Transcript Slide 1
Statistics for a Single Measure (Univariate) Dr. Michael R. Hyman, NMSU 2 Wrong Ways to Think About Statistics 3 Evidence 4 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 5 Analysis Begins with Tabulation 6 Tabulation • Tabulation: Orderly arrangement of data in a table or other summary format • Frequency table • Percentages 7 Frequency Table • Numerical data arranged in a row-andcolumn format that shows the count and percentages of responses or observations for each category assigned to a variable • Pre-selected categories 8 Sample Frequency Table 9 Sample SPSS Frequency Output 10 SPSS Histogram Output 11 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 12 Measures of Central Tendency • Mode - the value that occurs most often • Median - midpoint of the distribution • Mean - arithmetic average – µ, population – X , sample 13 Measures of Dispersion or Spread for Single Measure • • • • • Range Inter-quartile range Mean absolute deviation Variance Standard deviation 14 Low Dispersion 5 4 3 2 1 150 160 170 180 190 Value on Variable 200 210 15 High Dispersion 5 4 3 2 1 150 160 170 180 190 Value on Variable 200 210 16 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 17 Deviation Scores Differences between each observed value and the mean d x x i i 18 Average Deviation (X i X ) 0 n 19 Mean Squared Deviation ( Xi X ) n 2 20 Variance Population 2 Sample S 2 21 Variance X X ) S n 1 2 2 22 Variance • Variance is given in squared units • Standard deviation is the square root of variance S 2 Xi X n 1 23 Summary: Central Tendency and Dispersion Type of Scale Measure of Central Tendency Measure of Dispersion Nominal Ordinal Interval or ratio Mode Median Mean None Percentile Standard deviation 24 Distributions for Single Measures 25 Symmetric Distribution 26 Skewed Distributions 27 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 28 Normal Distribution 13.59% 2.14% 34.13% 34.13% 13.59% 2.14% 29 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 30 Normal Curve: IQ Example 70 85 100 115 145 31 Standardized Normal Distribution • Area under curve has a probability density = 1.0 • Mean = 0 • Standard deviation = 1 32 Standardized Normal Curve -2 -1 0 1 2 z 33 Standardized Normal is Z Distribution –z +z 34 Standardized Scores Used to compare an individual value to the population mean in units of the standard deviation z x 35 Linear Transformation of Any Normal Variable into a Standardized Normal Variable Sometimes the scale is stretched X Sometimes the scale is shrunk z -2 -1 0 1 2 x 36 Data Transformation • Data conversion • Changing the original form of the data to a new format • More appropriate data analysis • New variables 37 Data Transformation Summative Score = VAR1 + VAR2 + VAR 3 38 Index Numbers • Score or observation recalibrated to indicate how it relates to a base number • CPI - Consumer Price Index 39 Recap • • • • Count/frequency data Measures of central tendency Dispersion from central tendency Data distributions – Symmetric versus skewed – (Standardized) normal • Data transformation 40