1. Statistics for a Single Measure (Univariate)
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Dr. Michael R. Hyman, NMSU

3. Wrong Ways to Think About Statistics

4. Evidence

5. 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

6. Analysis Begins with Tabulation

7. Tabulation
Tabulation: Orderly arrangement of data in a table or other summary format
Frequency table
Percentages

8. 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

9. Sample Frequency Table

10. Sample SPSS Frequency Output

11. SPSS Histogram Output

12. 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

13. Measures of Central Tendency
Mode - the value that occurs most often
Median - midpoint of the distribution
Mean - arithmetic average
µ, population
, sample

14. Measures of Dispersion or Spread for Single Measure
Range
Inter-quartile range
Mean absolute deviation
Variance
Standard deviation

15. Low Dispersion 5 4 3 Frequency 2 1 150 160 170 180 190 200 210
Value on Variable

16. High Dispersion 5 4 3 Frequency 2 1 150 160 170 180 190 200 210
Value on Variable

17. 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

18. Deviation Scores
Differences between each observed value and the mean

19. Average Deviation

20. Mean Squared Deviation

21. Variance

22. Variance

23. Variance Variance is given in squared units
Standard deviation is the square root of variance

24. Summary: Central Tendency and Dispersion
Measure of
Central Measure of
Type of Scale Tendency Dispersion
Nominal Mode None
Ordinal Median Percentile
Interval or ratio Mean Standard deviation

25. Distributions for Single Measures

26. Symmetric Distribution

27. Skewed Distributions

28. 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

29. Normal Distribution 13.59% 13.59% 34.13% 34.13% 2.14% 2.14%
Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video.
2.14%
2.14%

30. 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. Normal Curve: IQ Example70 85
100
115
130

32. Standardized Normal Distribution
Area under curve has a probability density = 1.0
Mean = 0
Standard deviation = 1

33. Standardized Normal Curve
Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. z -2 -1 1 2

34. Standardized Normal is Z Distribution

35. Standardized Scores
Used to compare an individual value to the population mean in units of the standard deviation

36. Linear Transformation of Any Normal Variable into a Standardized Normal VariableX m
Sometimes the
scale is stretched
Sometimes the
scale is shrunk

37. Data Transformation Data conversion
Changing the original form of the data to a new format
More appropriate data analysis
New variables

38. Summative Score = VAR1 + VAR2 + VAR 3
Data Transformation
Summative Score =
VAR1 + VAR2 + VAR 3

39. Index Numbers
Score or observation recalibrated to indicate how it relates to a base number
CPI - Consumer Price Index

40. Recap Count/frequency data Measures of central tendency
Dispersion from central tendency
Data distributions
Symmetric versus skewed
(Standardized) normal
Data transformation