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.”

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)

LEARNING GOAL

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 numberThe number of textbooks owned by a

studentThe incomes of college graduatesThe 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 circulation

The voltage of electricity in a power lineThe number of eggs that hens layThe time it takes for a student to

complete a test

Levels of Measurement

Nominal Ordinal Interval Ratio Nominal and ordinal are qualitative

(categorical) levels of measurement. Interval and ratio are quantitative levels of

measurement.

TYPES OF QUALITATIVE MEASUREMENTS

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.

TYPES OF QUANTITATIVE MEASUREMENTS

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.

CLASSIFY THE FOLLOWING AS TO QUALITATIVE OR QUANTITATIVE MEASUREMENT. THEN STATE THE LEVEL OF MEASUREMENT.

Eye Color (blue, brown, green, hazel) Rating scale (poor, good, excellent) ACT score Salary Age Ranking of high school football teams in Missouri Nationality Temperature Zip code

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Figure 2.1 summarizes the possible data types and levels of measurement.

Figure 2.1 Data types and levels of measurement.

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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 temperatures.)

End of 2.1

2.2 Dealing with Errors

Mistakes are the portals of discovery.

James Joyce

LEARNING GOAL

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

http://www.socialresearchmethods.net/kb/measerr.php

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

1. Pilot test your instruments.2. Thoroughly train people taking measurements.3. Check and double check. If possible take multiple measurements.4. 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.

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TIME OUT TO THINK

Go to a Web site (such as www.time.gov) 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.

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 distance.

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.

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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:

relative error = x 100%absolute error

true value

Determine the absolute and relative error.

The true weight of a football player is 212 pounds but the program says 220 pounds.

You pay for 500 pounds of fish for a stand at the fair, but the true weight of the fish is 492 pounds.

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Describing Results: Accuracy and Precision

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 error.)

Precision describes the amount of detail in a measurement.

•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.

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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 precision.