methods and measurement in psychology. statistics the description, organization and interpratation...
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Methods and Measurement in Psychology
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Statistics
• THE DESCRIPTION, ORGANIZATION
AND INTERPRATATION OF DATA
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DESCRIBING DATA
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SCALING
The method by
which one puts
numbers to variables.
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1. NOMINAL
• The most Primitive of all scales and is included by definition in all other scales.
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Criteria
• 1. NAMING OR POINTATABLE VARIABLE
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Criteria
2. NO NUMERICAL ANALYSIS POSSIBLE
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EXAMPLES:
• Drivers License
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Examp.
• Social Security Number
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Examp.
• Numbers On The Backs of Football Players
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Scale 2
• ORDINAL
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• The objects of a variable set can be rank ordered on some operationally defined characteristic.
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Ordinal Scale
• Rank order in terms of the magnitude of the variables i.e.
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• More of, or less of, one variable with respect to another variable.
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Requires the use of the nominal scale.
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Examples• Positions in a race; 1st, 2nd etc.
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The Scale You Are Most Familiar With
• GRADES
• A > B > C > D > F
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Problems With Ordinal Scales
• 1. No Zero point
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• 2. What is the magnitude of the distance between units of the scale
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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.
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High Ordered Metric Scale
• Tries to measure the distance between two ordinal variables
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Ideally, grades are equal distance from one another
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A > B > C > D > F
• You can take the test and get one of two grades, A or C.
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• You don‘t have to take the test and get a B.
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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.
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Subjective loss less than the subjective gain
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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 .
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Choose B for sure
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One Can Make The Same Comparisons Between Grades BC
and CD.
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• When One Makes All Of The Possible Choices, One Sees That The Distances Do Not Rank Order Themselves In Terms Of Magnitude.
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Scale 3
• INTERVAL SCALE
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• 1. Possesses all of the characteristics of the Nominal and Ordinal scale especially rank-order
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• 2. Numerically equal distance on the an interval scale means equal distance on the property being measured
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• There Must Be An Arbitrary Zero.
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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
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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
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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.
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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
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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
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ORGANIZING DATA
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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).
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Histogram
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Histogram of a set of scores
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Frequency Polygon of the same set of scores
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Frequency polygon plus histogram
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Measures of Central Tendency
• How common are you?
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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!
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Median
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Where is the word Median Used in Common Parlance?
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• Keep Off The Median – used in Highway Driving
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Mean
• Average
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Positively Skewed Distribution
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Positively Skewed Distribution
• Note how the positive numbers pull the mean to the right.
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Measures of Variability
• How unique are you------How scores differ one from another.
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• Range
• Lowest to highest
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Deviation
• The difference between a score and some constant measure
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• The constant can be any measure, but that which makes most sense is one of the measures of central tendency
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Deviation score
• X - MEAN = DEVIATION
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• Σ = sum of
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SUM OF MEANS
• Σ (X – MEAN) = 0
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How do I get rid of negative deviation scores
• SQUARE THE DEVIATION SCORES
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• Σ (X – MEAN)2 = 0
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VARIANCE
• Σ (X – MEAN)2
• N
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• Is there a way to compare the same individual on two different tests?
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Standard Scores
• z scores are called Standard Scores
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• COMPARE THE DEVIATION SCORE OF EACH TEST TO ITS STANDARD DEVIATION
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• z = (X – MEAN)
S.D.
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Characteristics of a z distribution
“z” DISTRIBUTIONS ARE CHARACTERIZED BY THE PARAMETERS OF A NOTRMAL CURVE
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• THE S.D. OF A z DISTRIBUTION = 1
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• THE MEAN OF A “z” DISTRIBUTION = 0
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A NORMAL CURVES OF IQ
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Normal Curves
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How the mean, median and mode are effected by skewness
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Three types of normal curvesdepends on range of x values
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DESCRIBING THE RELATION BETWEEN TWO VARIABLES
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Correlation
• Correlation allows one to compare two different groups using parameters of a normal distribution.
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Correlation Coefficient
• Correlation coefficient “r” has a range from -1 to + 1
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Calculation formula
• r = Σ(zxzy) /N
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Assume the following data
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Use of correlation
• Correlation coefficient allows one to account for the variation of trait 1 to the variation of trait 2.
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Caveat (warning) of correlation data
• Does not allow for inferring causation
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INTERPRETING THE DATA
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Existing Data
• One has existing data that shows high blood pressure is a consistent problem within class X people.
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• 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.
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Causal Interpretation of Data
• Assert a hypothesis concerning the variable of interest.
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Hypothesis0 (null)
• Drug A does not causes a significant drop in blood pressure for those people who have chronic high blood pressure
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Hypothesis1 (experimental hypothesis)
• Drug A does causes a significant drop in blood pressure for those people who have chronic high blood pressure.
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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.
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Random Sample of 25 people from population X given Drug A
• Measure the drop in blood pressure of those 25 selected people.
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Results
• Mean drop in blood pressure after being given Drug A is 10 points with a S.D. 2.5.
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Question is Drug A effective?
• Test the mean difference between that for the population from that of the sample.
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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.
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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.
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One gains the unbiased estimate by correcting the S.D
• SE (standard error) =
• S.D./(N-1)-1/2
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Critical ratio
• Critical ratio = obtained mean
• SE
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Numerically our example
• SE = SD/(N-1)-1/2
• SE = 2.5/(24)-1/2 = 2.5/4.9 = 0.51
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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.
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Go back to the two Hypotheses
• Reject Hypothesis0
• Accept Hypothesis1
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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.