 when the reading was actually 1.00. If included in calculations, gross errors will tend to affect both accuracy and precision. Precision: describes the reproducibility of a result. If you measure a quantity several times, and the values agree closely with one another, your measurement is precise. If the values vary widely, the measurement is not precise. Accuracy: describes how close a measured value is to the “true” value Replicates What are they? Repeated measurement of the same sample / same parameter / identical conditions Central Value / Central Tendency • arithmetic mean • geometric mean • mean • mode ( ) Residuals: comparison between successive individual values within a set of measurements, show the quality of the data set Sample mean:
∑
Sample standard deviation:
√
Degrees of freedom: (d.o.f., v)
(
Relative standard deviation: Coefficient of Variation:
∑ (
)
)
k = parameter
During experiment: Result of extensive inter-laboratory trials Best attainable %RSD (CV) as function of analyte concentration Places limit on ability to quantify analytes
Analytical measurement: Error, Uncertainty, Canfidence (4.1-6) Population mean, standard deviation: difference between min and max values Precision of mean value: random error in the mean decrease and n increase Measurement uncertainty: the range within the value of a measured quantity is reasonably expected to lie with a stated level of confidence (probability)
Question 1. Systematic or random error a) A 25mL transfer pipette consistently delivers 25.032 + 0.009mL 25.032 is Systematic error + 0.009 is Random error b) A 10mL buret consistently delivers 1.98 + 0.01mL when drained from exactly 0 to exactly 2mL and consistently delivers 2.03 + 0.02mL when drained from 2 to 4mL 1.98, 2.03 are Systematic error + 0.01 and 0.02 are Random error c) A 10mL buret delivered 1.9839g of water when drained from exactly 0.00 to 2.00 mL. the next time I delivered water from the 0.00 to the 2.00mL. the delivery mass was 1.9900g Random error d) 4 consecutive 20.0uL injections of a solution into a chromatograph were made and the area of a particular peak was 4383, 4410, 4401, 4390 unit Random error CHM 217 TUT1
Questions
1. The analysis of a calcite sample yielded CaO percentage of 55.95 56.04 56.08 61.08. Mean = 57.03 standard deviation = 2.26. The last value is appeared suspicious should it be retained or rejected? 2. A standard home CO detector has a standard deviation of 0.21 ppm CO. A modification had been made to the existing detector. Is this modification significantly more precise then the original? Detector StdCO Modified CO
Standard Deviation 0.21 0.12
N 20 21
3. A new method was developed to determine sulfur content in kerosenes. A known sample containing 0.123% s was tested against this method. Resultant % S are as follows: 0.112, 0.118, 0.115, 0.119. Do the data indicate that there is a bias in the method at 95% confidence level? Mean = 0.116%S
Standard Deviation = 0.0032%S
F(19,11) = 2.54
4. Two barrels of wine analyzed for alcohol content to determine if they were from the same source (by our definition same source would have similar means) Do our data indicate a difference between the wines? *Spooled = 0.070 (n=10) Barrel 1 2
Mean%EtOH 12.61 12.53
N 6 4
