Chapter 12 Understanding Research Results: Description and Correlation SCALES OF MEASUREMENT: A REVIEW Nominal No numerical, quantitative properties Levels represent different categories or groups Ordinal – minimal quantitative distinctions Order the levels from lowest to highest Interval – quantitative properties Intervals between levels are equal in size Can be summarized using means No absolute zero Ratio – detailed quantitative properties Equal intervals Absolute zero Can be summarized using mean ANALYZING THE RESULTS OF RESEARCH INVESTIGATIONS Three basic ways of describing the results: Comparing Group Percentage Analyzing nominal level data with known frequencies Correlating Individual Scores Analyzing ordinal, interval, or ratio data to find relationships Comparing Group Means Analyzing interval or ratio data to find group mean differences FREQUENCY DISTRIBUTIONS Graphing Frequency Distributions Pie charts Bar graphs Frequency polygons Histograms Graphing the distribution- gives a visual representation of the distribution of scores. frequency distribution- indicates the number of individuals that receive each possible score on a variable. PIE CHART Pie Charts- shows percentages as slices of a whole unit. BAR GRAPH Bar Graphs- use a distinct bar for each piece of information; used for nominal level data. Frequency Polygon and Histogram frequency polygon-uses a line to represent frequencies. histogram- like a bar graph except the bars are touching. Let’s Do Some Stats! You ask participants to rate their opinion of a new soft drink you have developed on a 1-7 scale. Below are the results. n = 12 Participant # Response 1 5 2 3 3 4 4 5 5 7 6 6 7 5 8 5 9 7 10 2 11 5 12 1 Descriptive Statistics- the use of numerical indexes to describe either a population or a sample central tendency- a single number or value that describes the typical or central score among a set of scores; the three types are listed below. Mean ( )- the arithmetic average of a set of scores. For the example, the mean is 4.58 median- the middle score in a set of scores; the score that cuts the distribution in half. For the example, the median is 5; lay the numbers out in order and choose the middle number1 2 3 4 5 5 5 5 5 6 7 7 mode- the score that occurs most often in the set of scores. For this example, the mode is also 5. Variability- the amount of dispersion the scores have around the central value (usually the mean). x- sum all of the scores. Example 55. x2- square all of the values and then sum up. Example 289. Deviation Score- a method of telling how far a particular score is from the mean For our example, we will have 12 deviation scores. Formula Example for Participant 2 (3-5)= -2 If our mean is 4.58… Participant # Response Deviation Score 1 5 0.42 2 3 -1.58 3 4 -0.58 4 5 0.42 5 7 2.42 6 6 1.42 7 5 0.42 8 5 0.42 9 7 2.42 10 2 -2.58 11 5 0.42 12 1 -3.58 Sums of Squares (SS)- uses deviation scores to account for the total variability in the data set. This number can never be negative. Formula 1 Formula 2 For our example SS = 36.92 Variance (s2)- a measure of variability of scores that is more practical to use than SS; shows the amount of spread in the distribution. This number can also never be negative. Formula for population variance Formula for sample variance For our example (use sample equation) 3.077 Standard Deviation (s)- the measure of variability that is the easiest to interpret; indicates the average deviation of scores from the mean. This number can also never be negative. Formula (pop.) σ= Formula (samp.) s= For our example 1.754 Correlation Strength of the relationship- is the IV/DV relationship strong or weak? Pearson correlation coefficient- gives a numeric representation of the strength; only used for interval and ratio data; on a scale of 0.00 to ±1.00. The sign tells you whether the relationship is positive or negative. “0” indicates no relationship. (<.3 indicates little relationship, .3-.49 indicates medium relationship, ≥.5 indicates a large relationship) Scatterplot- used when there is an IV and a DV (unlike our example) each pair of scores (an IV value and a DV value) is plotted as a single point on a diagram. This gives a visual representation of the data points used for the correlation. Restriction of range- reduces the magnitude of the correlation coefficient. This is a bad thing. A Pearson correlation analysis revealed that female life expectancy was significantly correlated with literacy, r= +.86, n= 107, p<.05, two tails. effect size- the general term for the strength of association between variables. Advantage of Reporting Effect Size is that it Provides a Scale of Values that is Consistent Across All Types of Studies Differences in effect sizes Small effects near r = .15 Medium effects near r = .30 Large effects above r = .40 Squared value of the coefficient r² - transforms the value of r to a percentage Percent of shared variance between the two variables statistical significance- indicates that there is very high confidence in reporting an observed relationship between the IV and the DV Inferring whether the results will hold up if the experiment is repeated several times, each time with a new sample of research participants Regression- used to predict a person’s score on one variable when that person’s score on another variable is already known. Y= a + bX Y = Score we wish to predict X = Score that is known a = constant b = weighing adjustment Multiple correlation- the correlation between a large set of predictor variables and a single criterion variable; symbolized by R Partial correlation- a way of statistically removing a third variable by “partialling out” its influence. Structural Models- a method of creating a “structure” for IV/DV relationships; several DVs measured at one time. Path Analysis- a type of structural model that focuses on individual relationships between variables rather than the whole model.