Measurements in Statistics

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Transcript Measurements in Statistics

Measurements in Statistics
Chapter 2
2.1 Data Types and
Levels of Measurement
The goal is to transform data into information,
and information into insight.”
Carly Fiorina (Executive and president of Hewlett-Packard Co. in
1999. Chairwoman in 2000)
Be able to identify data as qualitative or quantitative,
to identify quantitative data as discrete or continuous,
and to assign data a level of measurement (nominal,
ordinal, interval, or ratio).
Data Types
Qualitative (or categorical) data consist of
values that can be separated into different
categories that are distinguished by some
nonnumeric characteristic.
Quantitative data consist of values
representing counts or measurements.
Determine whether the data described are qualitative or
quantitative and explain why.
A person’s social security number
 The number of textbooks owned by a
 The incomes of college graduates
 The gender of college graduates
Types of Quantitative Data
Continuous data can take on any value in a
given interval. Continuous data values
results from some continuous scale that
covers a range of values without gaps,
interruptions, or jumps.
Discrete data can take on only particular
distinct values and not other values in
between. The values in discrete data is
either a finite number or a countable number.
State whether the actual data are discrete or
continuous and explain why.
The number of 1916 dimes still in
 The voltage of electricity in a power line
 The number of eggs that hens lay
 The time it takes for a student to
complete a test
Levels of Measurement
Nominal and ordinal are qualitative
(categorical) levels of measurement.
Interval and ratio are quantitative levels of
Nominal level of measurement—classifies data into
names, labels or categories in which no order or
ranking can be imposed. Example—the number of
courses offered in each of the different colleges.
Ordinal level of measurement—classifies data into
categories that can be ordered or ranked, but precise
differences between the ranks do not exist.
Generally it does not make sense to do calculations
with data at the ordinal level. Example—letter grades
of A, B, C, D, and F.
Interval level of measurement—ranks data, precise differences
between units of measure exist, but there is no meaningful zero.
If a zero exists, it is an an arbitrary point.Example—IQ scores, it
makes sense to talk about someone having an IQ 20 points
higher than another person, but an IQ of zero has no meaning.
Ratio level of measurement—has all the characteristics of the
interval level, but a true zero exists. Also, true ratios exist when
the same variable is measured on two different members of the
population. Example—weight of an individual. It makes sense
to say that a 150 lb adult weighs twice as much as a 75 lb. child.
Eye Color (blue, brown, green, hazel)
Rating scale (poor, good, excellent)
ACT score
Ranking of high school football teams in Missouri
Zip code
Figure 2.1 summarizes the possible data types and
levels of measurement.
Figure 2.1 Data types and levels of measurement.
Copyright © 2009 Pearson Education, Inc.
By the Way ...
Scientists often measure temperatures on the Kelvin
scale. Data on the Kelvin scale are at the ratio level
of measurement, because the Kelvin scale has a true
zero. A temperature of 0 Kelvin really is the coldest
possible temperature. Called absolute zero, 0 K is
equivalent to about -273.15°C or -459.67°F.
(The degree symbol is not used for Kelvin
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End of 2.1
2.2 Dealing with Errors
Mistakes are the portals of discovery.
James Joyce
Understand the difference between random and
systematic errors, be able to describe errors by their
absolute and relative sizes, and know the difference
between accuracy and precision in measurements.
Two Types of Measurement Error
Random errors occur because of random
and inherently unpredictable events in the
measurement process.
Systematic errors occur when there is a
problem in the measurement system that
affects all measurements in the same way.
Measurement Error
T = True value of the observation
X = Measured value of the observation
Source: Research Methods Knowledge Base
What is random error?
Caused by any factors that randomly affect measurement of the
variable across the sample.
Each person’s mood can inflate or deflate their performance on
any occasion. In a particular testing, some children may be in a
good mood and others may be depressed. Mood may artificially
inflate the scores for some children and artificially deflate the
scores for others.
Random error does not have consistent effects across the entire
sample. If we could see all the random errors in a distribution, the
sum would be zero.
The important property of random error is that it adds variability to
the data but does not affect average performance for the group.
What is systematic error?
Systematic error is caused by any factors that systematically
affect measurement of the variable across the sample.
For instance, if there is loud traffic going by just outside of a
classroom where students are taking a test, this noise is liable to
affect all of the children's scores -- in this case, systematically
lowering them.
Unlike random error, systematic errors tend to be consistently
either positive or negative -- because of this, systematic error is
sometimes considered to be bias in measurement.
Reducing Measurement Error
Pilot test your instruments.
Thoroughly train people taking measurements.
Check and double check. If possible take multiple measurements.
You can use statistical procedures to adjust for measurement error.
These range from rather simple formulas you can apply directly to
your data to very complex modeling procedures for modeling the
error and its effects. Using multiple forms of measurement helps to
reduce systematic errors.
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Is the potential error systematic or random?
Amtrak passenger trains are most often
late in arriving at their destinations.
 A recipe for grape jelly calls for 4
pounds of grapes. The jelly maker
estimates the 4 pounds of grapes by
standing on a bathroom scale with and
without the grapes. The scale only
shows the weight to the nearest pound.
Go to a Web site (such as that gives the
current time. How far off is your clock or watch?
Describe the possible sources of random and systematic
errors in your timekeeping.
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Identify at least one likely source of random errors
and at least one likely source of systematic errors.
You need to measure 50 meters for a sprint
workout. You don’t have a tape measure, so
you use a meter stick to measure the
You are doing a survey about alcohol use
among college students. You ask students
to write down how many drinks they have
consumed in the last week.
Size of Error: Absolute versus Relative
The absolute error describes how far a claimed or
measured value lies from the true value:
absolute error = claimed or measured value – true value
The relative error compares the size of the absolute error
to the true value. It is often expressed as a percentage:
absolute error
relative error =
true value
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x 100%
Determine the absolute and relative error.
The true weight of a football player is
212 pounds but the program says 220
 You pay for 500 pounds of fish for a
stand at the fair, but the true weight of
the fish is 492 pounds.
Describing Results: Accuracy and
Accuracy describes how closely a
measurement approximates a true value. An
accurate measurement is close to the true
value. (Close is generally defined as a small
relative error, rather than a small absolute
Precision describes the amount of detail in a
Copyright © 2009 Pearson Education, Inc.
•Avogadro’s number is the number of molecules
of a substance in a quantity of the substance
measured in grams equal to its atomic weight.
•It can only be determined by chemistry or
physics experiments. It is named after Amadeo
Avogadro, who postulated in 1881 that this
number is the same for all substances.
•Various values for this constant have been
determined experimentally. Some of them are
6.02  1023, 6.022  1023, and 6.02214199 
1023. The 1023 means that you have to move the
decimal point 23 places to the right.
•Which of these values is the most accurate?
Which of these values is the most precise?
Compared to a scale that measures your
height to tenths of feet, a scale that measures
your height to the nearest inch is.
a. more precise and more accurate.
b. less precise, but may be more accurate.
c. more precise, but may be less accurate.
d. less precise and less accurate.
Summary: Dealing with Errors
• Errors can occur in many ways, but generally
can be classified into one of two basic types:
random errors or systematic errors.
• Whatever the source of an error, its size can be
described in two different ways: as an absolute
error or as a relative error.
• Once a measurement is reported, we can
evaluate it in terms of its accuracy and its
Copyright © 2009 Pearson Education, Inc.