Reasoning in Psychology Using Statistics 2015

Please complete the survey mentioned in President Dietz’s email. While at it, feel free to think about statistics: What scales of measurement? How would I graph these results? Etc.

Rolling the dice Let’s collect set of data I’ll pass out some pairs of dice Collect n=36 data points 12 people roll a pair of dice three times Write down the 36 numbers on the board Type numbers here: Rolling the dice

Distributions Distribution Type numbers here: The distribution of a variable is a summary of all the different values of a variable The set of all of the outcomes of rolling the dice Both type (each value) and token (each instance) Un-organized, the overall pattern and properties of the distribution are difficult to see A “picture” of the distribution is usually helpful Type numbers here: Distributions

Distributions Descriptive statistics Statistical tools/procedures to help organize, summarize, and simplify large sets of data (distributions) Important descriptive properties of distribution Center Where most of the data in the distribution are Spread (variability) How similar/dissimilar are the scores in the distribution? Shape Symmetric vs. asymmetric (skew) Unimodal vs. multimodal Distributions

Distributions A “picture” of the distribution is usually helpful Gives a good sense of the properties of the distribution Many different ways to display distribution Table Frequency distribution table Stem and leaf plot Graphs Distributions

Frequency distribution table The values of the variable The number of tokens of each variable The proportion of tokens at each value The percentage of tokens at each value Cumulative percentage p = f/N N=total Set-up for values of pair of dice Frequency distribution table

Cumulative percent Quiz: “What % got this score or worse?” 10 10% got a 1 or worse Simpler case oo a short quiz Cumulative percent

Cumulative percent Quiz: “What % got this score or worse?” 15% got a 2 & 10% got a 1 25 10 25% got a 2 or worse Cumulative percent

Cumulative percent Quiz: “What % got this score or worse?” 10% got a 3 & 15% got a 2 & 10% got a 1 35 25 10 35% got a 3 or worse Cumulative percent

Cumulative percent Quiz: “What % got this score or worse?” 35% got a 4 & 10% got a 3 & 15% got a 2 & 10% got a 1 70 35 25 10 70% got a 4 or worse Cumulative percent

Cumulative percent Quiz: “What % got this score or worse?” 20% got a 5 & 35% got a 4 & 10% got a 3 & 15% got a 2 & 10% got a 1 90 70 35 25 10 90% got a 5 or worse Cumulative percent

Cumulative percent Quiz: “What % got this score or worse?” 10% got a 6 & 20% got a 5 & 35% got a 4 & 10% got a 3 & 15% got a 2 & 10% got a 1 100 90 70 35 25 10 100% got a 6 or worse Cumulative percent

Frequency distribution: sum of 2 dice Fill in numbers from our class: Sample distribution n = 36 Frequency distribution: sum of 2 dice

Theoretical Frequency distribution: sum of 2 dice p = probability when predicting p = proportion when describing what you observed Think of this as defining our population distribution of the outcome of tossing two dice Theoretical Frequency distribution: sum of 2 dice

Theoretical Frequency distribution: sum of 2 dice Value D1+D2 D1 D2 frequency D1 D2 D1 D2 12 1 4 3 2 7 6 5 3 6 11 2 11 10 9 8 3 2 1 4 5 5 4 Total outcomes = 62 = 36 = 1+2+3+4+5+6+5+4+3+2+1 = 36 Theoretical Frequency distribution: sum of 2 dice

Theoretical Frequency distribution: sum of 2 dice

Theoretical frequency distribution & class sample (Actual fs are from a previous term.)

Distributions Important properties of distribution Center Where most of the data in the distribution are Spread (variability) How similar/dissimilar are the scores in the distribution? Shape Symmetric vs. asymmetric (skew) Unimodal vs. multimodal Distributions

Describing the distribution What is the most frequent score? 7 Describing the distribution

Describing the distribution What is the most frequent score? Where do most of the scores lie? Two-thirds of the data are here Describing the distribution

Describing the distribution What is the most frequent score? Maximum score: 12 Where do most of the scores lie? What was the range of scores? Minimum score: 2 Describing the distribution

Distributions A “picture” of the distribution is usually helpful Gives a good sense of the properties of the distribution Many different ways to display distribution Table Frequency distribution table Stem and leaf plot Graphs Distributions

Stem and Leaf Plots Distribution of exam scores (section 01): 5 7 6 4 67, 90, 92, 58, 76, 75, 84, 92, 78, 93, 89, 74, 62, 98, 75, 73, 75, 89, 89, 76, 65, 49 5 7 6 4 9 10 8 2 2 3 8 4 9 9 9 6 5 8 4 5 3 5 6 7 2 5 8 9 Stem and Leaf Plots

Stem and Leaf Plots Distribution of exam scores (section 01): 1 5 3 4 67, 90, 92, 58, 76, 75, 84, 92, 78, 93, 89, 74, 62, 98, 75, 73, 75, 89, 89, 76, 65, 49 1 5 3 4 2 8 9 6 5 7 6 4 9 10 8 5 7 6 4 9 8 3 2 Distribution of exam scores (section 03): 72, 90, 83, 58, 66, 65, 84, 95, 72, 93, 89, 70, 42, 100, 71, 73, 75, 62, 62, 74, 65 Stem and Leaf Plots

Distributions A “picture” of the distribution is usually helpful Gives a good sense of the properties of the distribution Many different ways to display distribution Table Frequency distribution table Stem and leaf plot Graphs Graphs types Continuous variable: histogram, line graph (frequency polygons) Categorical (discrete) variable: pie chart, bar chart Distributions

Graphs for continuous variables Histogram Line graph Graphs for continuous variables

Graphs for categorical variables Bar chart Pie chart Graphs for categorical variables

Distributions Important properties of distribution Center Where most of the data in the distribution are Spread (variability) How similar/dissimilar are the scores in the distribution? Shape Symmetric vs. asymmetric (skew) Unimodal vs. multimodal Distributions

Symmetric Asymmetric Positive Skew Negative Skew tail tail Shape

Shape Unimodal (one mode) Multimodal Major mode Minor mode Bimodal examples Shape

Descriptive statistics Coming up in future lectures: In addition to pictures of the distribution, numerical summaries are also presented. Numeric Descriptive Statistics Shape Skew (symmetry) & Kurtosis (shape) Number of modes Measures of Center Measures of Variability (Spread) In lab, create basic tables and graphs both by hand and using SPSS If time in lecture there are some SPSS show and tell slides Descriptive statistics

Drag & drop Drag & drop SPSS: Bar graph

Frequency distribution of different categories SPSS: Bar graph

SPSS: Cluster bar graph Drag & drop Drag & drop SPSS: Cluster bar graph

SPSS: Cluster bar graph Legend Frequency distribution of categories broken down into another category SPSS: Cluster bar graph

Drag & drop Drag & drop SPSS: Histogram

Frequency distribution SPSS Graphing