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Methods and Measurement in Psychology Statistics • THE DESCRIPTION, ORGANIZATION AND INTERPRATATION OF DATA DESCRIBING DATA SCALING The method by which one puts numbers to variables. 1. NOMINAL • The most Primitive of all scales and is included by definition in all other scales. Criteria • 1. NAMING OR POINTATABLE VARIABLE Criteria 2. NO NUMERICAL ANALYSIS POSSIBLE EXAMPLES: • Drivers License Examp. • Social Security Number Examp. • Numbers On The Backs of Football Players Scale 2 • ORDINAL • The objects of a variable set can be rank ordered on some operationally defined characteristic. Ordinal Scale • Rank order in terms of the magnitude of the variables i.e. • More of, or less of, one variable with respect to another variable. Requires the use of the nominal scale. Examples • Positions in a race; 1st, 2nd etc. The Scale You Are Most Familiar With • GRADES •A>B>C>D>F Problems With Ordinal Scales • 1. No Zero point • 2. What is the magnitude of the distance between units of the scale Example • • • • • • Grades A>B>C>D>F What is the last upper number What is the last lower number How much less is a B from an A. How much less is a C from a B etc. High Ordered Metric Scale • Tries to measure the distance between two ordinal variables Ideally, grades are equal distance from one another A>B>C>D>F • You can take the test and get one of two grades, A or C. • You don‘t have to take the test and get a B. If One Takes The Test • The subjective gain of getting a B is so small relative to getting a C that one would gamble for the A. Subjective loss less than the subjective gain If One Takes the Assured B • The subjective loss of the B by taking the test is too large relative to the gain of getting an A. One would not gamble for the A. • The distance AB is shorter than the distance BC . Choose B for sure One Can Make The Same Comparisons Between Grades BC and CD. • When One Makes All Of The Possible Choices, One Sees That The Distances Do Not Rank Order Themselves In Terms Of Magnitude. Scale 3 • INTERVAL SCALE • 1. Possesses all of the characteristics of the Nominal and Ordinal scale especially rank-order • 2. Numerically equal distance on the an interval scale means equal distance on the property being measured • There Must Be An Arbitrary Zero. The underling concept is mean molecular motion. • Centigrade scale starts at zero and has 100 equal intervals. • Fahrenheit scale starts at 32 and ends at 212 with 180 equal appearing intervals The Distances Between Rank Orders Is Equal • The distance from 20 degrees to 30 degrees is the same as the distance between 75 degrees and 85 degrees, or -75 degrees and -85 degrees. There Are Ten Degrees Of Difference One Can Use Most Of The Mathematical Operations With Interval Scales • ADD, Subtract, Multiply, Divide, Square, and Take Square Root. • Will be used in most of the statistical methods covered below. Ratio Scale • The most powerful of the scale. • An Absolute Zero. • Includes Nominal, Ordinal and Interval Scales • Equal Intervals. • The Ratio Between Intervals Are Equal Example • Kelvin or Absolute Zero Temperature scale. Defined as that point where all molecular motion (Brownian movement) stops. • There is no true Ratio scale in Psychology ORGANIZING DATA DATA ORGANIZATION • Frequency Distribution • A distribution that counts the number of individuals obtaining a given score and arranges those counts in a rank order from high to low or low to high (ordinal scale). Histogram Histogram of a set of scores Frequency Polygon of the same set scores of Frequency polygon plus histogram Measures of Central Tendency • How common are you? MODE • Common Use: Pie Ala Mode, the hump of ice cream on the pie! • Mode = The most frequently measured score! • Distribution of scores can have more than one hump! Median Where is the word Median Used in Common Parlance? • Keep Off The Median – used in Highway Driving Mean • Average Positively Skewed Distribution Positively Skewed Distribution • Note how the positive numbers pull the mean to the right. Measures of Variability • How unique are you------How scores differ one from another. • Range • Lowest to highest Deviation • The difference between a score and some constant measure • The constant can be any measure, but that which makes most sense is one of the measures of central tendency Deviation score • X - MEAN = DEVIATION • Σ = sum of SUM OF MEANS • Σ (X – MEAN) = 0 How do I get rid of negative deviation scores • SQUARE THE DEVIATION SCORES • Σ (X – MEAN)2 = 0 VARIANCE • Σ (X – N • 2 MEAN) • Is there a way to compare the same individual on two different tests? Standard Scores • z scores are called Standard Scores • COMPARE THE DEVIATION SCORE OF EACH TEST TO ITS STANDARD DEVIATION • z = (X – MEAN) S.D. Characteristics of a z distribution “z” DISTRIBUTIONS ARE CHARACTERIZED BY THE PARAMETERS OF A NOTRMAL CURVE • THE S.D. OF A z DISTRIBUTION = 1 • THE MEAN OF A “z” DISTRIBUTION = 0 A NORMAL CURVES OF IQ Normal Curves How the mean, median and mode are effected by skewness Three types of normal curves depends on range of x values DESCRIBING THE RELATION BETWEEN TWO VARIABLES Correlation • Correlation allows one to compare two different groups using parameters of a normal distribution. Correlation Coefficient • Correlation coefficient “r” has a range from -1 to + 1 Calculation formula • r = Σ(zxzy) /N Assume the following data Use of correlation • Correlation coefficient allows one to account for the variation of trait 1 to the variation of trait 2. Caveat (warning) of correlation data • Does not allow for inferring causation INTERPRETING THE DATA Existing Data • One has existing data that shows high blood pressure is a consistent problem within class X people. • With in the class of X people, high blood pressure has a mean of 50 points higher than normal and a S.D. of ±5 points. Causal Interpretation of Data • Assert a hypothesis concerning the variable of interest. Hypothesis0 (null) • Drug A does not causes a significant drop in blood pressure for those people who have chronic high blood pressure Hypothesis1 (experimental hypothesis) • Drug A does causes a significant drop in blood pressure for those people who have chronic high blood pressure. Draw a sample of X people with high blood pressure • Note here, one already has for their disposition the Mean and S.D. of higher blood pressure for the Population of X people. Random Sample of 25 people from population X given Drug A • Measure the drop in blood pressure of those 25 selected people. Results • Mean drop in blood pressure after being given Drug A is 10 points with a S.D. 2.5. Question is Drug A effective? • Test the mean difference between that for the population from that of the sample. Calculate a z score • Since one has sampled the population of X, one wants to assure oneself that one has an unbiased estimate of the population that is represented by the sample. What calculating the z score does • The calculation of the z score forces the assumption that the mean of the blood drop is 0 and a S.D. of 1. One gains the unbiased estimate by correcting the S.D • SE (standard error) = • S.D./(N-1)-1/2 Critical ratio • Critical ratio = obtained mean • SE Numerically our example • SE = SD/(N-1)-1/2 • SE = 2.5/(24)-1/2 = 2.5/4.9 = 0.51 Sample – Population mean divided by SE 10 – 0/SE = 10/0.51 = 19.61 From a z distribution if the ratio is larger than 1.96 one calls that change significant. Go back to the two Hypotheses • Reject Hypothesis0 • Accept Hypothesis1 Confidence interval • A confidence interval is saying that within ±2 SE of the mean difference 95 % of the time one would find the mean of the sample.