APPLICATION OF STATISTICAL CONCEPTS IN THE DETERMINATION OF WEIGHT VARIATIONS IN SAMPLES

M.J. COCAMAS1 and E.J. SEVILLA21DEPARTMENT OF SCIENCE EDUCATION, COLLEGE OF EDUCATION2DEPARTMENT OF PSYCHOLOGY, COLLEGE OF SOCIAL SCIENCES AND PHILOSOPHYUNIVERSITY OF THE PHILIPPINES, DILIMAN, QUEZON CITY 1101, PHILIPPINESDATE SUBMITTED: 26 AUGUST 2015DATE PERFORMED: 12 AUGUST 2015

I. ANSWERS TO QUESTIONS

1. Give the significance of Grubbs test.

The Grubbs test is used to detect a single outlier and used as a criterion to determine whether the suspected outlier is to be retained or taken out. The lowest or the highest observed value, or both, will be tested to determine if there is an outlier or there are outliers present. These outliers may result from human error, difference in composition of the material weighed or measured, loss of calibration of measuring instrument, etc. [1]

2. Give the significance of the mean and the standard deviation

In experiments, errors are unavoidable and it would be best if a scientist runs an experiment multiple times and average the results. The mean, or the average, can be acknowledged as an accurate value to the true value, although at times it may deviate if there is an outlier present. [2] While the standard deviation can be defined as the positive square root of the variance, that has the same unit of measurement as the observed values. The value of the standard deviation reflects on how close or spread the values are from the mean. If one data set has a smaller standard deviation compared to another data set, it can be interpreted that the observed values are closer to the mean. On the other hand, if a data set has a larger standard deviation than the other data set, it can be interpreted that the observed values are more distant from the mean. The value of standard deviation increases as the observed values become more dispersed and varied. [3] 3. Give the significance of the confidence interval.

A confidence interval is a range of values, obtained from a sample statistic, within which the value of an unknown population parameter is estimated to lie, with an associated degree of confidence. For example in the calculated confidence interval and confidence limits of data set 1, it can be interpreted that: there is a 0.95 probability that 3.5772 < < 3.6202. It can also be concluded that, the higher the value of the confidence level, the wider the confidence interval, and the lower the value of the confidence level, the narrower the confidence interval. [4]

4. How do the statistics calculated from data set 1 differ from those obtained from data set 2?

Data set 1 has a lower standard deviation and range compared to data set 2, which can be accounted for by the lower number of weights accounted for unlike data set 2. The range of the confidence interval, the difference between the upper and lower confidence limits, of data set 1 (0.043) is higher than data set 2 (0.042). Data set 1 was not able to properly represent the values of data set 2, and most likely the whole population, considering that it is a subset of data set 2. Therefore, it can be concluded that the average weight obtained from the 10 pieces of 25-centavo coin samples of data set 2 is more accurate and reliable than the average weight obtained from the six samples of data set 1.

II. DATATABLE I. Weight of all 25 centavo-coin samples used. Sample No.Weight, gDATA SET 2DATA SET 1

13.5833

23.6061

33.5922

43.6357

53.5788

63.5959

73.5322

83.6005

93.6209

103.6125

GRUBBS TEST

Data SetSuspected ValuesGtabGexpConclusion

1H: 3.63571.80241.8871Retained or Accepted at 95%

L: 3.57880.969411.8871Retained or Accepted at 95%

2H: 3.63571.41742.2900Retained or Accepted at 95%

L: 3.53222.25922.2900Retained or Accepted at 95%

REPORTED VALUES

Data SetsRSDRRRCL

13.59870.0205285.70430.056915.8 0.0215 = 3.5772-3.6202

23.59580.0281517.82860.103528.780.0201= 3.5757 3.6159

III. CALCULATIONS

1. Divided samples into data sets

TABLE II. Weight of samples in the Data Sets arranged from highest to lowest.Data Set 1Data Set 2

3.63573.6357

3.60613.6209

3.59593.6125

3.5922 3.6061

3.5833 3.6005

3.57883.5959

3.5922

3.5833

3.5788

3.5322

2. Grubbs Test

(1)If gcalc > gtab, then reject XsIf gcalc < gtab, then accept Xs

Data Set 1 Highest:Lowest:

= = = 1.802416212 1.8024= 0.969407638 0.96941gtab = 1.88711.8042 < 1.88710.96941 < 1.8871Retain/Accept at 95% Confidence LevelRetain/Accept at 95% Confidence Level

Data Set 2Highest:Lowest:

= == 1.417356399 1.41742.259244787 2.2592gtab= 2.29001.4174 < 2.29002.2592 < 2.2900Retain/Accept at 95% Confidence LevelRetain/Accept at 95% Confidence Level3. Statisticsa. Mean

(2)

Data Set 1Data Set 2

= = 3.598666667 3.5987= 3.59581 3.5958

b. Standard DeviationSample No. XiXi -(Xi -)2

13.5833-0.01540.00023716

23.60610.00740.00005476

33.5922-0.00650.00004225

43.63570.0370.001369

53.5788-0.01990.00039601

63.5959-0.00280.00000784

Total0.00210702

(3)

Data Set 1

= 0.020528127 0.020528

Sample No. XiXi -(Xi -)2

13.5833-0.01250.00015625

23.60610.01030.00010609

33.5922-0.00360.00001296

43.63570.03990.00159201

53.5788-0.0170.000289

63.59590.00010.00000001

73.5322-0.06360.00404496

83.60050.00470.00002209

93.62090.02510.00063001

103.61250.01670.00027889

Total0.00713227

Data Set 2

= 0.028150922 0.028151

c. Relative Standard Deviation

(4)

Data Set 1Data Set 2

= 5.7042822102 5.7043ppt= 7.828577785 7.8286ppt

d. RangeR = Xhighest Xlowest (5)

Data Set 1Data Set 2XHighest = 3.6357XHighest = 3.6357XLowest = 3.5788XLowest = 3.5322R = 3.6357 3.5788 = 0.0569R = 3.6357 3.5322 = 0.1035

e. Relative Range

(6)

Data Set 1Data Set 2

= 15.81126518 15.8ppt= 28.78358084 28.78ppt

f. Confidence Limits

(7)

Data Set 1Data Set 2

= 0.0215 =0.0201

REFERENCES

[1] Grubbs, F.E., 1974. Procedures for Detecting Outlying Observations in Samples. Maryland: Ballistic Research Laboratories. 10-13.

[2] Gilbert, T.R., et al., 2011. Chemistry: The Science in Context. New York: WW Norton & Company, Inc. 31-32.

[3] Almeda, J.V., et al., 2010. Elementary Statistics. Quezon City: UP Diliman. 236-239.

[4] Mode, E.B., Elements of Statistics. New Jersey: Prentice-Hall, Inc. 202-204.