Chapter 7Chapter 7

Understanding Numbers Understanding Numbers in Measurementsin Measurements

Types of NumbersTypes of Numbers

• Nominal numbersNominal numbers

• Ordinal numbersOrdinal numbers

• Interval (scalar) numbersInterval (scalar) numbers

• Ratio numbersRatio numbers

Significant Digits and PrecisionSignificant Digits and Precision

• Significant digits: The number of digits in a Significant digits: The number of digits in a measurement or calculation that have measurement or calculation that have meaning.meaning.– For example, the number 75.25 has For example, the number 75.25 has four four

significant digits.significant digits.

• The greatest danger in PE and athletic The greatest danger in PE and athletic applications is implying more precision applications is implying more precision (more significant digits) than we actually (more significant digits) than we actually have.have.

Normal vs. Non-NormalNormal vs. Non-Normal(Non-Standard) Distributions(Non-Standard) Distributions

• Normal distribution: Normal distribution: – A set of data in which the mean, median, and A set of data in which the mean, median, and

mode are identical and are right in the middle mode are identical and are right in the middle of the distribution.of the distribution.

• Non-normal distributions: Non-normal distributions: – May occur when the population being sampled May occur when the population being sampled

is not usual (for example, a group of athletes is not usual (for example, a group of athletes that includes gymnasts and basketball that includes gymnasts and basketball players).players).

Types of Non-Normal DistributionsTypes of Non-Normal Distributions

• Leptokurtic distributionLeptokurtic distribution

• Platykurtic distributionPlatykurtic distribution

• Bimodal distributionBimodal distribution

• Skewed negative distributionSkewed negative distribution

• Skewed positive distributionSkewed positive distribution

A Normal CurveA Normal Curve

Examples of Distribution CurvesExamples of Distribution Curves

Your ViewpointYour Viewpoint

• Think about the quote from George Think about the quote from George Carlin, shown on p. 141:Carlin, shown on p. 141:

““Think about how dumb the average Think about how dumb the average person is. Now just think, half the people person is. Now just think, half the people are stupider than that.”are stupider than that.”

• What do you think of references to “the What do you think of references to “the average person” (i.e., in the media)? average person” (i.e., in the media)? Do you consider yourself “average”?Do you consider yourself “average”?

Measures of DispersionMeasures of Dispersion

• RangeRange

• Standard deviationStandard deviation

RangeRange

• Shows us how widely scores are Shows us how widely scores are dispersed.dispersed.– From the point that is farthest to the right in a From the point that is farthest to the right in a

curve (highest score) to the point that is curve (highest score) to the point that is farthest to the left (lowest score).farthest to the left (lowest score).

• Tells us how far apart the extreme scores Tells us how far apart the extreme scores are, and how variable.are, and how variable.

• Mathematical range: Mathematical range: – Range = (highRange = (high–low) + 1–low) + 1

Standard DeviationStandard Deviation

• Uses:Uses:– To establish the value of a unit.To establish the value of a unit.– To make predictions about the population To make predictions about the population

from which the sample was drawn.from which the sample was drawn.

• Standard error of the estimateStandard error of the estimate::– Estimated standard deviation of the error in Estimated standard deviation of the error in

a prediction.a prediction.

Normal Curve Normal Curve Showing the Standard DeviationsShowing the Standard Deviations

Standard ScoresStandard Scores

• When a raw score has been When a raw score has been transformed by use of the standard transformed by use of the standard deviation; standardization.deviation; standardization.

• Types:Types:– Percentile rank Percentile rank – PercentilePercentile– Z-score Z-score – Z-tableZ-table

Percentile Ranks and Percentile Ranks and PercentilesPercentiles

• Percentile rank: Percentile rank: – The percentage of scores that are The percentage of scores that are

lower than or equal to a specified lower than or equal to a specified score.score.

• Percentile:Percentile:– A particular score, on an ordered list A particular score, on an ordered list

of scores, at or below which a given of scores, at or below which a given percent of other scores fall.percent of other scores fall.

Z-ScoresZ-Scores

• A standard score that allows us to compare A standard score that allows us to compare any score to the mean score and then express any score to the mean score and then express it as a fraction of the standard deviation.it as a fraction of the standard deviation.

• Finding z-scores:Finding z-scores:– When a high score is better (i.e., batting average): When a high score is better (i.e., batting average):

ZZhigh score better high score better = (score = (score – mean)/SD– mean)/SD

– When a low score is better (i.e., golf score): When a low score is better (i.e., golf score):

ZZlow score better low score better = (mean = (mean – score)/SD– score)/SD

Normal Distribution Normal Distribution Showing Z-ScoresShowing Z-Scores

Z-TablesZ-Tables

• A table used to show percentile ranks A table used to show percentile ranks or probabilities for a certain z-score for or probabilities for a certain z-score for normally distributed data.normally distributed data.

• Z-tables can be found online:Z-tables can be found online:www.stat.lsu.edu/EXSTWeb/statlab/Tables/TABLES98-Z.html

Checking aChecking aMeasurement TechniqueMeasurement Technique

• Think about the problem to get a rough Think about the problem to get a rough estimate of what the answer should be.estimate of what the answer should be.

• If the final measurement is very different If the final measurement is very different from the rough estimate, review from the rough estimate, review calculations carefully.calculations carefully.

• Remember that sometimes the same Remember that sometimes the same units can have different values.units can have different values.