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Statistical Methods (Daniels), Table Example, How do I figure this out? …

- - - - Sampling Error
        The naturally occurring difference (discrepancy) between a statistic and a parameter.
        Example
        The average verbal SAT score for the entire class of entering freshmen is 530. However, if you select a sample of 20 freshmen and compute their average verbal SAT score you probably will not get exactly 530.
        The concept used to explain this natural difference between a sample mean and the corresponding population is SAMPLING ERROR.
  - - - Hypothetical Constructs
        Intangible characteristics which cannot be directly observed (intelligent, anxious, hungry, smoker, etc)
        
        Operation Definition
        Describes a set of operations for measuring a construct via external behavior and defines the construct in terms of the resulting measurements.
    - - Variables
        A variable changes or has diff values for the characteristic/condition. In contrast, a constant does not vary in value for the c/c.
        Independent Variables (IV) are manipulated by the researchers.
        Dependent Variables are observed for changes (e.g. to assess effective treatment)
        Quasi-independent are used to create multiple groups of subjects (cannot randomly assign participants) and uses a non-manipulated variable to define conditions that are being compared; usually a subject or time variable.
        Confounding Variables are uncontrollable, not part of the experiment but will have impact on the study.
        
        Systematic differences between groups = evidence that changing the IV causes a
        change in the DV.
        
        Populations = Parameters
        Samples = Statistic
        Both usually numerical. May be obtained either from a single measurement or derived from a set of measurements
        
        Obtaining Samples
        
        Random Selection or sampling, the process of obtaining a sample that requires every individual a the same change of being selected.
        
        When individuals need to be put into groups:
        
        Random Assignment is the process of assigning members of sample to groups so that each member has an equal chance of being in any group
      - Research Designs
        
        Correlational Method: Two variables are observed to see if there is a relationship.
        
        Experimental Method: One (IV) is manipulated while changes are observed in another (DV).
        
        Experimental Treatments
        
        Control Condition DO receive treatment
        
        Experimental Condition DO NOT receive treatment
        
        Data (plural) are measurements or observations
        Datum (singular) a single measurement or observ, commonly called score/raw score
        Notations
        
        English letters used to represent stats.
        
        Greek letters used to represent parameters.
        
        X = raw scores
        N = how many scores in a population
        n = how many scores in a sample
        Σ = summation
- - - - Stem and Leaf Display
        
        The stem and leaf display takes the data
        and simplifies it.
        Ex: There are two numbers in the data starting with 3: 32 and 33. The first number goes in the stem side, and the second half goes on the leaf side.
        This allows you to get a sense of what the shape of the data is without a graph.
    - - Determining Percentile/Ranks
        Find the # of individuals who are located at/below each point in the distribution with a FD table.
        The resulting values are called cumulative frequencies (CF) because they represent the accumulation of individuals as you move up the scale.
        
        Interpolation: provides a method for finding estimated intermediate values, or values that are located between two specific numbers. p.76 example
        
        To find Cumulative Percentage
        C% = CF/N x 100
  - - - Shape of a F.D.
        Describes the distribution of the graph:
        
        Symmetrical: Mirrors each other when divided in half.
        
        Bi-modal: Contains 2 different frequency peaks that are still symmetrical.
        
        Skewed: Scores pile up on one end.
        
        Negatively Skewed: The tail of the distribution is piled up on the left/high side.
        
        Positively Skewed: The tail of the distribution is piled up on the right/low side.
- - - - *Ex: When data is already in a frequency distribution table, create a new column of fX
        Then add all the scores under f
        Then add all the scores under fX
        9:20 sec in first lecture vid
    - - 3. Mode
        The score or category that has the greatest frequency. Also useful bc it's the only measure of CT that corresponds to an actual score in the data. So on a FD table, look at the f column to find the highest #.
        (The mean & median are calculated values & often produce an answer that does not = a score in the distribution)
        Find the score that was repeated the most (highest frequency).
        It is possible to have more than one mode. 2 modes = bimodal, 3 modes = multimodal.
      - Reporting Measures of Central Tendency
        Measures of CT are commonly used in behavioral sciences to summarize and describe the results of a research study.
        
        These values may be reported in text describing the results, or presented in tables or in graphs.
        
        APA tables have no
        vertical lines.
        
        Undetermined Values
        A situation in which an individual has an unknown score. Often occurs when measuring number of errors/amount of time required to complete a task. Impossible to compute mean. Appropriate method: MEDIAN.
        Open-Ended Distributions
        The MEDIAN can calculate the CT.
        Ordinal Data
        The MEDIAN is always preferred to calculate the CT.
        Extreme Scores + Skewed Distr
        A situation when there are scores that are quite different in value from most others. Appropriate method: MEDIAN
        When to use MODE
        For nominal scales, discrete variables, describing shape. Always identifies an actual score. Usually reported in narrative text.
        
        Graphs
        
        Can be used to show means or medians using a line graph, histogram, or bar graph - depending on the scale of measurement used for the IV.
        
        *The height of a graph is 2/3 to 3/4 the length of the graph.
        X and Y intersect should be zero (unless zero is an actual score).
        
        If a graph is symmetrical:
        
        The mean = the median
        
        The median is exactly at the center bc exactly half of the area in the graph will be on the other side of the center. (Mdn)
        
        The mean is also exactly at the center bc each score on the left is balanced by a corresponding score on the right.
        
        If a symmetrical distribution has only 1 mode, it will also be in the center of the distribution.
        
        A bi-modal symmetrical distribution will have the mean & median together in the center, with the mods on each side.
        
        A rectangular distribution has no mode bc all X values occur with the same frequency.
        
        If the graph is ROUGHLY symmetrical, but not perfect, the mean and median will be close together in the center of the distribution.
        
        If a graph is skewed:
        The mode is located toward the side where the scores pile up and the mean is pulled toward the extreme scores in the trail. The median is usually located between these two values.
        
        In a positively skewed graph (to the left): most likely order of the 3 measures of CT from smallest to largest (left to right) is the mode, median, and mean.
        (Mean > Med > Mode)
        
        In a negatively skewed graph (to the right), the most probable order is mean, median, mode.
        (Mode > Med > Mean)
  - - - Open ended distributions A MEDIAN can be found to calculate the CT.
        Ordinal Data The median is always appropriate and preferred to find the CT.
        Extreme Scores / Skewed Distributions A situation when there are scores that are very different in value from most others. *Appropriate method: MEDIAN
        
        When to use MODE
        For nominal scales, discrete variables, describing shape.
        Always identifies an actual score.
        Usually reported in narrative text.
- - - - ♥ To calculate the standard deviation / standard distance from the mean:
        
        Computate the population mean: µ = EX/N
        
        Find the deviation for each score. (X-µ)
        
        Square each deviation and add them.
        
        Find the avg of the squared deviations (called "variance").
        
        Take the square root of the variance to finally get the standard deviation.
        
        Variance = mean squared deviation =
        sum of squared deviations / # of scores
        aka: Variance: M = Σ (X - µ)² / N
  - - - When Dealing with a SAMPLE
        The basic assumption of that samples represent populations. This poses a special problem for variability bc samples consistently tend to be less variable than populations.
        
        Bias in sample variability is consistent and predictable, so it can be corrected.
        
        Sample Variance = s²
        
        Calculate the sum of squares (SS) (lowercase n) using the computational formula.
        
        Find the sample variance using: s² = SS / n-1
        
        Finally, take S= square root of s²
        
        Quick Recap
        SS is the sum of squared deviations.
        Variance is the mean squared deviation.
        Standard Deviation is the square root of variance.
    - - Presenting Mean & Standard Deviation in a Graph
        
        In frequency distribution graphs, we identify the position of the mean by drawing a vertical line and labeling it with m or M.
        
        Bc the standard deviation measures distance from the mean, it would be presented by a line or an arrow drawn from the mean to outward for a distance equal to the S.D. and labeled with an s or an S.
        
        See: Box Plot
- - - - The z-Score can be used to establish a relationship between the score, mean, and standard deviation which can be used to answer other questions*
        
        Example 5.4. In a population with a mean of µ = 65, a score of X = 59 corresponds to z = -2.00. What is the standard deviation for the population?
        You can subtract the X of 59 from the mean µ = 65 to determine that it is located by a distance of 6 points (65-59 = 6).
        Since the X score is -2.00 s.d.'s from the mean, you can divide 6 points by 2, and see that 1 standard deviation is 3.
        Example 5.5. In a population with a s.d. of σ = 6, a score of X 33 corresponds to z = +1.50. What is the mean of the population?
        A z-score of +1.50 indicates the score is located above the mean by 1.5 deviations. With an s.d. of 6, this distance is (1.5)(6) = 9 points. Therefore, the score is located 9 points above the mean. The score x=33 is 9 points above the mean, so 33-9 is µ = 24.
        
        Example 5.6
        In a population distr, a score of X = 54 which corresponds to z = +2.00 and a score of X = 42 which corresponds to z = -1.00. What are the values for the mean and standard deviation for the population?
        Focus on the distance between the two scores. Notice the distance can be measured in points AND in standard deviations.
        In points/raw scores, the distance from X = 42 and X = 54 is 12 points.
        According to the two z-scores, X = 42 is located 1 standard deviation below the mean, and X = 54 is located 2 standard deviations above it. Therefore, the raw scores are separated by 3 standard deviations.
        So, if the distance is 12 points which equal 3 standard deviations, as an equation:
        3σ = 12 points
        Divide both sides by 3.
        Standard deviation σ = 4 points
        Finally, X=42 is 1 deviation below the mean.. so the mean must be µ = 46 (bc 42 - 4 = 42).
  - - - Note: Remember when calculating for samples rather than population, to add the n-1 to your SS formula. s = SS / n - 1
        
        In inferential stats, z-scores provide an objective method for determining how well a specific score represents its population. A z-score near 0 indicates that the score is close to the population mean, and therefore representative. A z-score beyond 2 indicates that the score is extreme and is noticeably different from others in the distribution.
- - - - Example 1
        In a FD of 10 scores: 1,1,2,3,3,4,4,4,5,6 What is the probability you would randomly select a score above 4?
        p(X > 4) = ?
        There are 2 scores meeting that criteria, so 2/10 ~ 0.2 ~ 20% chance.
        What is the probability you'll randomly select a score less than five?
        p(X < 5) = 8/10 = 0.8 = 80% chance
        Example 2
        What is the probability of randomly selectin an individual with an IQ score less than 120 (given the mean µ = 100 and sd σ = 15).
        
        Find the z-score of the raw score (120) using the z-score formula. z = 1.33
        
        Draw out the distribution, mark 1.33 and shade everything less than that. So we know we are looking for a body.
        
        Look at the Unit Normal Table, find a z-score of 1.33 and find the B column proportion = .9082
        
        Example 3
        If you start with the proportion and want to know what score corresponds with the top 10% of IQ tests. Start with proportion of probability, calc a z score, plug that into the table.
        
        Revisiting Percentile/Ranks:
        
        The percentile rank for a specific X value = the % of individuals with scores AT OR BELOW that value.
        
        When a score is referred by its rank, the score is called a percentile.
        
        Random sampling with replacement: When a sampling unit is drawn from a finite population and is returned to that population, after its characteristic(s) have been recorded, before the next unit is drawn, the sampling is said to be “with replacement”.
  - - - When you take a lot of different samples, you expect the means to "pile up" around the µ.
    - - The Binomial Distribution
        To answer probability questions about 2 points of data (e.g., male and female, heads or tails):
        
        The two categories are identified as A and B.
        
        The probabilities/proportions
        
        p = p(A) = the probability of A
        q = p(B) = the probability of B
        
        Notice that p + q = 1.00 because A and B are the only possible outcomes.
        
        The # of individuals/observations in the sample are identified by n.
        
        The variable X refers to # of times category A occurs in the sample.
        
        Notice that X can have any value from 0 to n (all the sample is in category A).
        
        Example
        Consider a coin toss. There are 2 possible outcomes:
        p = p(heads) = 1/2
        p = p(tails) = 1/2
        If we are looking at a sample of n = 2 coin tosses and the variable interest is x = number of heads up...
        4 possible outcomes when you toss 2 times:
        Heads / Heads - (both heads)
        Heads / Tails - (each sequence has 1 head / 1 tail)
        Tails / Heads - (each sequence has 1 tail / 1 head)
        Tails / Tails - (no heads)
        
        Only 1 of the outcomes has 2 heads, so the probability of obtaining 2 heads is p = 1/4.
        
        Only 2 of the outcomes leads to 1 head, so probability of 1 head is p = 2/4 = 1/2.
        
        The probability of no heads (x = 0) is p = 1/4.
        
        What is the probability of obtaining at least 1 head in 2 coin tosses? 3 times out of 4 sequence possibilities. 3/4
        
        Binomial Distribution tends to approximate a normal distribution, especially when n is large.
        The BD will be a nearly perfect normal distribution when:
        q(n) and p(n) are equal to or greater than 10.
        Mean: µ = pn
        S.D. σ = sqr rt of npq
        Within this normal distr, each value of X has a corresponding z score:
        z = x-µ/σ = x - pn / sqr rt of npq
        This means we can compute probability values directly from z-scores and the UNT.
      - Standard Error of M (aka variability): σM
        Determine this by σ / sqr rt of n
        The magnitude of the standard error is determined by 2 factors:
        
        The size of the sample - the law of large #s states the larger teh sample size (n), the more probable the sample mean will be close to the population mean.
        
        The standard deviation of the population where the sample is selected from: there is an inverse relationship between the sample size and standard error.
        
        Law of large numbers: The larger the sample size (n) the more probable the M will be close to the µ.
        Bigger samples have smaller errors. At the extreme, the smallest possible sample (n=1), the DOSM is identical to the original distribution of scores and the S.D. for the DOSM is identical to the S.D. for the distribution of scores. PUT SIMPLY:
        if n=1, the s.d. of σM is identical to σ.
        Standard error measures the standard distance between a sample mean (M) and the population mean (µ). It tells how much error to expect if you're using the M to represent a population.
        note that the standard error decreases in relation to the square root of the sample size
        The location of each M in the DOSM can be specified by a z-score:
        z = M - µ / σM
        
        Page 198 Example 6.10 to see translating a z-score into an x value and to remember that 1.28 z score is equal to 1.28 points above the sd and if 1 sd = 10, then (10)1.28 = 12.8 points which you then add to the mean.
- - - - Example 8.1 - Hypothesis: Men give waitresses who wear red shirts bigger tips.
        For a sample of n = 36 male customers.
        Is the mean of tips for waitresses in red shirts close to the population mean of 15.8% (the mean regardless of color shirt)?
        If yes: the hypothesis is incorrect. The red shirt had no effect. It is a null hypothesis, or H0:µ
        Or was it different? If there was an effect, whether positive or negative, it is known as a alternative hypothesis, or H1:µ
        To formalize the decision process, use the null hypothesis to predict:
        (A) the kind of sample mean that ought to be obtained to be consistent (very close) with the NH and
        (B) the sample means which would be at odds (very different) with the NH.
        
        The Alpha Level: α
        A probability value that is used to define the concept of "very unlikely" in a hypothesis test. Also known as the level of significance.
        
        Commonly used alpha levels are α = .05 (5%), .01 (1%), and.001 (0.1%).
        
        For example, with an α = .05, we separate the most unlikely 5% of the sample means from the most likely 95% (or central values).
        
        The critical region is a boundary composed of the extreme sample values defined by the alpha level. They are extremely unlikely values which are not consistent with the null hypothesis.
        Whenever data from a research study produces a sample mean µ that is located in the critical region, we can conclude the data are not consistent with the null hypothesis and we reject it.
        To determine the boundaries, use the alpha level and the UNT. (Ex: If the alpha is 5%, that is split between the two 2.5% extreme sides of the distribution. You would look up 2.5 or 0.0250 in the Tail column.
        
        (Step 2) We construct the DOSM that would be expected if the H0 were true (the entire set of sample means could be obtained if the treatment has NO effect).
        (Step 3) Collect Data and compute sample stats. In the red shirt example, compute a z-score that describes exactly where the sample mean is located in the hypothesized distribution (relative to the H0).
        z = M - µ / σM where M is obtained from the sample data and µ is obtained from the H0 null hypothesis. This can be restated as: z = sample mean - hypothesized pop mean / standard error between M and µ
        (Step 4) Make a decision using the z-score obtained in the previous step, there are 2 possible outcomes:
        
        Sample data located in critical region, very unlikely and not consistent with the H0 so we reject the null hypothesis. Meaning we are saying the H0 is incorrect - because the treatment DID have an effect (it is far away from the µ).
        
        Sample data NOT located in critical region, reasonably close to the pop mean specified in the H0, therefore we fail to reject the null hypothesis because the treatment DID NOT have an effect.
  - - - The variability of scores and the size of the sample can influence the outcome of a HT.
        (1) In HT, higher variability (i.e., a higher standard deviation, which produces a larger standard error and a smaller z-score value closer to zero) can reduce the chances of finding a significant treatment effect.
        (2) The # of scores in the sample. In general, increasing the sample size produces a smaller standard error and a larger value for the z-score, increasing the likelihood of finding a significant treatment effect.
        Assumptions for HT with z-Scores
        
        Random Sampling
        
        Independent Observations (one observation is not influenced by another, and sampling with replacement)
        
        The value of σ is unchanged by the treatment.
        
        Normal Sampling Distribution
    - - 8.5 Concerns About HT: Measuring Effect Size
        HT makes relative, rather than absolute, comparisons between the size of the treatment effect relative to the standard error. If the standard error is small, then the treatment effect can also be small and still large enough to be significant.
        A significant effect does not necessarily mean a big effect.
        A measure of EFFECT SIZE is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used.
        Cohen's d standardizes the measure of the mean difference in terms of the standard deviation. It is computed as: Cohen's d = mean difference / standard deviation ... or:
        Cohen's d = µ(treatment) - µ(no treatment) / σ
        This formula measures the distance between two means and is typically reported as a positive number even when the formula produces a negative value.
        Cohen's d describes the size of the treatment effect and is not influenced by the number of scores in the sample
        
        A larger standard deviation decreases the likelihood of rejecting the null hypothesis and decreases the value of Cohen’s d.
        A large sample and small value for σ is likely to reject the null hypothesis even if the treatment effect is very small.
        
        8.6 Statistical Power
        The power of a stats test is the probability that the test will correctly reject a false null hypothesis. That is, power is the probability that the test will identify a treatment effect if one really exists.
        Researchers calculate power as a means of determining whether a research study is likely to be successful before conducting the study.
        Power and effect size are related.
        Power is influenced by factors such as:
        
        Sample Size
        
        Alpha Level
        
        One-tailed vs Two-tailed Tests
- - - - Confidence Intervals
        Another technique used to describe the treatment effect. It creates an interval/range of values that extend around the sample statistic.
        Based on the Central Limit Theorem, your sample mean is the most reflective of the population mean. Therefore, we can create an interval around it that gives us some level of confidence that the pop mean should fall within a certain range.
        To calculate CI:
        
        The first step is to select a level of confidence and look up the corresponding t values in the t distribution table.
        
        This value, along with M and sM are plugged into the estimation formula:
        
        µ = M +/- t(sM)
        
        Reporting Confidence Intervals
        Value CI [LL, UL]
- - - - Cohen's d for an independent-measure design:
        estimated d = M1 - M2 / sqr rt of s2p
        Evaluating Effect Size:
        d = 0.2 is a small effect
        d = 0.5 is a medium effect
        d = 0.8 is a large effect
- - - - Create Source Table to organize calculations and keep track of various computations necessary to compute F. "Source" refers to source of variance to be analyzed.
    - - Sum of Squares
        SS(tot) = EX^2 - (EX)^2/N
        Correction Factor
        (EX)^2/N
        SS(bg)
        
        SS(wg)
        
        Mean Square
        
        MS(variance) = s^2 = SS / df
        
        F Statistic
        
        Then:
        Find the Crit Region using the F Distribution Table. .05 are in light text, .01 are in bold text. The numerator is df(BG) and the denominator is the df (WG)
      - Make a decision about the hypothesis.
        Compare the observed f-statistic with the critical region f-statistic.
        
        If no significance, stop here.
        If there IS a significance, calculate Tukey's HSD.
        
        Compare the Groups
        
        MAKE FINAL CONCLUSIONS: is there, or is there not, a significant difference between each group?
- - - - Sum of Products measures co-variability between two variables.
        Note: create a scatter plot to make a preliminary estimate of the correlation.
        
        SSx = ΣX^2 - (ΣX)^2/n
        SSy = ΣY^2 - (ΣY)^2/n
        SP = ΣXY - (ΣX)(ΣY)/n (sum of products)
        r = SP / sqr rt of SSx SSy
        
        r for a sample, rho or ρ for population
    - - Uses of correlations
        Prediction, Validity, Reliability, Theory Validation.
      - Interpreting Pearson's r
        Range restrictions (lowers r, sampling problem
        Outliers (can increase or decrease r)
        Strength of Relationship
        Coefficient of determination:
        
        r^2
        
        proportion of variability in one variable that can be determined from the relationship with another
        
        a correlation of r = .80 means r^2 = .64 or 64% of the variability in the Y scores can be predicted from the relationship with X.
        
        Cohen (1977):
        
        r = .50 is a strong correlation
        
        r = .30 is moderate
        
        r = .10 is small
        
        Regression
        A statistical procedure that determines the equation for the straight line that best fits a specific set of data. The resulting straight line is called a regression line.
        
        Steps for the Least Squares Solution
        Determines how well a line fits the data points, the first step is to define the distance between the line and each data point.
        
        Y - y hat to find the error/distance between the predicted and actual value
        
        Bc some distances may be negative, each distance should be squared to make a positive average.
        
        Total squared error = Σ(Y - Y hat)^2
        
        Y hat = bX + a to find the predicted value of y hat for any X.
        
        .
        
        Calculate standard error of estimate-
        SSresidual = Σ(Y - Y hat)^2
        The obtained SS value is then divided by its degrees of freedom.
        Variance = SS / df