Psyc 3000 Week 1 - Measurement Levels PDF

Measurement Levels Descriptive Statistics Normality Psyc 3000 Week 1  Statistics, Science and Observations “Statistics” means “statistical procedures” Uses of Statistics Organize and summarize information Determine exactly what conclusions are justified based on...

Measurement Levels Descriptive Statistics Normality Psyc 3000 Week 1  Statistics, Science and Observations “Statistics” means “statistical procedures” Uses of Statistics Organize and summarize information Determine exactly what conclusions are justified based on the results that were obtained Goals of statistical procedures Accurate and meaningful interpretation Provide standardized evaluation procedures  Populations and Samples Population The set of all the individuals of interest in a particular study Vary in size; often quite large Sample A set of individuals selected from a population Usually intended to represent the population in a research study Relationship between population and sample Variables and Data Variable – Characteristic or condition that changes or has different values for different individuals Data (plural) – Measurements or observations of a variable Data set –A collection of measurements or observations A datum (singular) –A single measurement or observation –Commonly called a score or raw score  Parameters and Statistics Parameter Statistic A value, usually a numerical value, A value, usually a numerical value, that describes a population that describes a sample Derived from measurements of Derived from measurements of the individuals in the individuals in the population the sample  Descriptive & Inferential Statistics Inferential statistics Descriptive statistics Study samples to make Summarize data generalizations about the Organize data population Simplify data Interpret experimental Familiar examples data Tables Common terminology Graphs “Margin of error” Averages “Statistically significant”  Sampling Error Sample is never identical to population Sampling Error The discrepancy, or amount of error, that exists between a sample statistic and the corresponding population parameter Example: Margin of Error in Polls “This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points” A demonstration of sampling error Role of statistics in experimental research  Data Structures, Research Methods, and Statistics Individual Variables A variable is observed “Statistics” describe the observed variable Category and/or numerical variables Relationships between variables Two variables observed and measured One of two possible data structures used to determine what type of relationship exists  Relationships Between Variables Data Structure I: The Correlational Method One group of participants Measurement of two variables for each participant Goal is to describe type and magnitude of the relationship Patterns in the data reveal relationships Non-experimental method of study Data structures for studies evaluating the relationship between variables  Correlational Method Limitations Can demonstrate the existence of a relationship Does not provide an explanation for the relationship Most importantly, does not demonstrate a cause-and-effect relationship between the two variables  Relationships Between Variables Data Structure II: Comparing two (or more) groups of Scores One variable defines the groups Scores are measured on second variable Both experimental and non-experimental studies use this structure Data structure for studies comparing groups  Experimental Method Goal of Experimental Method To demonstrate a cause-and-effect relationship Manipulation The level of one variable is determined by the experimenter Control rules out influence of other variables Participant variables Environmental variables The structure of an experiment  Independent/Dependent Variables Independent Variable is the variable manipulated by the researcher Independent because no other variable in the study influences its value Dependent Variable is the one observed to assess the effect of treatment Dependent because its value is thought to depend on the value of the independent variable  Experimental Method: Control Methods of control – Random assignment of subjects – Matching of subjects – Holding level of some potentially influential variables constant Control condition – Individuals do not receive the experimental treatment. – They either receive no treatment or they receive a neutral, placebo treatment – Purpose: to provide a baseline for comparison with the experimental condition Experimental condition – Individuals do receive the experimental treatment  Non-experimental Methods Non-equivalent Groups – Researcher compares groups – Researcher cannot control who goes into which group Pre-test / Post-test – Individuals measured at two points in time – Researcher cannot control influence of the passage of time Independent variable is quasi-independent  Variables and Measurement Scores are obtained by observing and measuring variables that scientists use to help define and explain external behaviors The process of measurement consists of applying carefully defined measurement procedures for each variable  Constructs & Operational Definitions Constructs Operational Definition –Internal attributes –Identifies the set of or characteristics operations required to that cannot be measure an external directly observed (observable) behavior –Useful for –Uses the resulting describing and measurements as both explaining behavior a definition and a measurement of a hypothetical construct  Discrete and Continuous Variables Discrete variable Has separate, indivisible categories No values can exist between two neighboring categories Continuous variable Have an infinite number of possible values between any two observed values Every interval is divisible into an infinite number of equal parts  Scales of Measurement Measurement assigns individuals or events to categories The categories can simply be names such as male/female or employed/unemployed They can be numerical values such as 68 inches or 175 pounds The complete set of categories makes up a scale of measurement Relationships between the categories determine different types of scales  Scales of Measurement Scale Characteristics Examples Nominal Label and categorize Gender No quantitative distinctions Diagnosis Experimental or Control Ordinal Categorizes observations Rank in class Categories organized by size or Clothing sizes (S,M,L,XL) magnitude Olympic medals Interval Ordered categories Temperature Interval between categories IQ of equal size Golf scores (above/below par) Arbitrary or absent zero point Ratio Ordered categories Number of correct answers Equal interval between Time to complete task categories Gain in height since last year Absolute zero point  Statistical Notation Statistics uses operations and notation you have already learned Statistics also uses some specific notation Scores are referred to as X (and Y) N is the number of scores in a population n is the number of scores in a sample  Summation Notation Many statistical procedures sum (add up) a set of scores The summation sign Σ stands for summation – The Σ is followed by a symbol or equation that defines what is to be summed – Summation is done after operations in parentheses, squaring, and multiplication or division. – Summation is done before other addition or subtraction  Defining Central Tendency Central tendency A statistical measure A single score to define the center of a distribution Purpose: find the single score that is most typical or best represents the entire group Locate Each Distribution “Center”  Central Tendency Measures Previous figure shows that no single concept of central tendency is always the “best” Different distribution shapes require different conceptualizations of “center” Choose the one which best represents the scores in a specific situation  The Mean The mean is the sum of all the scores divided by the number of scores in the data. Population: =  X N Sample: M=  X n  The Mean: Three Definitions Sum of the scores divided by the number of scores in the data The amount each individual receives when the total is divided equally among all the individuals in the distribution The balance point for the distribution Mean as Balance Point  The Weighted Mean Combine two sets of scores Three steps: Determine the combined sum of all the scores Determine the combined number of scores Divide the sum of scores by the total number of scores Overall Mean = M =  X 1 +  X2 n1 + n2  Computing the Mean from a Frequency Distribution Table Quiz Score (X) f fX 10 1 10 9 2 18 8 4 32 7 0 0 6 1 6 Total n = Σf = 8 ΣfX = 66 M = ΣX / n = 66/8 = 8.25  Characteristics of the Mean Changing the value of a score changes the mean Introducing a new score or removing a score changes the mean (unless the score added or removed is exactly equal to the mean) Adding or subtracting a constant from each score changes the mean by the same constant Multiplying or dividing each score by a constant multiplies or divides the mean by that constant Mean is Highly Sensitive to Changes in Scores  The Median The median is the midpoint of the scores in a distribution when they are listed in order from smallest to largest The median divides the scores into two groups of equal size  Locating the Median (odd n) Put scores in order Identify the “middle” score to find median 3 5 8 10 11 “Middle” score is 8 so median = 8  Locating the Median (even n) Put scores in order Average middle pair to find median 1 1 4 5 7 9 (4 + 5) / 2 = 4.5  Median, Mean, and “Middle” Mean is the balance point of a distribution Defined by distances Often is not the midpoint of the scores Median is the midpoint of a distribution Defined by number of scores Often is not the balance point of the scores Both measure central tendency, using two different concepts of “middle”  The Mode The mode is the score or category that has the greatest frequency of any score in the frequency distribution Can be used with any scale of measurement Corresponds to an actual score in the data It is possible to have more than one mode Bimodal Distribution  Selecting a Measure of Central Tendency Measure of Appropriate to choose if … Could be misleading if… Central Tendency Mean You can calculate ∑X Extreme scores You know the value of every Skewed distribution score Undetermined values Open-ended distribution Ordinal scale Nominal scale Median Extreme scores Nominal scale Skewed distribution Undetermined values Open-ended distribution Ordinal scale Mode Nominal scales Interval or ratio data, except Discrete variables to accompany mean or median Describing shape Showing Large Gaps in Data Means or Medians in a Line Graph Means or Medians in a Bar Graph  Central Tendency and the Shape of the Distribution Symmetrical distributions Mean and median have same value If exactly one mode, it has same value as the mean and the median Distribution may have more than one mode, or no mode at all Figure 3.10  Central Tendency in Skewed Distributions Mean, influenced by extreme scores, is found far toward the long tail (positive or negative) Median, in order to divide scores in half, is found toward the long tail, but not as far as the mean Mode is found near the short tail. If Mean – Median > 0, the distribution is positively skewed. If Mean – Median < 0, the distribution is negatively skewed Skewed Distributions  Overview of Variability Variability can be defined several ways A quantitative distance measure based on the differences between scores Describes distance of the spread of scores or distance of a score from the mean Purposes of Measure of Variability Describe the distribution Measure how well an individual score represents the distribution Population Distributions  Three Measures of Variability The Range The Variance The Standard Deviation  The Range The distance covered by the scores in a distribution From smallest value to highest value For continuous data, real limits are used range = URL for Xmax — LRL for Xmin Based on two scores, not all the data An imprecise, unreliable measure of variability  Standard Deviation and Variance for a Population Most common and most important measure of variability is the standard deviation A measure of the standard, or average, distance from the mean Describes whether the scores are clustered closely around the mean or are widely scattered Calculation differs for population and samples Variance is a necessary companion concept to standard deviation but not the same concept  Defining the Standard Deviation Step One: Determine the Deviation Deviation is distance from the mean Deviation score = X — μ Step Two: Find a “sum of deviations” to use as a basis of finding an “average deviation” Two problems Deviations sum to 0 (because M is balance point) If sum always 0, “Mean Deviation” will always be 0. Need a new strategy!  Defining the Standard Deviation (continued) Step Two Revised: Remove negative deviations First square each deviation score Then sum the Squared Deviations (SS) Step Three: Average the squared deviations Mean Squared Deviation is known as “Variance” Variability is now measured in squared units Population variance equals mean (average) squared deviation (distance) of the scores from the population mean  Defining the Standard Deviation (continued) Step Four: Goal: to compute a measure of the “standard” (average) distance of the scores from the mean Variance measures the average squared distance from the mean—not quite our goal Adjust for having squared all the differences by taking the square root of the variance Standard Deviation = Variance Variance and Standard Deviation Variance and Standard Deviation  Population Variance Formula sum of squared deviations Variance = number of scores SS (sum of squares) is the sum of the squared deviations of scores from the mean Two formulas for computing SS  Two formulas for SS Definitional Formula Computational Formula Find each deviation Square each score and sum score (X–μ) the squared scores Square each deviation Find the sum of scores, score, (X–μ)2 square it, divide by N Sum up the squared Subtract the second part deviations from the first SS =  ( X −  ) 2 SS =  X − ( 2 X) 2 N  Caution Required! When using the computational formula, remember… SS   X 2 SS  ( X ) 2 X 2  ( X ) 2  Population Variance: Formula and Notation Formula Notation Variance is the average of squared deviations, so we SS identify population variance = variance with a lowercase N Greek letter sigma SS squared: σ2 standard deviation = N Standard deviation is the square root of the variance, so we identify it with a lowercase Greek letter sigma: σ  Standard Deviation and Variance for a Sample Goal of inferential statistics: Draw general conclusions about population Based on limited information from a sample Samples differ from the population Samples have less variability Computing the Variance and Standard Deviation in the same way as for a population would give a biased estimate of the population values Population of Adult Heights  Sample Variance and Standard Deviation Sum of Squares (SS) is computed as before Formula for Variance has n-1 rather than N in the denominator Notation uses s instead of σ SS variance of sample = s = 2 n −1 SS standard deviation of sample = s = n −1 Histogram for Sample of n = 8  Degrees of Freedom Population variance Mean is known Deviations are computed from a known mean Sample variance as estimate of population Population mean is unknown Using sample mean restricts variability Degrees of freedom Number of scores in sample that are independent and free to vary Degrees of freedom (df) = n – 1  More About Variance and Standard Deviation Mean and standard deviation are particularly useful in clarifying graphs of distributions Biased and unbiased statistics Means and standard deviations together provide extremely useful descriptive statistics for characterizing distributions  Showing Mean and Standard Deviation in a Graph For both populations and samples it is easy to represent mean and standard deviation Vertical line in the “center” denotes location of mean Horizontal line to right, left (or both) denotes the distance of one standard deviation Showing Means and Standard Deviations in Graphs  Sample Variance as an Unbiased Statistic Unbiased estimate of a population parameter Average value of statistic is equal to parameter Average value uses all possible samples of a particular size n Corrected standard deviation formula (dividing by n-1) produces an unbiased estimate of the population variance Biased estimate of a population parameter Systematically overestimates or underestimates the population parameter  Biased and Unbiased Estimates Sample Statistics (Unbiased) Biased Unbiased Sample 1st Score 2nd Score Mean Variance Variance 1 0 0 0.00 0.00 0.00 2 0 3 1.50 2.25 4.50 3 0 9 4.50 20.25 40.50 4 3 0 1.50 2.25 4.50 5 3 3 3.00 0.00 0.00 6 3 9 6.00 9.00 18.00 7 9 0 4.50 20.25 40.50 8 9 3 6.00 9.00 18.00 9 9 9 9.00 0.00 0.00 Totals 36.00 63.00 126.00 Mean of 9 unbiased sample mean estimates: 36/9 = 4 (Actual μ = 4) Mean of 9 biased sample variance estimates: 63/9 = 7 (Actual σ2 =14) Mean of 9 unbiased sample variance estimates: 126/9=14 (Actual σ2 =14)  Standard Deviation and Descriptive Statistics A standard deviation describes scores in terms of distance from the mean Describe an entire distribution with just two numbers (M and s) Reference to both allows reconstruction of the measurement scale from just these two numbers (Sample) n = 20, M = 36, and s = 4  Transformations of Scale Adding a constant to each score The Mean is changed The standard deviation is unchanged Multiplying each score by a constant The Mean is changed Standard Deviation is also changed The Standard Deviation is multiplied by that constant  Variance and Inferential Statistics Goal of inferential statistics is to detect meaningful and significant patterns in research results Variability in the data influences how easy it is to see patterns High variability obscures patterns that would be visible in low variability samples Variability is sometimes called error variance Experiments with high and low variability  Equations Concepts? 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