Application of Statistical Concepts in the Determination of Weight Variation in Samples Emmalin S. Mesina
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Department of Mining, Metallurgical Metallurgical and Materials Materials Engineering, Engineering, College of of Engineering Engineering University of the Philippines, Diliman, Quezon City, Philippines Date Due: June June 28, 2013 2013 Date Submitted: Submitted: June 28, 2013
Abstract
The application of statistical concepts in analytical chemistry was observed in the experiment when the replicate measurements gathered were calculated and had undergone the various statistical tools in order to achieve reliable data. The replicate measurements were the sample weights of the ten one-peso coins weighed using an analytical balance. The calculated mean of data set 1 was 5.3919 ± 0.0001 g, while the data set 2 had 5.39877 ± 0.00009 g. The standard deviation of data sets were 0.04 and 0.03, respectively, which showed high precision in the measurements. The 95% confidence level that incorporates about two standard deviation was used in estimates of validity of analytical measurements. The confidence limit calculated for both data sets were 5.3919 ± 0.04 and 5.39877 ± 0.02, respectively, which implied that the larger number of replicate values, the more precise the results would be. The experiment performed was effective in showing how statistical analysis can help in determining reliable results with respect to accuracy and precision which analytical chemistry is up to. Keywords: statistical analysis, analytical chemistry, accuracy, accuracy, precision precision
INTRODUCTION In analytical chemistry, it is important to have reliable data in order to achieve useful results in experiments and calculations. In the experiment, replicate measurements are being evaluated with respect to accuracy and precision by the use of various statistical parameters to be able to gather quantitative measures of data.
̅
The first statistical concept calculated in the experiment is the mean, one one of the most common measures of central tendency [4]. It is defined as the sum of the individual measurement values X i, divided by the number of measurements n [3],
̅ =
∑ Xi
measurements [4]. It is theoretically the square root of the mean of square of the difference between the individual measured value xi, and the mean , of the infinite number of [2] measurement ,
s = s =
(2).
When several small sets have the same sources of indeterminate error, the standard deviations of the individual sets may be pooled to more accurately determine the standard deviation of the analysis. Pooled standard deviation or s pooled is a weighted average of the individual estimates [3] . It is given by
(1).
Another concept used is the standard deviation, s, s, which expresses the precision of the replicate
̅ ∑ ( )
S pooled =
( ) ∑( )
(3)
where n1 is the number of data in set 1, n 2 is the number of data in set 2, and so forth. The term n s is the number of data sets that are being pooled. [4]
Moreover, there is a probability that the true value falls within a range defined by the experimental mean and standard deviation. This range is called the confidence interval and the limits of the range are called the confidence limit. The confidence limit is given by Confidence Limit =
̅
±
√ √
(4)
where t is the statistical factor that depends on the number of degrees of freedom (n-1) and the confidence of desired level [2]. In the experiment, the 95% level that incorporates about two standard deviation was used in estimates of validity of analytical measurements. In this experiment, the accuracy and precision of measurements which lead to reliable results were studied through the use of statistical analysis. Hence, this will help to the study of analytical chemistry.
METHODOLOGY Ten one-peso coins produced in the same year were first cleaned with lint-free tissue paper to ensure that no other weight will contribute to the sample. These were placed one by one in a 50mL beaker with the use of forceps. Using the weighing by difference technique, an analytical balance was used to get the weights of the samples. In the balance room, the area where the analytical balance was located, was cleaned and ensured that it was free of materials not needed in the weighing procedure. The analytical balance was then leveled by adjusting its leveling feet until the bubble, the level indicator, appeared in the center of the circle. Before weighing anything in the analytical balance, it was first calibrated calibrated to 0.0000 0.0000 g. In the the weighing process, the ON TARE switch was pressed to turn on the balance. balance. Only one of the three chambers of the balance was opened when placing the beaker inside to ensure that no
weight will be added by the humidity of air. Light movements were also done to prevent producing vibrations vibrations that will be detected by the balance. The weight was recorded when the stability indicator of the balance appeared. Then one by one, the coins were taken out of the beaker with with the use of forceps.
RESULTS AND DISCUSSION
In the experiment, the coins were handled with the use of forceps to ensure that there will be no fingerprints that can affect the weighing results. This was done to minimize the adsorption of humidity of the samples, most especially when they are weighed by high-resolution balance. After weighing, the sample weights were recorded. With these data, it is possible to have values that differ significantly from the others, which are called outliers, and were considered to be invalid. So So before proceeding proceeding to the statistical statistical analysis, Q-Test was done to suspected values, usually the data with the highest and the lowest value, to determine if these were to be accepted or rejected for future calculations. For the suspected value to be accepted, it should not exceed the tabulated critical values for the certain confidence level. In the experiment, the test was performed at 95% confidence level and the Q-Test experimental values for the two data sets did not exceed the Q tab. Therefore, there were no outliers in the experiment which showed high precision in the results. The precision of the replicate measurements is expressed in terms of standard deviation. Standard deviation serves as the basis for calculating the uncertainty of measurements [1]. Furthermore, standard deviation is related to Gaussian/normal distribution. To understand the probability factors of a normal normal distribution, there there are some ‘rules’: 1. The total area under the curve is equal to 1 (100%) 2. About 68% of the area under the curve falls within 1 standard deviation.
3. About 95% of the area under the curve falls within 2 standard deviations. deviations. 4 About 99.7% of the area under the curve falls within 3 standard deviations. deviations. Items 2,3 and 4 are sometimes referred to as the ‘empirical rule’ or the 68-95-99.7 68 -95-99.7 rule. In terms of probability, once the data is determined to be normally distributed (bell curved) and the mean and standard deviation are calculated, it is then possible to determine determine the probability that a single data point will fall within a given range of possibilities. possibilities. [6]
Confidence limits, as mentioned earlier, is also significant in the study because it is the likely range of the true value. Another important concept embodied in confidence limits is precision of estimation. estimation. The wider the confidence interval, the less the precision. [5] The statistical parameters calculated in both data sets are shown below: Table 1. Reported Values DATA SET 1
PARAMETERS
5.3714 ± 0.0003 5.4213 ± 0.0003 5.4199 ± 0.0003 5.3707 ± 0.0003 5.3419 ± 0.0003 5.4266 ± 0.0003
It is inevitable in every experiment to have errors. Thus, when there are several sets of indeterminate errors, the standard deviation may be pooled. Pooled standard deviation is significant because it gives more accurate standard deviation of the analysis despite the errors. There are three types of errors that were possible done in the experiment: systematic, indeterminate, and gross errors. In systematic or determinate errors, there are three subtypes: (1) personal/operative, personal/operative, which is because of ineptitude of the experimenter, like touching the samples that are to be weighed, (2) method, which is due to non-ideal chemical or physical behaviour of analytical analytical systems, like the instability of the balance, and (3) instrumental, which refers to the failure of measuring device to perform in accordance to the required standards. These errors affect the accuracy of the measurements. On the other hand, the indeterminate or random errors affect the precision of the measurements. This type of error was present in the experiment. Gaussian distribution was used to still determine the precision of measurements despite this kind of error. Lastly, the gross error differ from the determinate and indeterminate errors. They usually occur occasionally and often large. It is the most serious type of error that the entire experiment has to be repeated. [3]
Mean SD RSD Range RR CL
5.3919 ± 0.0001 0.04 7 0.0847 ± 0.0004 15.708 ± 0.07 5.3919 ± 0.04
DATA SET 2 5.3714 ± 0.0003 5.4213 ± 0.0003 5.4199 ± 0.0003 5.3707 ± 0.0003 5.3419 ± 0.0003 5.4266 ± 0.0003 5.4059 ± 0.0003 5.4074 ± 0.0003 5.4113 ± 0.0003 5.4113 ± 0.0003 5.39877 ± 0.00009 0.03 5 0.0847 ± 0.0004 15.689 ± 0.07 5.39877 ± 0.02
It was found out that Data Set 1, with less replicate measurements (six measurements), has a wider confidence interval of 5.3519 to 5.4319, thus, less precise than Data Set 2 with a confidence interval interval of 5.37877 to 5.41877. CONCLUSION
Calculations showed good results in the experiment. The mean for data set 1 was calculated to have the value of 5.3919 ± 0.0001 g, while the data set 2 had 5.39877 ± 0.00009 g. The calculated standard deviations in both data sets were less than 0.1 which indicate high precision in the measurements. measurements. There was no outlier in the sample weights which tells that all measurements are valid.
Moreover, the concepts, namely, the mean, standard deviation and confidence limits, were used to determine the approximation of the true value, the variability of the data gathered and the range where the true value likely lies. It can be concluded that from the experiment that the larger number of replicate measurements, the more precise the results were. Hence, it can be suggested to gather more replicate values in order to have a more reliable results. Also, in order to have more accurate sample weights, the use of forceps was really required. The samples should be ensured of cleanliness to avoid the possible contributed weight from different factors such as moisture. The objectives of the experiments such as the gain of understanding of some concepts of statistical analysis and its application to analytical chemistry had been met by the end of it.
REFERENCES
[1] Brynn Hibbert, S. Quality Assurance in the Analytical Chemistry Chemistry Laboratory; Laboratory; Oxford University Press, 2007; pp. 150-153. [2] Charan, D. Analytical Chemistry; Chemistry; PHI Learning Pvt. Ltd., 2011. [3] Skoog, Douglas, West d. M, Holler F. J. and S. R. Crouch, Fundamentals Fundamentals of Analytical Chemistry, 8th ed .; .; Brooks/Cole, Thomson Learning, Shenton Way, Singapore. [4] Institute of Chemistry, UP Diliman. Quantitative Inorganic Analysis Laboratory Manual , 2007. [5] [5]http://www.sportsci.org/resource/stats/gen eralize.html [6] http://math.about.com/od/glossaryofterms/g/ Bell-Curve-Normal-Distribution-Defined.htm
