Mathematical Exercises Name:_________________

Do not round answers.

Show your work!

The problems below will be covered during the Balance Calibration Seminar and are Recommended to 

be completed prior to the seminar. 

If you wish feedback on your preparedness for these topics submit the problems in either hand written 

or MSExcel workbook (any version) form. MSExcel is preferred. Title the workbook file:{Your 

name} Balance Math Exercises Recommended {Date}.

Return the completed exercises by e-mail with the subject line 

“Balance Math Exercises Recommended” to aid in tracking your message.

Balance Math Exercises Rev 20150923 Recommended.doc 1 of 5 September 23, 2015

Metrology-Related Examples

Given the following values:

100, 100.5, 100.4, 99.9, 99.8, 100.1, 99.6, 99.9, 100.3, 100.2,

Calculate

R1. Mean (

x

): ___________________

R2. Range: ___________________

R3. Standard Deviation: ___________________ 

R4. Average: ___________________ 

 Identify your choice of method for computing the average. Circle the one that applies: 

Mean, Mode, Median, Other (explain)___________________________
The problems below will be covered during the Balance Calibration Seminar and are 

Recommended to be completed prior to the seminar. 

If you wish feedback on your preparedness for these topics submit the problems in either hand 

written or MSExcel workbook (any version) form. MSExcel is preferred. Title the workbook 

file:{Your name} Balance Math Exercises Recommended {Date}.

Return the completed exercises by e-mail with the subject line 

“Balance Math Exercises Recommended” to aid in tracking your message.

Balance Math Exercises Rev 20150923 Recommended.doc 2 of 5 September 23, 2015

R5: Given:

Old mean (

x0

) (nO=10): 99.5

New mean (

xn

) (nN=15): 100.07

Old standard deviation (

O

s

): 0.09954

New standard deviation (

N

s

): 0.283039

Calculate the value of the t-test using: 

N

N

O

O

n

n

s

n

s

x x

t

2 2

0







t =______________________________

R6:Given:

New value (xi): 100.5

Reference value (μref): 100.2

New uncertainty (Ui) (k=2): 0.2

Reference value uncertainty (Uref) (k=2): 0.16

Find the E-normal value for the data using the equation: 

 

 

2 2

i ref

i ref

n

U U

x μ

E







.

En =______________________________
The problems below will be covered during the Balance Calibration Seminar and are 

Recommended to be completed prior to the seminar. 

If you wish feedback on your preparedness for these topics submit the problems in either hand 

written or MSExcel workbook (any version) form. MSExcel is preferred. Title the workbook 

file:{Your name} Balance Math Exercises Recommended {Date}.

Return the completed exercises by e-mail with the subject line 

“Balance Math Exercises Recommended” to aid in tracking your message.

Balance Math Exercises Rev 20150923 Recommended.doc 3 of 5 September 23, 2015

R7:Given:

Old standard deviation (

O

s

): 0.09954

New standard deviation (

N

s

): 0.283039

Calculate the F value using the equation: 

 

 

2

2

n

O

s

s

F 

F =______________________________

R8:Given:

Tolerance: 2.3 g

Measurement uncertainty (

Uk2

) : 150 mg

Calculate the Pn value using the equation: 

Tolerance

U

P

k

n

3

1

2



Pn =______________________________
The problems below will be covered during the Balance Calibration Seminar and are 

Recommended to be completed prior to the seminar. 

If you wish feedback on your preparedness for these topics submit the problems in either hand 

written or MSExcel workbook (any version) form. MSExcel is preferred. Title the workbook 

file:{Your name} Balance Math Exercises Recommended {Date}.

Return the completed exercises by e-mail with the subject line 

“Balance Math Exercises Recommended” to aid in tracking your message.

Balance Math Exercises Rev 20150923 Recommended.doc 4 of 5 September 23, 2015

R9:Given:

Uncertainty of standard (Us k=2): 0.01 mg

Standard deviation of process (sp): 0.05 mg

Calculate the expanded uncertainty (U) using the equation: 

 

2

2

2

2 P

s

s

U

U  













U =______________________________
The problems below will be covered during the Balance Calibration Seminar and are 

Recommended to be completed prior to the seminar. 

If you wish feedback on your preparedness for these topics submit the problems in either hand 

written or MSExcel workbook (any version) form. MSExcel is preferred. Title the workbook 

file:{Your name} Balance Math Exercises Recommended {Date}.

Return the completed exercises by e-mail with the subject line 

“Balance Math Exercises Recommended” to aid in tracking your message.

Balance Math Exercises Rev 20150923 Recommended.doc 5 of 5 September 23, 2015

Units, Conversions, and Related Problems

Use units instructions from NIST Special Publication 811 (attached).

R10: Select the correct form of the abbreviation for the unit: grams

gms g gm gns gr G

R11: Select the correct form of the abbreviation for the unit: micro-grams

mg μg Mg mgr mgrs mcgs.

R12: Select the correct form of the abbreviation for the unit: kilograms

KG Kg Kgr kg KGs kgs

R13: Select the correct form of the abbreviation for the unit: liters

L ls Ls ltr ltrs l

R14: Select the correct form of the abbreviation for the unit: milligrams

Mgs Mg mG μg μgs mg

R15: Select the correct form of the abbreviation for the unit: milliliters

μL μl mL ml. cc ml

R16: Select the acceptable forms for writing the value: 1⁄4

.25 0.25 1⁄4

Rounding values

Rounding instructions can be found in NISTIR 6969 GLP 9 found at

http://www.nist.gov/pml/wmd/labmetrology/upload/GLP_9_20140911.pdf

R17: Round the value 4.9459 g to two significant digits: ________________ g

R18: Round the value 0.2 kg to two significant digits: ________________ kg

R19: Round the value 1059 mg to two significant digits: ________________ mg

Time taken to complete these RECOMMENDED math exercises: ___________ minutes