# Measurement Errors

Learning Goals:

• To define the types of measurement errors that exist and ways to lessen them;
• To be able to choose appropriate glassware for solution preparation using error propagation methods

## Measurement Errors

All measurements have errors associated with them.  These errors fall into two categories:

1. Systematic errors (determinate errors) – affect the accuracy of the measurement, or the closeness of the result to the “true” value;
2. Random errors (indeterminate errors) – affect the precision of the measurements, or the closeness of the results to each other;

The goal of any analyses is to obtain accurate, precise data.  So how does one ensure that errors are kept to a minimum during the measurement process?

### Systematic Errors

The mean (or average) of a set of values is considered representative of those values when the values exhibit a Gaussian distribution (bell-shaped curve).  In this special case, the mean value is considered the “true” value for the set of measurements.  Systematic errors are characterized by measurements that are consistently higher than the true value (positive systematic bias) or lower than the true value (negative systematic bias).  Systematic errors fall into four categories:

1. Sampling errors arise when a collected sample does not represent the environment being sampled.
2. Method Errors result from not being able to accurately measure the sample’s concentration or mass. This usually occurs because another substance present in the sample matrix is interfering in the measurement process.
3. Instrument Errors involve using the proper glassware/equipment for analyses and knowing the errors involved in its use;
4. Personal Errors are also known as human errors and arise negligence like over-titrating an endpoint or using a dirty cuvette for analysis.

Ways to minimize systematic errors

1. Collect representative samples
2. Analyze standard reference materials whose concentrations are known
3. Analyze blank samples
4. Use multiple methods to make measurements
5. Participate in a round-robin study with other labs
6. Vary the sample size

### Random Errors

Random errors are always present during any measurement and can’t be controlled for.  They arise from minor differences in sampling between different people, from how people read a buret or interpret an endpoint color, and even from electrical noise.  The contribution of random error to a measurement can be estimated through error propagation methods, by measuring the standard deviation of a series of measurements that show a normal distribution, and by monitoring the behavior of a piece of equipment over time.  In general, the magnitude of systematic errors greatly exceeds those of random errors, so that is where analysts typically focus their attention to minimize overall error in a process.

## Propagation of Error Using Glassware

Note

If  (A ± a) + (B ± b) = S ± s   then:  S =  A + B       and    $$s=\sqrt{a^2+b^2}$$

If  (A ± a) – (B ± b) = S ± s    then:  S =  A - B        and   $$s=\sqrt{a^2+b^2}$$

Rules for Multiplication and Division

If (A ± a) (B ± b) = P ± p     then:    P = (A)(B)        and     $$p=(A)(B)\sqrt{\left(\dfrac{a}{A}\right)^2+\left(\dfrac{b}{B}\right)^2}$$

If (A ± a) / (B ± b) = P ± p      then:    P = A/B           and      $$p=(A)/(B)\sqrt{\left(\dfrac{a}{A}\right)^2+\left(\dfrac{b}{B}\right)^2}$$

Refer to the Table of Glassware Tolerances at the end to answer these questions

1) Assuming that your goal is to minimize your uncertainty should you measure out 25 mL of your solvent using a class A volumetric flask or graduated cylinder?

2) Assuming that your goal is to minimize your uncertainty should you measure out 5 mL of your solvent using a class A volumetric flask or a class A transfer pipette?

3) You need 23 mL of solvent.  Should you use your graduated cylinder or a 20 and 3 mL transfer pipette?

4) I broke my 3-mL transfer pipette so I decided to use my 25-mL transfer pipette but then remove 2 mL using another transfer pipette.  What is my final uncertainty?

5) I also broke my 5-mL transfer pipette (It’s been a tough day in lab!) and I need five mL of my reagent.  Should I use my 1 mL pipette five times or should I use my graduated cylinder?

6) I need ~2 g of a solid reagent for my experiment.  I tare the balance (BIG HINT HERE!!!) and measure 1.9762 g.  How do I report the mass with uncertainty?

7) If the formula weight of the solid in the problem above is 382.981 g/mol, how many moles do I have? (You may treat molecular weights reported to three decimals as exact values, i.e. a value not having an associated uncertainty.)

8) I make a solution from the solid reagent from the problem above added to a 500-mL volumetric flask and brought to volume.  What is the molarity of the solution?

9) Which of the following is the best way to dispense 100.0 mL of a reagent: (a) use a 50‑mL pipet twice; (b) use a 25-mL pipet four times; or (c) use a 10-mL pipet ten times?

Table of Glassware Tolerances

Information in Table from Sigma-Aldrich.com

Capacity (mL) ≤

Graduated Cylinder, Class A, tc, (±mL)

Volumetric Flasks, Class A, tc, (±mL)

Burets, Class A, (±mL)

Pipets, Transfer, Class A, (±mL)

0.1

0.2

1

0.010

0.006

2

0.015

0.006

3

0.015

0.01

4

0.020

0.01

5

0.05

0.02

0.01

0.01

10

0.08

0.02

0.02

0.02

15

0.03

20

0.03

25

0.14

0.03

0.03

0.03

50

0.20

0.05

0.05

0.05

100

0.35

0.08

0.10

0.08

200

0.65

0.10

0.10

250

0.65

0.12

500

1.1

0.15

1000

2.0

0.30

2000

3.5

0.51

4000

6.5

Analytical Balance Uncertainty: 0.0002 g for tare and measurement, capacity 120 g

Toploader Balance Uncertainty: 0.02 g for tare and measurement, capacity 4200 g

• Amy Witter, Dickinson College (witter@dickinson.edu)
• Modification of “Glassware-uncertainty propagation.dox” uploaded by Dabrina Dutcher on June 13th, 2017 under #early-stage-materials