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

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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

Value on Variable Frequency Low Dispersion

Frequency Value on Variable High Dispersion

17 Range as a Measure of Spread Difference between smallest and largest value in a set Range = largest value – smallest value Inter-quartile range = 75 th percentile – 25 th 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 Measure of CentralMeasure of Type of ScaleTendencyDispersion NominalModeNone OrdinalMedianPercentile Interval or ratioMeanStandard deviation Summary: Central Tendency and Dispersion

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

% 13.59% 34.13% 13.59% 2.14% Normal Distribution

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

Normal Curve: IQ Example

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

z Standardized Normal Curve 1

34 Standardized Normal is Z Distribution –z+z

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 Variable Sometimes the scale is stretched Sometimes the scale is shrunk    X

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

38 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