Application of Statistical Concepts in the Determination of Weight Variation in Samples Jaimie P. Loja Institute of Chemistry, University of the Philippines, Diliman, Quezon City June 19, 2013 June 26, 2013 I.
II.
Methodology The materials used in the experiment were watch glass, ten 25 centavo coins, forceps, and analytical balance. Using “weighing by difference,” the students took the weight of each coin. This process was done by putting a watch glass with ten 25 centavo coins on the analytical balance and pressing “on tare.” When the reading turned zero, the students carefully took one coin out of the watch glass using forceps. The absolute value of the given weight by the analytical balance was recorded. Arrange the other coins left in such a way that their weight is concentrated on the middle. Again, press “on tare” that would make the reading zero. Take another coin, and record the absolute value of the reading. This process was repeated until there is no coin left. The weights of each coin were recorded in the data sheet. Results and Discussion The experiment used the process the process called “weighing by difference,” it was considered as the most accurate method of measuring the weight of a solid sample. The data acquired were used to make two data sets.
Weight of Samples One to Six Data Set 1 Samples Weight 1 3.6230 ± 0.0002 2 3.5574 ± 0.0002 3 3.6184 ± 0.0002 4 3. 5882 ± 0.0002 5 3. 5725 ± 0.0002 6 3. 6640 ± 0.0002
Table I.
The table above shows the obtained weight from the first six coins, and its tolerance 0.0002.
Table II.
Samples 1 2 3 4 5 6 7 8 9 10
Weight of All Samples Data Set 1 Weight 3.6230 ± 0.0002 3.5574 ± 0.0002 3.6184 ± 0.0002 3. 5882 ± 0.0002 3. 5725 ± 0.0002 3. 6640 ± 0.0002 3.5669 ± 0.0002 3.5881 ± 0.0002 3.5609 ± 0.0002 3.5896 ± 0.0002
The table above shows the obtained weight from all the coins, and its tolerance 0.0002. Using these two data sets, several statistical treatments were done to test the weight variation for each. The first test was the Q-test.
= questionable value = closest value to R = range
This test was done to determine if the questionable value from the set was very different from the rest of the data. The computed Q exp exp was compared to Q tab tab in order to reject or accept a data. If Q exp exp > Q tab tab, the questionable value should be rejected, and accepted if Q exp exp < Q tab tab. In the experiment, the computed results of Q exp exp for the highest and lowest questionable value were lesser than the Q tab tab value 0.625 for Data Set 1, and 0.468 for Data Set 2, which means that there was no data rejected.
In order to perform other successive statistical tests, computing the mean for each data set was needed.
X=
The computed range for Data Set 1 was 0.1066 ± 0.0003, and 0.1066 ± 0.0003 for Data Set 2. In relation with range, the relative range was computed.
∑
= individual values making up the set = population of the set
Relative range is the percentage ratio of the range to the average value in the set. The computed RR for Data Set 1 was 2.958 ppt ± 0.008, and 2.969 ppt ± 0.008. Lastly, confidence limit was computed for each set in the experiment.
One set can be represented by just a number using the mean. It can be described as the center of a data distribution. The computed mean for Data Set 1 is 3.6039 ± 0.0005, and 3.5900 ± 0.0006 for Data Set 2. Standard deviation can now be solved by using the mean.
CL
∑( )
= individual values making up the set = mean = population of the set
This treatment was done to find out how much dispersion from the mean there is inside the set. The computed value for Data Set 1 is 0.038940 ± 0.001, and 0.035361 ± 0.002 for Data Set 2. Both values were small, which means that the data from each were very close to the mean. In relation with standard deviation, the relative standard deviation was computed.
This test was done to compare SD of different measurements. Relative standard deviation is the percentage ratio of the standard deviation to the average value in the set. The computed RSD for Data Set 1 was 10.805 ppt ± 0.3, and 9.8499 ppt ± 0.6 in Data Set 2. Both values were low, which means that there is a low inconsistency in the set. In order to perform other successive statistical tests, computing the range for each data set is needed. R
= XHighest - XLowest
√
X = mean t = tabulated value s = standard deviation n = population of the set
s=
=X±
The obtained confidence limit for Data Set 1 was 3.6039 ± 0.0409, and 3.5900 ± 0.0253 for Data Set 2. Both values have a considerable low value for the confidence interval, which means that it has a high precision estimate for the parameter. III.
Conclusion The computed values for each statistical test used in the experiment proved that there’s not much weight variation in each sample. Due to these results, it can be concluded that the experiment is a success. This validates that statistical concepts can be used to determine the weight variation in samples. Aside from the easy procedure, the determination of variation is easy due to the provided formula.
IV. References th [2] Petrucci. General Chemistry 10 edition. Canada: Pearson Education, Inc. 2011 [1] Silberberg. Principles of General Chemistry. New York: McGraw-Hill Companies, Inc. 2010. [3] Wilbraham, Antony, Dennis Staley, Michael Matta, and Edward Waterman. Chemistry. Jurong: Pearson Education Asia Ptd Ltd, 2000.
V.
Appendices
r
Answers-to-Questions: 1. Give the significance of Q-test. - Q-test is a significance test used to determine if a data is very different from the rest 2. Give the significance of the mean and standard deviation. - The mean is the representation of a data set by using just one number. It can be describe as the center of distribution and the estimate value of dispersion. Meanwhile, standard deviation shows how much variation from the mean there is in the set. The lower the value of SD, the closer the data to the mean.
| | | | √ √
r
r
= ±0.03
X=
∑
=
( )
= 3.6039 ±0.0005
Range R = XHighest - XLowest = 3.6640 – 3.5574 = 0.1066 ± 0.0003
√ √ (()) √ ()(√ ) r
=
= = ±0.0003 Relative Range (in ppt)
=
= 2.958% ± 0.008 r
=
= 2.958
=
Mean
=
= 10.805
= 0.14165 ± 0.0003
= = ±0.001 Relative Standard Deviation (in ppt)
= 10.805 ppt ± 0.03
Lowest value
= = ±0.0003
=
=
= 0.38462 ± 0.0003
r
(∑( ))() (())(( )) √ √ (()) = 0.038940 ± 0.001
Q-test Highest value
= = ±0.0005 Standard Deviation
=
Sample Calculations
√ √
s=
3. Give the significance of the confidence interval. - Confidence interval is a measure of confidence around a point estimate. It is used to describe a reliability of an estimate. 4. How do the statistics calculated from data set 1 differ from those obtained from data set 2 ? - In the computations of the statistical treatments, the degree of reliability for each test increases with increasing number of observations. From this, it can be said that the statistics calculated from data set 2, is more reliable than statistics calculated from data set 1.
=
= ±0.008 Confidence Limit CL
=X±
= 3.6039 ± 0.0409
Name:
Jaimie P. Loja
Date Performed:
June 19,2013
Co-worker:
Monica B. Mabituin
Date Finished:
June 19, 2013
Lizette S. Lamella Application of Statistical Concepts in the Determination of Weight Variation in Samples DATA SHEET EXP. 1 Data Set 1
Weight of Samples One to Six Data Set 1 Samples Weight 1 3.6230 ± 0.0002 2 3.5574 ± 0.0002 3 3.6184 ± 0.0002 4 3. 5882 ± 0.0002 5 3. 5725 ± 0.0002 6 3. 6640 ± 0.0002
Table I.
Q exp w/ the highest value : 0.38462±0.0003 Q exp w/ the lowest value : 0.14165±0.0003 3.6039 ± 0.0005 Mean: 0.038940 ± 0.001 Standard Deviation: Relative Standard Deviation (ppt): 10.805 ppt ± 0.3 Range: 0.1066 ± 0.0003 Relative Range (ppt): 2.958 ppt ± 0.008 3.6039 ± 0.0409 Confidence Limit: Q-test:
Data Set 2
Weight of All Samples Data Set 1 Samples Weight 1 3.6230 ± 0.0002 2 3.5574 ± 0.0002 3 3.6184 ± 0.0002 4 3. 5882 ± 0.0002 5 3. 5725 ± 0.0002 6 3. 6640 ± 0.0002 3.5669 ± 0.0002 7 8 3.5881 ± 0.0002 3.5609 ± 0.0002 9 10 3.5896 ± 0.0002
Table II.
Q exp w/ the highest value : 0.38462±0.0003 Q exp w/ the lowest value : 0.032833±0.0003 Mean: 3.5900 ± 0.0006 Standard Deviation: 0.035361 ± 0.002 Relative Standard Deviation (ppt): 9.8499 ppt ± 0.6 0.1066 ± 0.0003 Range: 2.969 ppt ± 0.008 Relative Range (ppt): 3.5900 ± 0.0253 Confidence Limit: Q-test:
