Application of Statistical Concepts in the Weight Variation of Samples

Application of Statistical Concepts in the Weight Variation of Samples Gliezl Allison G. Imperial Teshia Faye T. Josue  Institute of Chemistry, College of Science, University of the Phi lippines, Diliman, Quezon City 1101 Philippines  Department of Chemistry, College of Science, University of the Philippines, Di liman, Quezon City 1101 Philippines

Experimental Detail The main objectives of this experiment are to determine the significance of statistical concepts in the field of analytical chemistry or more specifically (based on the experiment) the weight variation of samples, the 25 centavo coins. This experiment also aims to teach the  proper usage usage of the analytica analyticall balance.

that seems to be outside of the range. If the Qtest was not performed during the experiment, undetected gross error might hinder in getting accurate and precise values. Also, this test makes sure that all data rightfully belongs to the set and not discarded due to leniency in setting the limits. Equation 1 shows how Qexp was obtained where Xq is the suspected value, X n is the value closest to the suspected value, and R is the range.

In order to determine the weight variation, ten 25 centavo coins were placed on a watch glass using forceps and positioned inside an analytical  balance. Each coin is weighed using “weighing  by difference” method. By pressing the tare on  button of the instrument, the balance was set to zero and the coins were removed one by one until the weight of each coin was obtained. All weights gathered from the experiment were recorded for tabulation. Each weight recorded was considered as a single sample which was then grouped into two data sets wherein the first data set contained 6 samples while the second data set contained samples 1-10.

When QtabQexp, the Qexp value calculated is accepted. Below is a table showing the calculated Q exp values which are lesser than the Q tab value at 95% confidence level of 0.625 and 0.468 for data set 1 and 2, respectively.

Data and Results

Table 1. Q exp vs. Qtab

Slight variation was obtained from weighing the ten 25 centavo coins using the analytical  balance. (Refer (Refer to Appendix A for the table with the corresponding samples and their value)

Data Set

The Q-test, which is a simple, widely used statistical test for deciding whether a suspected result should be retained or rejected (Dean, Dixon.1951), was performed to determine which of the weights that were recorded is an outlier. This test is significant because there are times when a set of data contains an outlying result

2

Equation 1. Q test formula Qexperimental=Qexp= | Xq – X  – Xn | R 

1

Suspect Values 3.6410 3.5793 3.6410 3.5793

Qexp

0.17 0.41 0.17 0.41

Qtab

Conclusion

0.625

Accepted

0.468

Accepted

The results show that the value for the Q exp in the first data set is lesser than the Q tab. Moreover, the same comparison is made for the second data set. This shows that all the weights recorded are

all accepted and were made part of further  computations. After making sure that all values are part of the range, other statistical computations were made such as the mean, range, relative range, standard deviation, relative standard deviation, and confidence limits (at 95% confidence level). One of the most commonly used measures of  central tendency is the mean. The mean is the average or the sum of all measured values divided by the number of samples in a data set. Acquiring the value of the mean gives the best estimate central value of the set and with this, the set becomes more reliable than any of the individual result (Skoog. 2004). The equation of  the mean is shown below. Equation 2. Mean formula n

  X 

i

 X 



i 1

 N 

Below is the tabulated data of the calculated values from the experiment. As shown in the table, two values are recorded for data set 1 and 2. The mean value for the first data set is 3.61 and for the second, 3.61 as well. When compared to the standard weight of a 25 centavo coin presented by the Bangko Sentral ng Pilipinas (BSP), this shows a 0.19 difference in the weight wherein the official weight should be 3.8 g. A plausible reason for this difference is the deterioration in the percent material composition of the coins. Another cause of the difference is the year that the coins were manufactured. This may cause the variations of  the weights of the coins that were issued during the year 1995 and 2004. The coins manufactured in the year 1995 may have more material composition as compared to the coins from 2004. In the experiment, the year when the coins were minted were not taken note off.

Parameter

Data set 1

Data set 2

Mean Standard Deviation Relative Standard Deviation Range Relative Range

3.61 g 0.021583 g

3.61 g 0.019017 g

0.60%

0.53%

0.0617 1.71% 3.63 3.59

0.0617 1.71% 3.62 3.59

Confidence Limits

The next statistical parameters are the range and standard deviation which shows how close the values are obtained in the same way, under the measures of precision. Standard deviation (s) measures how closely the data are clustered about the mean. The smaller the s value, the more closely the data are clustered around the mean and vice versa. Referring to the table above, the standard deviation values for data set 1 and 2 are 0.021583 g and 0.019017 g, respectively. These values show that the data of  the second set is more clustered around the value of 3.61 as compared to the first data. No values are deviating from the value 3.61. Equation 3 shows the equation for the standard deviation. Equation 3. Standard deviation formula n

 ( X  s 

i

 X )2

i 1

n 1

The third statistical parameter is the relative standard deviation (RSD) which allows standard deviations of varying measurements to be compared more meaningfully. To evaluate the uncertainty between unlike measurements of  varying absolute magnitude, the RSD is used. The equation below shows the relative standard deviation formula. Equation 4. Relative standard deviation formula  RSD 

 s  X 

100

 Next, the range and relative range were calculated. The range is the difference between the highest and lowest sample values in the data set. In the case of the experiment, the range is the same in both data set 1 and 2 with a value of  0.0617 g. This is because the highest and lowest sample values of both the data sets are the same.

Equation 5 shows the formula in getting the range of the data set.

Equation 5. Range formula R= Xhighest-Xlowest On the other hand, the relative range refers to the percentage ratio of the range to the average value in the set. In the experiment, a relative range of 1.71% was obtained for both data set. The formula for relative range is stated below. Equation 6. Relative range formula   R   100   X  

 RR  

Lastly, the confidence limits are the probability that the true mean lies within a certain interval. This is significant because it show which certain interval does the mean 3.61 falls under. Based on the calculations, the interval for data set 1 is 3.63g and 3.59g and for data set 2, 3.62g and 3.59g. The formula for the confidence limit is stated below.

variation is because of the change in material composition based on the year it was manufactured. Those coins minted in 2004 have a different composition than that of the ones issued earlier. Also, all the values are relative to the standard official weight of the coin.

Appendix Appendix A. Table for data set

1 2 3 4 5 6 7 8 9 10

ts n

The statistics calculated from data set 1 is somehow the same in terms of the mean, range, and relative range. This means that the individual values obtained for both are precise with each other. The obtained values for the 2 data sets are close to each other.

Data Set 2 3.6078 3.6410 3.6305 3.6093 3.6048 3.5793 3.6116 3.5844 3.6197 3.6226

Calculations:

Qexperimental=Qexp= | Xq – Xn | R 

Equation 7. Confidence Limit formula ConfidenceLimit  X  

Data Set 1 3.6078 3.6410 3.6305 3.6093 3.6048 3.5793

Qexp= | 3.6078 – 3.6410 | = 0.17 0.0617 n

  X 

i

 X 



i 1

 N 

X=3.6078+3.6410+3.6305+3.6093+3.6048+3.5793 / 6 = 3.61 g

Conclusion In conclusion, the weight of the coins varies from each other with a close value from each other. All independent sample were taken into calculations since the Q-test showed that all are significant sample value. The reasons for the variation may be due to the decomposition of the material composition of the coins. As the time  passes, the percent material composition deteriorates. Also, another reasons for the

n

 s 

 ( X

i

 X )2

i 1

n 1

S= square root of (3.61-Xi)2 + …. / 5 =0.021583 g

Range=3.6078-3.5793=0.0617

 RSD 

 s  X 

100

RSD= 0.021583 g / 3.21 x 100=0.60%   R   100   X  

 RR  

RR= 0.0617g / 3.61 g = 1.71% ConfidenceLimit  X  

ts n

CL= 3.61 x [(2.45) (0.021583) / 6] = 3.63

References

Harris, D. C. Quantitative Chemical Analysis. Nixon, R. D. (1951). Analytical Chem. Skoog, e. (2004). Fundamentals of Analytical Chemistry.