Introductory Statistics Options, Spring 2008 Stat 100: MWF, 11:00 Science Center C. Stat 100: MWF, 11:00 Science Center C. –General intro to statistical methods in social and political science, public health and economics. –Text: Introduction to the Practice of Statistics Stat 101: MWF, 3:00 Science Center A Stat 101: MWF, 3:00 Science Center A –Similar to 100, but emphasis on applications and statistical reasoning in psychology and related disciplines. –Text: Statistics for Psychology Stat 104: MWF, 11:00 Science Center D Stat 104: MWF, 11:00 Science Center D –Similar to Stat 100, but with emphasis on applications to economics. Slightly faster pace than 100 and 104 –Same text as 100

Overview What can statistics be used for? What can statistics be used for? Our first quantitative example Our first quantitative example Frequency tables Frequency tables Graphical presentation of data: polygons, histograms, pie charts, bar charts Graphical presentation of data: polygons, histograms, pie charts, bar charts Types of Distributions Types of Distributions Summarizing Data – Central Tendency Summarizing Data – Central Tendency –Mean, Median, Mode Summary Summary

Chapter 1 Statistics can be used to accomplish many different things – such as: Statistics can be used to accomplish many different things – such as: –Quantification & measurement –Organizing and describing information –Systematic comparisons –Theoretical modeling –Weighing evidence and evaluating models

First Example How stressed have you been in the last 2 ½ weeks? Scale: 0 (not at all) to 10 (as stressed as possible) from Aron & Aron’s text, Statistics for Psychology

Frequency Tables A frequency table shows how often each value of the variable occurs A frequency table shows how often each value of the variable occurs Stress rating Frequency

Frequency Polygon A visual representation of information contained in a frequency table A visual representation of information contained in a frequency table Align all possible values on the bottom of the graph (the x-axis) Align all possible values on the bottom of the graph (the x-axis) On the vertical line (the y- axis), place a point denoting the frequency of scores for each value On the vertical line (the y- axis), place a point denoting the frequency of scores for each value Connect the lines Connect the lines (Typically add an extra value above and below the actual range of values) (Typically add an extra value above and below the actual range of values)

Histograms Another way of visually representing information contained in a frequency table Another way of visually representing information contained in a frequency table Histograms are kind of like bar charts; bars are used instead of connected points Histograms are kind of like bar charts; bars are used instead of connected points The bars typically cover “intervals” of values. The first bar here covers scores > 0 and 0 and < 1.

Pie Charts and Nominal Data Pie charts are commonly used to represent the frequency of scores for nominal data Pie charts are commonly used to represent the frequency of scores for nominal data Here, frequency of referents in a letter written by a subject in a psychological study. Here, frequency of referents in a letter written by a subject in a psychological study. 70% of the pronouns are in reference to the writer; 10% are in reference to the person being written to. 70% of the pronouns are in reference to the writer; 10% are in reference to the person being written to.

Barcharts and Nominal Data Barcharts are sometimes used to represent the frequency of scores for nominal data Barcharts are sometimes used to represent the frequency of scores for nominal data Here, frequency is expressed as a percentage of the total number of males and females Here, frequency is expressed as a percentage of the total number of males and females (78% and 68%) (78% and 68%)

Shapes of Distributions These representational aides all describe frequency distributions: the way score frequencies are distributed with respect to the values of the variable These representational aides all describe frequency distributions: the way score frequencies are distributed with respect to the values of the variable Distributions can take on a number of shapes or forms Distributions can take on a number of shapes or forms

Unimodal Distributions The mode of a distribution refers to the most frequently occurring score The mode of a distribution refers to the most frequently occurring score In a unimodal distribution, one score occurs much more frequently than others In a unimodal distribution, one score occurs much more frequently than others

Multimodal Distributions In multimodal distributions, more than one mode exists (or approximately so) In multimodal distributions, more than one mode exists (or approximately so) In a bimodal distribution, two modes exist In a bimodal distribution, two modes exist

Rectangular or Uniform Distributions In a uniform distribution, all values are observed equally often In a uniform distribution, all values are observed equally often

Symmetrical and Skewed Distributions A symmetrical distribution is balanced: if we cut it in half, the two sides would be mirror images of one another A symmetrical distribution is balanced: if we cut it in half, the two sides would be mirror images of one another normal distribution: a particular kind of distribution that resembles a bell (bell-shaped distribution) normal distribution: a particular kind of distribution that resembles a bell (bell-shaped distribution)

Skewed Distributions A skewed distribution is unbalanced; there may be a cluster of scores piling on one end of the scale A skewed distribution is unbalanced; there may be a cluster of scores piling on one end of the scale

Skew negative skew positive skew reasons for skew?

Question How can we summarize a distribution of scores efficiently using quantitative (as opposed to graphical) methods?

Measures of Central Tendency Central tendency: most “typical” or common score Central tendency: most “typical” or common score (a) Mode (b) Median (c) Mean

Measures of Central Tendency 1. Mode: most frequently occurring score 10, 20, 30, 40, 40, 50, 60 Mode = 40

Measures of Central Tendency 2. Median: the value at which 1/2 of the ordered scores fall above and 1/2 of the scores fall below Median = 3Median = 2.5

Measures of Central Tendency x = an individual score N = the number of scores Sigma or  = take the sum Note: Equivalent to saying “sum all the scores and divide that sum by the total number of scores” Note: Equivalent to saying “sum all the scores and divide that sum by the total number of scores” 3. Mean: The “balancing point” of a set of scores; the average

A B CDE  (-1)  (-2) (+4)  (+1)  (– 1) + (– 2) + (– 2) = 0

A B CDE (+2) (-1) (-3) (- 4) (– 1) + (– 3) + (– 4) + (– 4) + 2 = –10

Summary What can statistics be used for? What can statistics be used for? Our first quantitative example Our first quantitative example Frequency tables Frequency tables Graphical presentation of data: polygons, histograms, pie charts, bar charts Graphical presentation of data: polygons, histograms, pie charts, bar charts Types of Distributions Types of Distributions Summarizing Data – Central Tendency Summarizing Data – Central Tendency –Mean, Median, Mode