Analytical Chemistry ISCIT 1543

Statistical Evaluation of Analytical Data

Learning Objectives

Basic Treatment of Data  Video Lecture

Uses of Statistics .

Statistics is a valuable tool in the analytical laboratory.  Here are some of the major uses of statistics.

• PROBABILITY OF FINDING THE POPULATION MEAN
• NUMBER OF REPLICATE MEASUREMENTS REQUIRED
• DEALING WITH OUTLIERS
• SIGNIFICANT DIFFERENCE IN A SAMPLE ANALYZED BY DIFFERENT METHODS:    Was the difference due to indeterminate error or was it a real difference?
• DIFFERENCE IN PRECISION
• DETECTION LIMITS
• TREATING CALIBRATION DATA

Reliability of Data

The reliability of data is indicated through proper use of significant figures, rounding and the confidence interval.

Significant Figures

One "simple" method of indicating the probable uncertainty in an experimental measurement is to round the result to that it contains only significant figures.  By definition, significant figures are all of the digits that are "certain" and the first unknown digit.  Care must be taken to read and record measurements to the level of precision indicated by the instrument.

When performing calculations, care must be taken to determine the appropriate number of significant figures needed in the final result must be determined and result rounded.

Round the result to the same number of decimal places as the number with the smallest number of decimal places.

• Multiplication and Division

Round the result to the number of significant figures as the number with the smallest number of significant figures.

• Logarithms and Antilogarithms

In a logarithm of a number, keep as many digits to the right of the decimal point as there are significant figures in the original number.

In an antilogarithm of a number, keep as many digits as there are digits to the right of the decimal point in the original number.

Rounding Data

Postpone rounding until all calculations are completed. At least one digit beyond the significant digits should be carried through all of the calculations to avoid rounding error. Some prefer to use all digits available during these calculations. These extra digits are called "guard" digits.

The standard rounding rule (The Odd-Even Rule) for numbers ending in "5" is to round to an even number.    For example, 61.555 rounded to four significant figures would be 61.56. On the other hand, 23.465 rounded to four significant figures would be 23.46.

Some organizations, such as the US Phamacopea (USP), recommend rounding all "5’s" up to the next number. Using the previous examples, 61.555 would be rounded to 61.56, and 23.465 would be rounded to 23.47.

Use the rounding guidelines of your laboratory or organization.

Confidence Interval

The confidence interval is dependent on the standard deviation. It gives the probability that the true value will be within the defined limits.

The value, t, is called the Students’ t Value. It depends on the desired confidence value and the number of degrees of freedom in calculating the standard deviation, s.

 Student's t Distribution (t Values for Calculating Confidence Limits) Confidence Level (%) Degrees of Freedom (N-1) t 50% t 80% t 90% t 95% t 99% 1 1.000 3.078 6.314 12.706 63.657 2 0.816 1.886 2.920 4.303 9.925 3 0.765 1.638 2.353 3.182 5.841 4 0.741 1.533 2.132 2.776 4.604 5 0.727 1.476 2.015 2.571 4.032 6 0.718 1.440 1.943 2.447 3.707 7 0.711 1.415 1.895 2.365 3.500 8 0.706 1.397 1.860 2.306 3.355 9 0.703 1.383 1.833 2.262 3.250 10 0.700 1.372 1.812 2.228 3.169 15 0.691 1.341 1.753 2.131 2.947 20 0.687 1.325 1.725 2.086 2.845 ¥ 0.674 1.282 1.645 1.960 2.576

Detecting Outliers

Outliers are the result of gross errors.  When a set of data contains an outlier, a decision must be made as to retaining or rejecting the questionable data.  This decision must be done very carefully.  There is no universal statistical test or rule that can be used to definitively answer this question.  The ONLY valid reason for rejecting data is that there is clear and documented evidence that a mistake was made in making the measurement.   Common sense and taking the time to repeat a questionable experiment are usually more reliable than any statistical test!

Should a small set of data contain a questionable or suspect value, follow  these recommendations.

• Re-examine all data to see if a gross error may have affected the value. This requires keeping a carefully maintained and documented notebook.
• If possible, estimate the precision that can be reasonably expected from the procedure to be sure that the result is actually questionable.
• Repeat the analysis if you have enough time and sample.
• Appy one of the statistical tests (e.g. the Q Test) to see if the value should be retained or rejected.
• If the statistical test indicates retention, consider reporting the median rather than the mean. The median allows the inclusion of all of the data set without much influence from the outlier.

Statistical Tests for Outliers - The Q Test

The Q Test is a simple statistical test for outliers.   It is fairly stringent and is not particularly helpful for small sets of data (N<5).

If Q Calc < Q Crit , then Xq may be an outlier.

 Critical Values for the Rejection Quotient, Qcrit Number of Observations 90% Confidence 95% Confidence 99% Confidence 3 0.941 0.970 0.994 4 0.765 0.829 0.926 5 0.642 0.710 0.821 6 0.560 0.625 0.740 7 0.507 0.568 0.680 8 0.468 0.526 0.634 9 0.437 0.493 0.598 10 0.412 0.466 0.568