1 Prepared for SSAC by Bill Thomas Colby-Sawyer College, New London NH © The Washington Center for Improving the Quality of Undergraduate Education. All.

1

Prepared for SSAC byBill ThomasColby-Sawyer College, New London NH© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2005

SSAC2005:QD451.WT1.1

• How accurate is your pipetting?

• What are the sources of error? • What does it mean to calibrate a

piece of equipment, and how do you do it?

Calibrating a Pipettor

Core Quantitative IssueVariability: Precision vs. accuracy

Supporting Quantitative ConceptsData Analysis: Mean, standard deviationVisualizing data: bar and scatter graphsRelative error; percent errorSize: Mass vs. volume

2

Because error is so common in measurement, we have developed general

approaches, first to minimize error, and then to assess how much remains.

Slide 3 introduces the concept of equipment calibration. Slide 4 explains mass-to-volume conversion. Slide 5 discusses accuracy and reproducibility; standard deviation; and percent error. All are tools that help us measure the quality of what we think we know.Slides 6-8 spell out the problem and the approach. Slides 9-12 develop spreadsheets to treat model data. Slides 13-15 consider accuracy and reproducibility. Slide 16 asks you to think about your work in a new context.

Every measurement, no matter how carefully done, has some error associated with it. Is the

error great or small? Knowing the answer is important, for it tells us

how much confidence we can have in the values obtained, and that knowledge shapes decisions

that we make. Every measurement can be done more carefully, but to do so, we have to know the source of the

error. Is it due more to technique or to limitations of the equipment? The answer helps us to improve

our results. It would be ineffective to concentrate upon technique if the problem were really a faulty

tool!

Overview of Module

3

How do you know that your equipment is functioning correctly?

Typically we use the equipment in question to measure something whose attributes (size, volume, etc.) we can determine independently with considerable accuracy, and

then we adjust our equipment to yield the value that we “know” to be correct. This process is known as “calibrating” the equipment.

In this case, you are calibrating a micropipettor by determining the mean

(average) mass of multiple aliquots (volume samples) using an analytical balance, then converting that mass to a volume using the density () of water, which is a known value. You will compare the calculated volume with the delivery volume preset on the pipettor to

determine the accuracy of the pipettor.

Mean =

Sum of sample valuesNumber of samples

Equipment Calibration

4

You use the formula for density to determine volume delivered from mass measured.

Density ():

Density is simply mass per unit volume (g/cm3 or

g/mL) as expressed in the following formula:

= m/V

Temperature (ºC)

Density (g/mL)

15 0.999103

16 0.998946

17 0.998778

18 0.998599

19 0.998408

20 0.998207

21 0.997996

22 0.997774

23 0.997542

24 0.997300

25 0.997048

26 0.996787

27 0.996516

28 0 996237

Density of Water at 1 atmosphere

Mass to Volume Conversion

(from: Handbook of Chemistry and Physics, CRC press, 64th Ed.)

5

Accuracy measures how close a measured value comes to a

predetermined target value (the set volume on your pipettor).

Reproducibility (precision) measures how close repeated

values are to one another. These concepts can be visualized using

these cartoon (idealized) bulls-eye diagrams. Notice that accuracy

and precision can vary independently, so they can be

evaluated independently, as well.

not accurate precise

accuratenot precise

accurate precise

not accurate not precise

Accuracy vs. Reproducibility

6

You have just been appointed Quality Control Officer in a company responsible for critical medical assays. The company has recently been having difficulty with consistency in its medical tests, and it is your task to get to the bottom of the problem as soon as possible. People’s lives hang in the balance, and mistakes could have serious legal repercussions for the company.

Pipettors are a mainstay of the laboratory work, and as they have not been calibrated in a while, you suspect that they might be a source of the variability. On the other hand, there are some new employees in the company, and their pipetting technique might be part of the problem.

You decide to have these new employees carry out a standard calibration of the lab pipettes. Based on the outcome of these tests, you should be able to determine the source of the error.

•How can you evaluate the error in the pipetting? •Is it due to poor technique, a faulty pipettor, or some of both? In the next few slides you will see how an Excel spreadsheet and graphs can help you to answer these questions. You will be asked to make the determination in Question 6 of the end of module assignments.

PROBLEM

7

1. Choose a pipettor and arbitrarily select a volume that lies in the middle of the pipette’s range to deliver (Why in the middle?). Be sure that you have the appropriate tip for the pipette.

2. After taring the analytical balance with a plastic or aluminum “boat” on the pan of the balance, deliver in quick succession 10 aliquots of water to the boat, recording the increasing cumulative weight of water on the balance after each addition. Remember to record the ambient temperature for the density calculations.

3. Use a spreadsheet to evaluate the accuracy and reproducibility of the pipetting. In particular, what are the mean, standard deviation and percent error of the data sets?

Strategy

Click here for definitions of the words in bold, if needed.

8

In any cell you can create almost any equation that you need:

OR………..

= C3+C4

1. Before treating your own data, you will work with four sets of sample data to help you set up the Excel spreadsheet and “learn the ropes”. The first of these four sets of data will be in the model spreadsheet (Cells C3 – C12) on Slide 10.

2. The model spreadsheet provides all the values that you need. Your job is to create the equations that produce the given values in each cell. Once you have these equations in place, you can easily treat any new data set or any new conditions by simple substitution in the appropriate cells (as for the remaining three data sets).

= value is given, or chosen.

= calculate; use cell equation.

Note: Cells are color-coded according to function

Strategy (cont.)

9

Excel can provide you with equations (functions)

that you can use.

Using equations of your own construction or those provided by Excel, follow the steps in the following slide to calculate the mean and standard deviation for the data set given.

Strategy (cont.)

10

Cell C2 contains the preset volume for the pipette.

Block C4:C13 contains the measured weights.

Block D4:D13 contains equations to calculate the weight of each aliquot. Hint: use the copy and

paste commands where you can.

Block D15:D17 contains equations to calculate the mean by summing the aliquot weights and dividing by the number of them. Hint: for N, you can use

Excel’s COUNT function.

Recreate this spreadsheet in an Excel file of your own. Insert the equations necessary to calculate the values in the orange

boxes using the given values in the yellow boxes.

Creating a Spreadsheet

Cell C20 calculates the mean using Excel’s built-in AVERAGE function.

B C D2 Set Volume 6 ul

3Sample Number

Cumulative Weight (mg)

Aliquot weight (mg)

4 1 6.001 6.0015 2 12.003 6.0026 3 18.001 5.9987 4 23.998 5.9978 5 29.999 6.0019 6 35.998 5.999

10 7 41.996 5.99811 8 47.996 6.00012 9 53.997 6.00113 10 59.996 5.9991415 Sum= 59.99616 N= 1017 Mean= 5.9996181920 Mean= 5.999621

11

Block E4:E13 contains equations that subtract the mean from the respective

aliquot weights.

Block F4:F13 contains equations that square those

differences.

Block F15:F18 calculates the variance and standard deviation

of the aliquot weights. The variance is the sum of the

differences divided by COUNT-1, and the standard deviation is the

square root of the variance.

Cell F20 calculates the standard deviation using Excel’s built-in

STDEV function.

Creating a Spreadsheet (cont…)

Complete the spreadsheet. Add Columns E and F

B C D E F2 Set Volume 6 ul

3Sample Number

Cumulative Weight (mg)

Aliquot weight (mg)

Difference (D ) from mean D 2

4 1 6.001 6.001 0.0014 1.96E-065 2 12.003 6.002 0.0024 5.76E-066 3 18.001 5.998 -0.0016 2.56E-067 4 23.998 5.997 -0.0026 6.76E-068 5 29.999 6.001 0.0014 1.96E-069 6 35.998 5.999 -0.0006 3.60E-07

10 7 41.996 5.998 -0.0016 2.56E-0611 8 47.996 6.000 0.0004 1.60E-0712 9 53.997 6.001 0.0014 1.96E-0613 10 59.996 5.999 -0.0006 3.60E-071415 Sum= 59.996 Sum = 2.44E-0516 N= 10 N -1= 917 Mean= 5.9996 variance= 2.71E-0618 std dev= 0.0016471920 Mean= 5.9996 Std Dev= 0.00164721

12

Build a spreadsheet that calculates the mean and

standard deviations of these three additional data sets

together with the data set of the previous slides. Copy and

Paste Columns B, C, and D from your spreadsheet of Slide 11,

and Copy and Paste Columns C and D three times to make

space for the new data. Then insert the new values in Columns E, G, and J.

= value is given, or chosen.

= calculate; use cell equation.

Remember:

Using what you have learned

Sample Data

Data set 2 Cumulative

Wt (mg)

Data set 3 Cumulative

Wt (mg)

Data set 4 Cumulative

Wt (mg)6.501 6.1 6.613.003 12.3 13.319.501 18.1 19.625.998 23.8 25.832.499 29.9 32.438.998 35.8 38.845.496 41.6 45.151.996 47.6 51.658.497 53.7 58.264.996 59.996 64.6

We will use the spreadsheet in the

rest of the module to compare the four calibration runs.

B C D E F G H I J2 Set Volume 6 ul

3Sample Number

Data set 1 Cumulative

Wt (mg)

Data set 1 Aliquot Wt

(mg)

Data set 2 Cumulative

Wt (mg)

Data set 2 Aliquot Wt

(mg)

Data set 3 Cumulative

Wt (mg)

Data set 3 Aliquot Wt

(mg)

Data set 4 Cumulative

Wt (mg)

Data set 4 Aliquot Wt

(mg)4 1 6.001 6.001 6.501 6.1 6.65 2 12.003 6.002 13.003 12.3 13.36 3 18.001 5.998 19.501 18.1 19.67 4 23.998 5.997 25.998 23.8 25.88 5 29.999 6.001 32.499 29.9 32.49 6 35.998 5.999 38.998 35.8 38.8

10 7 41.996 5.998 45.496 41.6 45.111 8 47.996 6.000 51.996 47.6 51.612 9 53.997 6.001 58.497 53.7 58.213 10 59.996 5.999 64.996 59.996 64.9961415 Mean 5.999616 Std Dev 0.0016465

13

Right! For these calculations, use a density of 1 g/mL for water.

For your lab data (later), you will use a density that will depend on the temperature at which you did

the work. So, be prepared to add a row to your spreadsheet to convert the mass of the aliquot to volume.

•Organize the results of your calculations so you can easily compare them. Compare the statistics to the data themselves (Slide 12).•What do you notice? (You should notice something!)

Looking at your results

“But I have weights, not volumes. I still don’t know anything about the

accuracy of the pipettes!”

Remember, you will be using a spreadsheet like this to calculate several sets of values, so be sure you understand all of the steps.

B C D2 Results3 Data set Mean Std Dev4 15 26 37 4

14

Look at the four means from Slide 13 and determine which of the results are the most accurate (closest to the preset aliquot volume). Which are least accurate?

To create a bar graph, highlight your spreadsheet and then enter Chart Wizard in the menu bar of Excel to create your graph.

2. Use a spreadsheet to create a bar graph to visualize the difference in accuracy amongst the four data sets.

How can you compare the accuracies?1. Calculate the relative error for the sample data (use your spreadsheet). (Relative error is the magnitude of the difference between the measured and set values divided by the set value; it can be expressed as a per cent).

Evaluating the results: Accuracy

The data sets are so different that you will need two graphs with different vertical scales.

0

1

2

3

4

5

6

7

8

9

DataSet 1

DataSet 2

DataSet 3

DataSet 4

Rel

ativ

e E

rro

r (%

)0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

DataSet 1

DataSet 2

DataSet 3

DataSet 4

Rela

tive E

rro

r (%

)

15

Look again at the four data sets (your spreadsheet from Slide 12). Which data seem to vary most about the mean. Which vary the least? How can you visualize the variability?

Standard deviation provides a more compressed way to visualize the variability. Note, however, that standard deviation has limited validity for small data

sets (< 5 individual samples).

Use your spreadsheet from Slide 12 to make an XY scatter plot like this one.

Evaluating the results: Reproducibility

How do the two data sets marked by blue symbols differ

from the two marked by red symbols? What do the two

data sets marked by the circles and squares have in common? What do the two

data sets marked by the triangles and diamonds have in common? How do the four data sets match up with the

bulls-eyes on Slide 5?

5.6

5.8

6.0

6.2

6.4

6.6

6.8

0 1 2 3 4 5 6 7 8 9 10 11

Sample Number

Ali

qu

ot

Mas

s (m

g)

16

Look at the four standard deviations that you have calculated and use them to determine which of the four sets of results are the most precise (most clustered about the mean). Which are least precise? How does your assessment here compare with that based on the graph in the previous slide?

How can you visualize the standard deviations? Use your spreadsheet from Slide 13 to draw an X,Y scatter plot showing the means with error bars that correspond to the standard deviations. Create the graph by first plotting the means against the number of the data set. Then double click on a data point and specify the appropriate standard deviation. Click here for help on adding the error bars.

Evaluating the results: Reproducibility

The standard deviations for Data Sets 1 and 2 plot within the icon

locating the mean

5.75

6.00

6.25

6.50

6.75

0 1 2 3 4 5

Data Set

Aliq

uo

t M

ass

(mg

)

17

• Sketch the bulls-eyes of Slide 5 and label them with the number of the data set (Use your spreadsheet of Slide 12 and consult the graph in Slide 15.)

• What is the utility of a mean when dealing with data sets? Limitations?

• What is the utility of the standard deviation? What does it show?

• What is the utility of a spreadsheet for solving computational problems?

• What is the utility of graphs in general? In particular, what is the utility of an XY (scatter) plot as opposed to a bar graph.

• In your capacity as Quality Control Officer (see Slide 6), how can you determine whether the recent variability in your lab’s results is due more to poor pipetting (operator problem) or faulty equipment (instrument problem)?

• Now that you have your spreadsheet(s) fully formatted, use it (them) to carry out a similar analysis of your own lab data.

• A last reflection: Can you see situations in daily life where error/variability/uncertainty can be considered in similar ways and where such a consideration might affect your personal decisions? Justify your points with specific examples.

End of Module Assignments

18

Definitions

Mean:sum of all values total # of values

Standard Deviation

1

2

n

xxn

o i

Percent Error: (measured – set) x 100 set

• taring – this is a procedure used by scientists to factor out the weight of a container (or “boat”) when weighing a substance. The scientist measures the weight of the boat on a balance, then sets the balance equal to zero so that any subsequent measurements will not include the weight of the boat.

• aliquot – a equal fractional part of a whole. For example, if there are 10 aliquots of a mass of water, then each aliquot will be 1/10 the mass of the original mass.

• cumulative – an increase in some value resulting from successive additions• ambient temperature – the temperature of the surrounding environment • mean – the average value (see the equation below)• standard deviation – roughly the average distance between the mean of a

set of numbers and the individual values in that set (see the equation below)• percent error – the difference between the measured and the set values as

a fraction of the set value times 100 (see the equation below)

BACK

19

How to add error bars

• Once you create your x-y scatter plot, click once on any of the data points. This should highlight them all.

• Under the “Format” menu, select “Selected data series…”

• Select the “Y Error Bars” tab• Under the “Display” menu, highlight “Both”• Under the “Error amount” menu, select “Custom”

(ignore any values in the boxes above it). This will allow you to select your standard deviations as the plus and minus amounts.• Click on the graph symbol at the right end of the

“+” text bar. This will temporarily close the “Format Data Series” activity box, but leave open the text bar.

• Highlight the four cells with the standard deviation (beginning with the first and dragging down to the 4th…it is important that you highlight them in order so Excel can apply them in order to your graph).

• Click on the icon at the right end of the text bar (there should now be text within the bar). This will maximize the “Format Data Series” again.

• Because your standard deviation values applies to the plus and minus portion of the error bar, you will repeat this process using the “-” text bar.

• When you’re done, click “Okay”.BACK

20

4. This spreadsheet shows the cumulative weights of ten successive samples of powder added one by one to a weighing pan. The orange cells are intended to show the weights of the individual samples, the mean weight, and the standard deviation of the sample weights. To complete the spreadsheet, what cell equations do you need to place in (a) Cell D10, (b) Cell D14, and (c) D15?

5. Explain how you would Cut and Paste to simplify putting the equations into Cells D3 through D12.

1. What does it mean to calibrate a piece of laboratory equipment?

2. Distinguish between accuracy and precision by describing the difference between an inaccurate and an imprecise piece of laboratory equipment.

3. What does a standard deviation measure?

Pre-Test

B C D

2 SampleCumulative Wt (g)

Weight of sample (g)

3 1 0.5014 2 1.0035 3 1.4986 4 1.9987 5 2.498 6 2.9889 7 3.495

10 8 3.99611 9 4.48612 10 4.9911314 Mean

15Standard Deviation