introduction to measurement error: part 1jse.amstat.org/v22n3/casleton/lecture-part1.pdfintroduction...

Introduction to Measurement Error: Part 1

Introduction to Measurement Error: Part 1 1 / 24

Introduction

Metrology–the science of measurement (Wikipedia)

I People get paid money to sit around and think about how to measurethings.

Measurement error–important subset of metrology

At the end of this material, you should know:I What is measurement error? . . . probably not as bad as you would

guessI When does measurement error occur? . . . more often than you thinkI Why does measurement error occur?I How does measurement error affect the experimentation process?I Why is measurement error important to consider? . . . why we should

pay those peopleI How to report measurement error? . . . depends on how many

measurements you have


http://en.wikipedia.org/wiki/Metrology

Motivating Example

You have a family history of high blood pressure, so yours is watchedclosely at every doctor’s visit. At one visit your blood pressure reads145/95, slightly over the 140/90 cutoff for Stage 1 Hypertension. Basedon this you should immediately be placed on expensive medicine, that yourhealth insurance does not cover, that once you start taking, you must takefor the rest of your life.

Reluctant to start you on this regime, your doctor proposes an experiment.You return to the doctor’s office every day the following week. Due toyour class schedule on MWF you go in at 8:00 am, but on TH you cannotmake it in until 3:00 pm. At each visit your blood pressure is measured 3times with 3 different sphygmomanometers. On two of the days you havethe same nurse, but on the other three you have a different nurse eachday. The results of the 15 measurements are on the following slide.


Motivating Example continued

Day Nurse Time Device 1 Device 2 Device 3

Monday Jackie 8:00 am 136/90 130/84 120/82Tuesday Paul 3:00 pm 146/96 134/80 128/84Wednesday Denise 8:00 am 136/86 130/80 110/74Thursday Jackie 3:00 pm 140/88 140/90 138/86Friday Amy 8:00 am 132/82 130/80 118/82

Would you take the medicine if. . .

your appointment was only on Thursday?

device 3 was the only used?

the only reading was on Tuesday usingdevice 1?

Why aren’t all the numbers the same?



Day Nurse Time Device 1 Device 2 Device 3Monday Jackie 8:00 am 136/90 130/84 120/82Tuesday Paul 3:00 pm 146/96 134/80 128/84Wednesday Denise 8:00 am 136/86 130/80 110/74Thursday Jackie 3:00 pm 140/88 140/90 138/86Friday Amy 8:00 am 132/82 130/80 118/82








Day Nurse Time Device 1 Device 2 Device 3

Monday Jackie 8:00 am 136/90 130/84 120/82Tuesday Paul 3:00 pm 146/96 134/80 128/84Wednesday Denise 8:00 am 136/86 130/80 110/74Thursday Jackie 3:00 pm 140/88 140/90 138/86Friday Amy 8:00 am 132/82 130/80 118/82







Definition: Variability

The degree to which a set of measured values fluctuates under fixedmeasurement conditions

How spread out or closely clustered a set of data isI Closely clustered → small variabilityI Spread out → large variability

NOTE: Cannot measure variability of only one data point

Example: Which data set has the larger variability: A or B?


Examples: Variability

1 How much variability does the following set of number(s) have?

1, 1, 1, 1, 1, 1, 1

2 How much variability does the following set of number(s) have?

1

3 Everyday for lunch Mary eats a peanut butter and grape jellysandwich and an apple. How much variability is there in Mary’s lunchdiet?

4 Which set of data would have more variability?1 Randomly select 100 students and find out how many drinks they had

last Saturday night.2 Randomly select 100 students and find out how many cars they own.


Definition: Measurand

Quantity for which one hopes to establish a true or correct value

Goal of data collection is to learn as much as possible about themeasurand

It is important to clearly define your measurand-and keep it consistentfor repeated measurements.

Examples:I Surface roughness of metal surfaceI Weight of wooden blockI Amount of fuel pumped into a carI Sodium content of a 6” turkey sub

on wheat bread with cheese, mayo,onion, lettuce and tomatoes


Examples: Measurand

Identify the measurand in each of thefollowing:

1 A researcher is interested in theweight, at birth, of a Komodo dragon

2 A lab worker needs to determine theamount of caffeine in a cup of coffee

3 A trainer is interested in the time ittakes a race horse to run one laparound a track


Definition: Measurement

The experimental assignment of a potential value to a measurand

Measurand and measurement are two distinct things:I Measurand: Weight of wooden blockI Measurement: 2.7 ounces

No measurement is meaningful without specifying meaningful units.Units are needed to explain what the number is representing.

I What is wrong with the following statement:The advisory (suggested) speed limit on the German Autobahn(interstate) is 130

km/hr


Definition: Measurement

The experimental assignment of a potential value to a measurand

Measurand and measurement are two distinct things:I Measurand: Weight of wooden blockI Measurement: 2.7 ounces

No measurement is meaningful without specifying meaningful units.Units are needed to explain what the number is representing.

I What is wrong with the following statement:

The advisory (suggested) speed limit on the German Autobahn(interstate) is 130 km/hr


Examples: Measurement

Suppose the measurand is the weight ofthis woman. What is the measurement?

Assume the scale reports weights inpounds.


Examples: Measurement

Suppose the measurand is the volume of the anti-freezein this graduated cylinder. What is the measurement?

Note the cylinder reports measurements in milliliters.


Example of meaningful units

Assume your friend wants to know thelength of a pencil. Which method do youthink would be best and why?

Place your thumb next to the penciland tell your friend how many“thumbs” the pencil is long

Place the pencil next to a ruler andtell your friend how many inches thepencil is long


Definition: device and operatordevice–the equipment and procedures used to produce themeasurementoperator–person reading the measurement from the deviceSometimes operator is considered part of the device. Here theoperator will be considered separately from the device.

Examples:I RulerI StopwatchI ScaleI Sphygmomanometer

and stethoscope


Examples of measurand, measurement & device

1 Measurand: The speed of a car.Measurement: 64.2 miles/hourPossible device: Laser speed gun

2 Measurand: The time of sunrise in Ames, IA on June 10Measurement: 05:39:01 amPossible device: Wall clock

3 Measurand: The caloric count of a marshmallowMeasurement: 60.45 caloriesPossible device: Calorimeter


Example of measurand, measurement & device

A quality assurance employee of Quaker has been asked to weigh thecontents of a randomly selected 18 ounce container of oatmeal to ensurethe filling machine is loading the correct amount of oatmeal into eachcontainer. He dumps the contents onto a digital scale which reads 17.94ounces.

Match the definition on the left with the appropriate description on theright.

1. Measurand A. Ounces2. Measurement B. Digital Scale3. Device C. Weight of oatmeal4. Units D. 17.94 ounces


What is measurement error? Single measurement definition

The amount by which each observed measurement differs from the“true, but unknown value”

“True value”- value that would be attained by a perfect measurement

The true value of the measurand is never known

Measurements are only approximations to the true value of themeasurand


Example

A quality assurance employee of Quaker has been asked to weigh thecontents of a randomly selected 18 ounce container of oatmeal to ensurethe filling machine is loading the correct amount of oatmeal into eachcontainer. He dumps the contents onto a digital scale which reads 17.94ounces.

Which statement do you most closely agree and why?

1 I think that the weight of the oatmeal is exactly 17.94 ounces.

2 I think that the weight of the oatmeal is approximately 17.94 ounces.

3 I dont agree with either.


Why does measurement error occur?

A measurement is the best possible information obtained from themeasurement device which has finite precision

Measurement error is different from a mistake by the researcher. It isan inevitable consequence of the fact that the true value of themeasurand is never known


When does measurement error occur?

ALWAYS! . . . anyone guess that?

In order to know the true value of a measurand, i.e. no measurementerror, we need an infinite amount of information!


Definition: reference standard

A realization of a specified quantity used as a measurement unit.

Typical reference standards are multiples of meaningful measurementunits, such as a kilogram, an inch, or a second.

National reference standards in United States are developed andmaintained by the National Institute of Standards and Technology(NIST) (www.nist.gov)

There are also many international organizations that work tocoordinate measurement standards across countries


www.nist.gov

Example Reference Standards for. . .

Time:I Second is defined to be the time required for 9,192,631,770 vibrations

of a cesium atom.I Cesium atom is unaffected by changes in temperature, humidity, and

air pressure.I NIST maintains an accurate atomic clock in Boulder, CO

(www.time.gov) and broadcasts the results, with some loss intransmission†.

Weight:I Kilogram is currently defined as the mass of one liter of water at 4◦CI Possible redefinition: Mass of 2.15× 1025 Silicon-28 atomsI YouTube video (11:44 in length) on the redefinition of the kilogram

† Moore, David S. and William I. Notz. Statistics: Concepts and Controversies. W.H. Freeman

and Company (2006)


www.time.gov

https://www.youtube.com/watch?v=ZMByI4s-D-Y

How important is measurement error?. . . depends

Would you need to use the NIST clock in the following? A group ofstudents are interested in the percentage of times CyRide is late tothe Sawyer Elementary School stop during the morning rush(8:00-11:00 am).

Imagine a measurement is off by 2 grams in each of the following.What are the consequences? Which is more serious?

I Measuring a cup of sugar for a cookie recipeI Measuring a company’s supply of anthrax

Imagine a measurement is off by 10 inches in each of the following.What are the consequences? Which is more serious?

I Production of a car doorI Measuring the route of a marathon (26.2 miles)


How to report a single measurement?Since measurements are not exact (they are approximations to the truevalue of the measurand), they should be reported as an interval, ratherthan a single value.

The stated rating/calibration for themeasurement device as set by themanufacturer or NIST

I Readings made with a set of calipers haveerror of plus and minus 0.005 inches

I Example: Calipers read the thickness of atire is 0.78. We report it to be between0.775 and 0.785 inches

When a rating is not specified, assume thelast digit of the measurement is estimatedand thus inaccurate

I Record measurements as plus and minus 1in the last digit

I Common practice in chemistry, physics, andother disciplines


Example of reporting measurement error

Assume that there is no manufacturer stated error for the measuringdevice. What is the appropriate way to report the followingmeasurements?

1 Balance indicates 56 pounds

We report the measurement is between 55 and 57 pounds

2 Ruler reads 12.8 feet

We report the measurement is between 12.7 and 12.9 feet

3 Scale reads 4.09 grams

We report the measurement is between and 4.10 grams

4 Thermometer reads 75.6 degrees

We report the measurement is between and degrees


introduction to measurement error: part 1jse.amstat.org/v22n3/casleton/lecture-part1.pdfintroduction...

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