Dr. Michael R. Hyman, NMSU

Statistics for a Single Measure (Univariate)

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Wrong Ways to Think About Statistics

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Evidence

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Descriptive Analysis

Transformation of raw data into a form that make them easy to understand and interpret; rearranging, ordering, and manipulating data to generate descriptive information

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Analysis Begins with Tabulation

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Tabulation

• Tabulation: Orderly arrangement of data in a table or other summary format

• Frequency table

• Percentages

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Frequency Table

• Numerical data arranged in a row-and-column format that shows the count and percentages of responses or observations for each category assigned to a variable

• Pre-selected categories

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Sample Frequency Table

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Sample SPSS Frequency Output

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SPSS Histogram Output

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Base

• Number of respondents or observations (in a row or column) used as a basis for computing percentages

• Possible bases

– All respondents

– Respondents who were asked question

– Respondents who answered question

• Be careful with multi-response questions

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Measures of Central Tendency

• Mode - the value that occurs most often

• Median - midpoint of the distribution

• Mean - arithmetic average– µ, population– , sampleX

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Measures of Dispersion or Spread for Single Measure

• Range

• Inter-quartile range

• Mean absolute deviation

• Variance

• Standard deviation

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150 160 170 180 190 200 210

5

4

3

2

1

Value on Variable

Fre

quen

cy

Low Dispersion

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150 160 170 180 190 200 210

5

4

3

2

1

Fre

quen

cy

Value on Variable

High Dispersion

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Range as a Measure of Spread

• Difference between smallest and largest value in a set

• Range = largest value – smallest value

• Inter-quartile range = 75th percentile – 25th percentile

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Deviation Scores

Differences between each observed value and the mean

xxd ii

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Average Deviation

0)(

n

XX i

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Mean Squared Deviation

n

XXi 2)(

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Variance

2

2

S

Sample

Population

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Variance

1)2

2

nXX

S

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Variance

• Variance is given in squared units

• Standard deviation is the square root of variance

1

2

n

XX iS

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Measure ofCentral Measure of

Type of Scale Tendency Dispersion

Nominal Mode NoneOrdinal Median PercentileInterval or ratio Mean Standard deviation

Summary: Central Tendency and Dispersion

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Distributions for Single Measures

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Symmetric Distribution

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Skewed Distributions

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Normal Distribution

• Bell shaped• Symmetrical about its mean• Mean identifies highest

point• Almost all values are within

+3 standard deviations• Infinite number of cases--a

continuous distribution

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2.14%

13.59% 34.13% 34.13% 13.59%

2.14%

Normal Distribution

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Standard Normal Curve

• The curve is bell-shaped or symmetrical

• About 68% of the observations will fall within 1 standard deviation of the mean

• About 95% of the observations will fall within approximately 2 (1.96) standard deviations of the mean

• Almost all of the observations will fall within 3 standard deviations of the mean

31 85 115100 14570

Normal Curve: IQ Example

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Standardized Normal Distribution

• Area under curve has a probability density = 1.0

• Mean = 0

• Standard deviation = 1

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0-1-2 2 z

Standardized Normal Curve

1

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Standardized Normal is Z Distribution

–z +z

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xz

Standardized Scores

Used to compare an individual value to the population mean in units of the standard deviation

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Linear Transformation of Any Normal Variable into a Standardized Normal

Variable

-2 -1 0 1 2

Sometimes thescale is stretched

Sometimes thescale is shrunk

X

xz

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Data Transformation

• Data conversion

• Changing the original form of the data to a new format

• More appropriate data analysis

• New variables

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Data Transformation

Summative Score =

VAR1 + VAR2 + VAR 3

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Index Numbers

• Score or observation recalibrated to indicate how it relates to a base number

• CPI - Consumer Price Index

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Recap

• Count/frequency data

• Measures of central tendency

• Dispersion from central tendency

• Data distributions

– Symmetric versus skewed

– (Standardized) normal

• Data transformation