Error and Statistics

You have already seen in our discussion of glassware that there are indicators of error on some measurement instruments. We are going to take these ideas further and discuss how to discuss error in mathematical terms. This will build on your knowledge from CHEM103 and our recent laboratory experiences.

Significant Figures

In Measurement

We follow the same algorithm for ALL measurements:

If you are measuring from a digital readout (such as an analytical balance), write down ALL the digits.
If you are reading from a piece of single-volume calibrated glassware (such as a volumetric pipet or volumetric flask), use the variance (the +/- value) to tell you how many zeros to record after the decimal. If you filled a 100 mL volumetric flask with a +/-0.08 variance exactly to the calibration line, you would record 100.00 mL in your notebook and note the variance in your observations.
If you are reading from a scale with numbers and lines:
1. Note the divisions in numbers on the scale
2. Note any lines on the scale
  1. Consider what number place these lines go to
  2. Think hundreds, tens, ones, tenths, etc
3. Read digits that you can get from the scale
4. Then, always estimate ONE MORE DIGIT
  1. This is our uncertain digit!
  2. Different people in your group may estimate the uncertain digit differently!

place value - Sheet1.pdf

Place Value Chart

You may not have seen one of these in a while, but here's a handy visual of how we look at place value of digits in a number before and after the decimal point.

Graduated Cylinder

This graduated cylinder has numbers every 1 mL and lines by the tenth of a milliliter. We will read from the bottom of the meniscus. We can say for sure that this graduated cylinder contains 5.8 mL of blue liquid, but we must estimate one digit beyond the scale. Since the meniscus falls right on the line, we say that this is 5.80 mL.

What Does "Uncertain Digit" Mean?

The uncertain digit is generally the last digit of a measurement. It is uncertain because different people may estimate it slightly differently when reading from a scale with numbers and lines. Take a look at the blue liquid in the graduated cylinder above. Do you agree with my measurement, or are you off by a hundredth of a milliliter?

On a digital balance, the uncertain digit (the last digit of the readout) may vary slightly as you record the value.

Counting Sig Figs for Math

We think about two types of numbers:

Exact numbers: we do not need to count their significant figures, because they will not change/limit significant figures in calculations.
1. Those that come from definitions
  1. Exactly 2.54 cm = 1 in
  2. 1000 m = 1 km
  3. 12 cupcakes in 1 dozen
2. Those that come from counting
  1. 5 fingers on 1 hand
  2. 485 spectators in the stadium
  3. 22 students in a class
Numbers that come from measurements: we need to know how many significant figures these numbers have when we do calculations with them. The significant figures in a measurement DIRECTLY tell our audience about the measurement instrument because the last digit is the uncertain, or estimated, digit.

The general rules for counting significant figures are:

All non-zero digits are significant.
1. 3.14 has three significant figures
2. 598.2 has four significant figures
Zeros can vary:
1. A leading zero, or any zeros that come BEFORE a non-zero digit, is not significant
  1. 0.00125 has three significant figures
  2. 0.27 has two significant figures
  3. 0.00000095 has two significant figures
  4. Consider: When we use scientific notation, the number of significant figures will not change!
2. A zero between two non-zero digits is significant.
  1. 404 has three significant figures
  2. 50.0034 has five significant figures
  3. 0.0708 has three significant figures
3. A trailing zero, or any zeros that come AFTER a non-zero digit, is only significant if there is a decimal
  1. 40.00 has four significant figures
  2. 500. has three significant figures
  3. 500.0 has four significant figures
  4. Consider: When we use scientific notation, the number of significant figures will not change!

Mathematical Operations

Addition and Subraction

When adding or subtracting two (or more) numbers, we round the final answer to the lowest decimal place of the starting numbers.

For example: 1.47 + 1.2 = 1.67 in a calculator, but we would round to 1.7.

Multiplication and Division

When multiplying or dividing two (or more) numbers, we round the final answer to the lowest number of significant digits of our starting numbers.

For example: 20.00 x 2.30 = 46.0 because 20.00 has four significant digits and 2.30 has three significant digits.

Logs and Antilogs

First, we need to define a term. The mantissa is the number to the right of a decimal point in a logarithm. We will base our sig figs on the mantissa. When taking a log, the final result's mantissa will have the same number of significant digits as the number of significant digits in the number of the log.

For example: log(12.43) = 1.094471129 in a calculator, but we would round it to 1.0945 for the four significant digits into the mantissa.

For example: antilog(-1.56) = 0.027542287 in a calculator, but we would round to 0.028 for to have two significant digits to match the mantissa.

Reporting Error

Mean (average)

When we take measurements in the lab, we often do what is called replicate analysis. We take multiple measurements or do multiple trials of a reaction or process to collect data. The first step in understanding your data is to understand the mean of your data points.

The mean will then be used in a number of additional operations to analyze your data.

Mean of Ungrouped Data

You have used this before in calculating the mean of a set of multiple measurements of the same property.

Mean of Grouped Data

This is similar to a weighted average you might use to calculate a grade in a course or to determine the average atomic mass of an element given abundance and mass of its isotopes.

Variance

Variance helps us understand a group of data points by calculating the average square of the difference between any given point and the mean.

Standard Deviation

Standard deviation then puts the variance into a single dimension. By taking the square root of the variance, the standard deviation shows us the average between any given point and the mean. This helps us better understand how far apart our data points are spread.

When we report standard deviation, we round it to the same decimal place as our mean. Use the usual rules for subtraction and division to determine the significant figures in your mean.

Percent Yield

We did percent yield calculations in CHEM103. Not every reaction will go to completion, nor is it possible to recover all of a product. Percent yield allows us to compare our theoretical yield (the amount we calculated based on starting materials and stoichiometry) to our actual yield (the amount we experimentally measured in lab after purification). This is useful when comparing synthesis results to literature values, when trying to optimize a process, or when trying new methods such as green chemistry.

Percent Error

When we compare an experimental value to an actual value, we often report percent error.

For example, if a student made and calibrated a spectroscope to determine an emission line for sodium at 593 nm, but the actual value is 589 nm, the student could calculate:

593 - 589 = 4

4/589 = 0.00679

0.00679 x 100% = 0.679%

Round to 0.7% due to the subtraction significant figure rule earlier in the calculation.

Statistical Analysis

Degrees of Freedom

p Value

Student's t-Test

References

OpenStax. Chemistry Atoms First 2e. "1.5 Measurement Uncertainty, Accuracy, and Precision." https://openstax.org/books/chemistry-atoms-first-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision (Accessed September 10, 2024)

LibreTexts. Chemistry. "Significant Figures" https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_(Analytical_Chemistry)/Quantifying_Nature/Significant_Digits/Significant_Figures (Accessed September 10, 2024)

Hartland, G. V. Statistical Analysis of Physical Chemistry Data: Errors Are Not Mistakes. The Journal of Physical Chemistry A 2020, 124 (11), 2109-2112. DOI: 10.1021/acs.jpca.0c01403.

OpenStax. Chemistry Atoms First 2e. "7.4 Reaction Yields." https://openstax.org/books/chemistry-atoms-first-2e/pages/7-4-reaction-yields (Accessed September 10, 2024)

University of Iowa Physics and Astronomy. Imagining the Universe. "Percent Error Formula" https://itu.physics.uiowa.edu/glossary/percent-error-formula (Accessed September 10, 2024)

Geeks for Geeks. Statistics Formulas. https://www.geeksforgeeks.org/statistics-formulas/ (Accessed September 10, 2024)

Statistics How To. "T Test (Student’s T-Test): Definition and Examples." https://www.statisticshowto.com/probability-and-statistics/t-test/ (Accessed September 10, 2024)

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