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Small-N designs & Basic Statistical Concepts Psych 231: Research Methods in Psychology What are they? Historically, these were the typical kind of design used until 1920’s when there was a shift to using larger sample sizes Even today, in some sub-areas, using small N designs is common place • (e.g., psychophysics, clinical settings, expertise, etc.) Small N designs One or a few participants Data are not analyzed statistically; rather rely on visual interpretation of the data Observations begin in the absence of treatment (BASELINE) Then treatment is implemented and changes in frequency, magnitude, or intensity of behavior are recorded Small N designs Baseline experiments – the basic idea is to show: 1. When the IV occurs, you get the effect 2. When the IV doesn’t occur, you don’t get the effect (reversibility) Before introducing treatment (IV), baseline needs to be stable • Measure level and trend • Level – how frequent (how intense) is behavior? • Are all the data points high or low? • Trend – does behavior seem to increase (or decrease) • Are data points “flat” or on a slope? Small N designs AB design (baseline, treatment) – Can look for levels and trends before and after the treatment A B Steady state (baseline) | Transition steady state AB design ABA design (baseline, treatment, baseline) – The reversibility is necessary, otherwise something else may have caused the effect other than the IV (e.g., history, maturation, etc.) A B A Steady state (baseline) | Transition steady state | Reversibility ABA design ABAB design (baseline, treatment, baseline, treatment) – The second AB part essentially provides a replication of the first (and you can see the reversal effect) – A useful control for ruling out potential threats to internal validity (e.g., history) A B A Steady state (baseline) | Transition steady state | ABAB design B Reversibility | Transition2 Advantages Focus on individual performance, not fooled by group averaging effects Focus is on big effects (small effects typically can’t be seen without using large groups) Avoid some ethical problems – e.g., with non-treatments Allows to look at unusual (and rare) types of subjects (e.g., case studies of amnesiacs, experts vs. novices) Often used to supplement large N studies, with more observations on fewer subjects Small N designs Disadvantages Effects may be small relative to variability of situation Some effects are by definition between subjects • Treatment leads to a lasting change, so you don’t get reversals Difficult to determine how generalizable the effects are Small N designs Mistrust of statistics? It is all in how you use them They are a critical tool in research Statistics Why do we use them? Descriptive statistics • Used to describe, simplify, & organize data sets • Describing distributions of scores Inferential statistics • Used to test claims about the population, based on data gathered from samples • Takes sampling error into account, are the results above and beyond what you’d expect by random chance Statistics Recall that a variable is a characteristic that can take different values. The distribution of a variable is a summary of all the different values of a variable Both type (each value) and token (each instance) How much do you like psy231? 5 values (1, 2, 3, 4, 5) 1-2-3-4-5 Hate it Love it 1 5 5 Distribution 7 tokens (1,1,2,3,4,5,5) 4 1 3 2 Properties of a distribution Shape • Symmetric v. asymmetric (skew) • Unimodal v. multimodal Center • Where most of the data in the distribution are • Mean, Median, Mode Spread (variability) • How similar/dissimilar are the scores in the distribution? • Standard deviation (variance), Range Distribution A picture of the distribution is usually helpful Gives a good sense of the properties of the distribution Many different ways to display distribution Graphs • Continuous variable: • histogram, line graph (frequency polygons) • Categorical variable: • pie chart, bar chart Table • Frequency distribution table Distribution A frequency histogram Example: Distribution of scores on an exam Frequency 20 18 16 14 12 10 8 6 4 2 0 18 17 12 11 10 8 7 5 3 1 5054 5559 60- 6564 69 70- 7574 79 80- 8584 89 9094 95100 Exam scores Graph for continuous variables Bar chart Pie chart Cutting Doe Missing Smith Graphs for categorical variables VAR00 003 Va lid 1.00 Fre quen cy 2 Percent 7.7 Va lid Perce nt 7.7 Cumu lati ve Percent 7.7 2.00 3.00 4.00 3 3 5 11 .5 11 .5 19 .2 11 .5 11 .5 19 .2 19 .2 30 .8 50 .0 5.00 6.00 7.00 8.00 4 2 4 2 15 .4 7.7 15 .4 7.7 15 .4 7.7 15 .4 7.7 65 .4 73 .1 88 .5 96 .2 9.00 To tal 1 26 3.8 10 0.0 3.8 10 0.0 10 0.0 Values Counts Percentages (types) Frequency distribution table Symmetric Asymmetric (skewed) Negative Skew Positive Skew tail Properties of distributions: Shape tail Unimodal (one mode) Multimodal Minor mode Major mode Bimodal examples Properties of distributions: Shape There are three main measures of center Mean (M): the arithmetic average • Add up all of the scores and divide by the total number • Most used measure of center Median (Mdn): the middle score in terms of location • The score that cuts off the top 50% of the from the bottom 50% • Good for skewed distributions (e.g. net worth) Mode: the most frequent score • Good for nominal scales (e.g. eye color) • A must for multi-modal distributions Properties of distributions: Center The most commonly used measure of center The arithmetic average Computing the mean – The formula for the population mean is (a parameter): X N – The formula for the sample mean is (a statistic): X X n The Mean Divide by the total number in the population Add up all of the X’s Divide by the total number in the sample How similar are the scores? Range: the maximum value - minimum value • Only takes two scores from the distribution into account • Influenced by extreme values (outliers) Standard deviation (SD): (essentially) the average amount that the scores in the distribution deviate from the mean • Takes all of the scores into account • Also influenced by extreme values (but not as much as the range) Variance: standard deviation squared Spread (Variability) Low variability High variability The scores are fairly similar The scores are fairly dissimilar mean Variability mean The standard deviation is the most popular and most important measure of variability. The standard deviation measures how far off all of the individuals in the distribution are from a standard, where that standard is the mean of the distribution. • Essentially, the average of the deviations. Standard deviation Our population 2, 4, 6, 8 X 2 4 6 8 20 5.0 N 4 4 1 2 3 4 5 6 7 8 9 10 An Example: Computing the Mean Our population 2, 4, 6, 8 Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. X 2 4 6 8 20 5.0 N 4 4 X - = deviation scores 2 - 5 = -3 -3 1 2 3 4 5 6 7 8 9 10 An Example: Computing Standard Deviation (population) Our population 2, 4, 6, 8 Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. X 2 4 6 8 20 5.0 N 4 4 X - = deviation scores 2 - 5 = -3 4 - 5 = -1 -1 1 2 3 4 5 6 7 8 9 10 An Example: Computing Standard Deviation (population) Our population 2, 4, 6, 8 Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. X 2 4 6 8 20 5.0 N 4 4 X - = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 1 1 2 3 4 5 6 7 8 9 10 An Example: Computing Standard Deviation (population) Our population 2, 4, 6, 8 Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. X 2 4 6 8 20 5.0 N 4 4 X - = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 3 1 2 3 4 5 6 7 8 9 10 Notice that if you add up all of the deviations they must equal 0. An Example: Computing Standard Deviation (population) Step 2: So what we have to do is get rid of the negative signs. We do this by squaring the deviations and then taking the square root of the sum of the squared deviations (SS). X - = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 SS = (X - )2 = (-3)2 + (-1)2 + (+1)2 + (+3)2 = 9 + 1 + 1 + 9 = 20 An Example: Computing Standard Deviation (population) Step 3: ComputeVariance (which is simply the average of the squared deviations (SS)) So to get the mean, we need to divide by the number of individuals in the population. variance = 2 = SS/N An Example: Computing Standard Deviation (population) Step 4: Compute Standard Deviation To get this we need to take the square root of the population variance. X 2 standard deviation = = 2 N An Example: Computing Standard Deviation (population) To review: Step 1: Compute deviation scores Step 2: Compute the SS Step 3: Determine the variance • Take the average of the squared deviations • Divide the SS by the N Step 4: Determine the standard deviation • Take the square root of the variance An Example: Computing Standard Deviation (population) To review: Step 1: Compute deviation scores Step 2: Compute the SS Step 3: Determine the variance • Take the average of the squared deviations • Divide the SS by the N-1 Step 4: Determine the standard deviation • Take the square root of the variance An Example: Computing Standard Deviation (sample)