RMS II Lecture 1: Review and Stats Intro

Central Limit Theorem

"The cornerstone for inferential statistics." Parametric tests such as t-tests, count on the Central Limit Theorem to be applied properly. *For any population with mean (u) and standard deviation alpha (o), the distribution of sample means for a sample size of n will approach a normal distribution with a mean of (u) and a standard deviation of alpha (o)/square root n as n approaches infinity. -This describes the distribution of sample means for any population, no matter what shape or mean or SD. -The distribution of sample means approaches a normal distribution very rapidly. The central limit theorem can be described in terms of shape, central tendency, and variability.

Transformation of the Scale for the SDTransformation of the Scale for the SD

(See formula, not memorize p. 16 Lec. 1 notes) 1. Adding or subtracting a constant to each score in a distribution will not change the standard deviation but will change the mean by the same constant. 2. Multiplying or dividing each score by a constant causes the standard deviation and the mean to be multiplied or divided by the same constant

Be able to read frequency distribution graphs!

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Symmetrical distribution

-A distribution in which the left side of the distribution mirrors the right side of the distribution. In this case, the mean, median and mode are all in the same point and lie in the middle. -No skew. (see graphs and histogram examples)

Skewed Distribution

-A distribution is skewed if one of its tails is longer than the other. The first distribution shown has a positive skew. That means that it has a long tail in the positive direction (to the right, towards higher numbers). The distribution has a negative skew if it has a long tail in the negative direction (towards the left, negative or smaller numbers)

Sampling Error

-The discrepancy or amount of error between the statistics calculated from the sample and its corresponding population parameters. -If you take multiple samples from same population, different samples should have different statistics: mean, standard deviation etc.

SD Indicates

-the average deviation from the mean -the consistency in the scores -how far scores are spread out around the mean

Bimodal distribution

4 types of Non-normal distributions: 1. Positively Skewed 2. Negatively Skewed 3. Bimodal: 2 modes (see example charts p. 18 Lecture 1) 4. Rectangular distribution

A sampling distribution

A distribution of statistics obtained by selecting all the possible samples of a specific size from the population. -Characteristics of sampling distributions: 1. Sample means tend s to pile up around the population mean. 2. Distribution of sample means is approximately normal in shape. 3. Distribution of sample means can be used to answer probability questions about sample means.

Platykurtic

A distribution with negative kurtosis is called platykurtic. In terms of shape, a platykurtic distribution has a smaller "peak" around the mean, that is a lower probability than a normally distributed variable of values near the mean. Has "thin tails", (that is a lower probability than a normally distributed variable of extreme values.

Parameter (Population parameter)

A number that describes a characteristic of a population of scores, symbolized by a Greek letter (alphabet)

Statistics (Sample statistics)

A number that describes a characteristic of a sample of scores, symbolized by an English letter (alphabet)

Leptokurtic

A positive kurtosis distribution. In terms of shape, leptokurtic distribution has a more acute "peak" around the mean, aka a higher probability than normally distributed variable of values near the mean. Has "fat tails", (which means a higher probability than a normally distributed variable of extreme values).

Experiment (true experiment)

A research procedure in which 1 independent variable is actively manipulated, the scores on the outcome variable (dependent variable) are measured, and all other variables are held constant to determine whether there is a relationship. It can determine a cause-effect relationship!

Correlational study

A research procedure in which subjects' scores on 2 variables are measured without manipulation of either variable, to determine whether there is a relationship. It CANNOT determine a cause-and-effect relationship!

Quasi-experiment

A research procedure in which the researcher does not directly manipulate the independent variable, nor randomly assign subjects to treatment conditions. It examines differences between preexisting groups of subjects or differences between preexisting conditions. The variable used to differentiate the groups is called the quasi-independent variable, and the score obtained for each individual is the dependent variable. Examples comparing PTSD patients versus healthy controls.

Sample

A subset of the population that is intended to represent. We want a representative sample—a sample that accurately represents the population.

Examples: T-test

ANOVA, regression.

Cumulative frequency

Accumulation of individual scores as one moves up the scale. You start with the lowest score (i.e. 4) and then sum all the frequencies of a particular score and all scores below it. Thus, the highest (i.e. 10) should have a cumulative frequency that equals the total sample size (i.e. 20).

Cumulative percentile rank

Accumulation of percent of all the scores as one moves up the scale. Percentage of individuals in the distribution with scores at or below the particular value. You start with the lowest score (i.e. 4) then divide the cumulative frequency of a particular score by the total sample size (i.e. 20). Note: sample SIZE = How many scores IN THE SAMPLE (NOT TOTAL OF ALL SCORES)

Transformation of the Scale for the Mean

Adding subtracting, multiplying, and dividing by a constant to each score in a set of scores (distribution) will always change the mean by the same constant. For more info: See Common symbols in descriptive stats (Lecture 1 p. 10)

Discrete variables

Consists of separate, indivisible categories. No values can exist between 2 neighboring categories

Measures of Central Tendency

Describe the center or midpoint of a distribution. The three most common measures of central tendency are Mean Median Mode. MEAN- (aka the average) Sum of scores divided by the number of scores. (see notes for Mean formula and symbols). MEDIAN- (aka the middle number) the score that divides the distribution exactly in half. Exactly 50% of individuals in a distribution have scores at or below the median. It is equivalent to 50th percentile. (see notes for formula). MODE- (aka the most commonly occurring score; score that occurs most often). The score or category in the distribution that has the greatest frequency. Mode works best for nominal classifications (i.e. favorite colors, political affiliation). Mode can be unimodal, bimodal, or multimodal. (i.e. if two scores occur with the same highest amount of times (i.e. a score of 9 and 10 each occur 12 times in the sample) (see notes for formula)

Range

Difference between most extreme (highest and lowest) scores in a distribution. R = 16-4 or | 4-16 | =12 OR sometimes people report the entire range, R = 4 - 16

Mesokurtic

Distributions with zero kurtosis (k = 0) are called mesokurtic. The most common example of a mesokurtic distribution is the normal distribution.

Type II error

False acceptance of the null hypothesis! Opposite error to Type I. Saying there is no relationship between IV and DV when in fact it DOES exist! False negative pregnancy test example.

Type I error

False rejection of the null hypothesis, and falsely accepting the alternative (hoped for) hypothesis as true. False positive HPT example. A stats decision-making error in which a large amount of sampling error causes the rejection of the null hypothesis when the null hypothesis is true. Saying a relationship exists between the independent and dependent variable when in fact it does NOT exist!

Rectangular distribution

Has mean, median and no Mode! Why? Because scores all have same frequency (of occurrence). Flat across top rectangular shape. (see graph p. 19)

Know the order of Mode

Mean, and Median for positive and negative skew, For positive skew, the order of measures of central tendency from left to right is mode, median, and mean. For negative skew, the order of measures of central tendency from left to right is mean, median, and mode (the opposite of positive skew).

Kurtosis

Measure of the "peakedness" or "flatness" of the distribution. Describes how fat or thin is the distribution. The degree to which a frequency distribution is flat (low kurtosis) or peaked (high kurtosis). Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly sized deviations. Thus, a high kurtosis distribution has a sharper "peak" and fatter "tails" while a low kurtosis distribution has a more rounded peak with wider "shoulders."

Alpha Level

Minimizing risk of a Type I Error Ho: The sample mean is the same as the population mean. -Example: depressed patients will score the same on mood ratings as the general population. If there is a big difference between data (sample mean) and the Ho (mean) we will likely reject Ho.

4 Types of Measurement Scales

NOMINAL SCALE Discrete; No true zero. Observations are labeled and categorized (names) (Random race bib numbers assigned to runners). Research example: Different diagnoses, Ethnicity. ORDINAL SCALE: Discrete; No true zero. Ranking observations in terms of size or magnitudes (rank order of race winners: 1st 2nd 3rd). Research example: Military or Starfleet rank, Ranks of professors. INTERVAL Scale: Continuous; No true zero. Equal differences (or intervals) between numbers on the scale reflecting differences in magnitude. (Ratios of magnitudes are not meaningful). Performance on a 0 - 10 scale. (i.e. Olympic judging 8.2; 9.1, 9.6). IQ Score, GPA. RATIO SCALE: Continuous. Ratio of numbers that do reflect reflect ratio of magnitudes. Has absolute zero point. Examples: height, number of items correct on a research comp test!

Frequency distribution

Organized tabulation of the number of individuals located in each category on the scale of measurement. Ex: List of scores: 8, 9,8,7,10,9,8,7,8,8,10,9,8,6,9,9,7,8,8, [ See frequency distribution table lecture 1 p. 5]

Why use df in sample SD?

Over many random samples the sample will underestimate the population variance. Sx and S2X are biased estimators and do not accurately estimate the true population variability. In fact, it underestimates the population parameter. We can correct this bias by subtracting 1 from the sample size; this relates to Degrees of Freedom (v or f).

Power

Power is the probability of correctly rejecting the null hypothesis when the null is false! It is the probability that a Type II error is not committed. The probability of power is 1-beta. It is ideally set at .80. Why? We want a high chance (at least 80%) of detecting a significant treatment or relationship.

Inferential statistics

Procedures for determining whether the sample data represents a particular relationship in the population. Examples: T-test, ANOVA, regression.

Descriptive Statistics

Procedures for organizing and summarizing data so that the important characteristics of the data can be described and communicated. Examples: Mean, Standard Deviation, Range

Semi-Interquartile range

Quartile means a quarter of the distribution. The semi-interquartile range is one-half the distance between the scores at the 25th and 75th percentile. Because the median and the semi-interquartile are based on percentiles they tend to be associated or described together.

Difference between population and sample SD formulas

Remember the ultimate goal is to describe the population of scores. But we can only estimate it using the sample based on the population.

**Definitional formula for Unbiased Sample Standard Deviation

Remember this formula ***see Lecture 1, p. p 13

Computational Formula for the Unbiased Sample Standard Deviation

Remember this formula also! (p. 13)

Formula for Semi-interquartile Range

Score at 75th percentile - Score at 25th percentile / divide by 2

Statistical Model

Statistical representation of the real world -In statistics we fit models to our data (use stats model to represent real world happenings). -General mathematical form: outcome = (model) + error

Distribution of sample means

The collection of sample means for all possible random samples of a particular size (n) that can be obtained from the population. (Most common type of sampling distribution). The shape of the distribution of sample means. The shape will always be a normal distribution if one of the following is true: 1. The population from which the samples are selected is a normal distribution 2. The number of scores (n) in each sample is relatively large, around 30 or more

Population

The entire group of interest. All members of the group that is to be studied.

Law of Large numbers

The larger sample size, the more probable it is that the sample mean will be close to the population mean.

A Simple Statistical Model: The Mean

The mean is a hypothetical value (it doesn't have to be a value that actually exists in the data set). As such the mean is a simple statistical model

Measuring the 'Fit' of the Model

The mean is a model of what happens in the real world: the typical score. It is not a perfect representation of the data. How can we assess how well the mean represents reality? Calculating Error -A deviation is the difference between the mean and an actual data point -deviation + X1 - Xbar (mean). Assessments of model fit: -Deviance -Sum of Squared errors (SS) -Variance -Standard deviation

Degrees of Freedom (ν or df)

The number of values used to estimate a parameter minus the number of parameters to be estimated. -Refers to the fact that all scores in a set of scores are free to vary EXCEPT for the last score. The last score is restricted if the mean (or the sum) and the number of scores are known. Thus, the CORRECT way to get the unbiased estimators of the population is to divide the squared deviations by N-1

Dependent variable

The response or outcome measure of an experiment. It is the selected behavior which is measured to gauge the effect of the independent variable. Aka criterion variable; Y variable.

The Standard Error of X

The standard deviation of the distribution of sample means is called the standard error of X (with a bar over it) = alpha symbol times X (with bar over x) (see p. 23 notes). Specifies precisely how well a sample mean estimates its population mean. -Numerical value of Standard Error is Determined by: 1. Standard deviation of the Population. When (alpha) is small, each individual score is close to the mean (U) (aka reverse u with tail on left) and the average score (X) is also close to the mean (u).. With a large SD, likely to obtain extreme scores that can increase the distance between X and u. 2. Sample size. The larger the sample size, the more accurately the sample represents the population.

Alternative Hypothesis

The stats hypothesis describing the population parameters that the sample data represent if the predicted relationship DOES exist. What we hope to accept! The symbol is H1

Null Hypothesis

The stats hypothesis describing the population parameters that the sample data represent if the predicted relationship does NOT exist. The symbol is Ho. This is what we assume at the outset of a study. This is what we hope to reject.

Independent variable

The variable manipulated or controlled by the experimenter. Also known as predictor and the X variable.

Factors Affecting Power

There are Many! 1. The alpha level. Increasing the alpha level will increase the power of a statistical test (i.e. moving alpha from .01 to .05). 2. One tailed versus 2- tailed alpha test. If you predict a directional hypothesis i.e. sample mean should be greater than the population mean) so that you predict the t-statistic score would be in the right side of the curve, you put the area of rejection all in the right side of the curve at .05 probability (versus in a non-directional two-tailed t-test when it is set at 0.25 (on either side). IF you predict correctly then you increase (double) the area of rejection and thus increase power. If you are wrong and the sample mean is in reality less than the population mean, you have a 0% chance of rejecting the null hypothesis. Thus, be every sure of your distribution and conduct a 2-tailed test. 3. Sample size. As sample increases, so does power! (Error is reduced with more subjects). 4. Reducing error variance will increase power! (i.e. test everyone in same quiet room instead of different rooms with different levels of noise). This would reduce the error thus increasing power. 5.Increasing effect size of the Independent Variable increases power! (i.e. doubling dose of an antidepressant would increase the difference between drug group versus placebo group (on depression scores) ... hopefully). 6. Greater subject variability DECREASES power (More differences between subjects results in more error). 7. Using match, versus within subject, versus between subject, versus mixed design in analyses with more than one independent variable. Why? Because we reduce error variance due to subject variability and within-subject variables. Within subject variables = when subject receive all conditions of that variable. Between subject variables are when subject gets only one (treatment) condition of the variable.

Distribution of scores by Standard Deviations on a normal distribution graph

Units stay true to the type of data (s in our data set will be buckets of popcorn sold at the movies). Using a standard normal distribution as seen in graph (see graph example notes p 12)

Measures of Variability (Range

Variance, Standard Deviation, Semi-interquartile range), Indicate how spread out the scores are. Measures of central tendency indicate the location of a distribution of scores. Measures of variability indicate the distances among the scores in the distribution. The common measures of variability are range, variance, standard deviation, and less common the semi-interquartile range.

The Relationship of Hypothesis Testing by Chart and by Diagram

[See lecture 1 p. 28-30 for full Chart & Diagram!] α = .025 α = .025 (region of rejection for extreme high values). Any value between the 2 vertical lines results in accepting H0. Extreme values (Probability = alpha if H0 is true). Possible but "very unlikely" outcomes.

Variance

a measure of the variability of the scores in a set of data. Computed as the average of the squared deviations of the scores around the mean. Finding the variance is not as easy as finding the mean. You cant simply add up the deviations (X - X) and then divide them by the number of scores. Why? Always = 0 Variance = SD squared! See Lecture 1, p. 11 for Sum of Squares formula: Sum of Squares (SS) =

Confounding variable (aka nuisance variable)

an UNCONTROLLED variable that is unintentionally allowed by the experimenter that affects the dependent variable.

f (Frequency)

frequency of a particular score

Continuous variables

have infinite number of possible values that fall between any two observations. There is a divisible into an infinite number of fractional parts.

Dichotomous variable

only 2 categories of a variable.

Standard Deviation

s for the sample, alpha for the population. A summary score describing how all scores vary from the mean (averagely). Also is the square root of the Variance (for the sample). The square root of the average square deviations for scores around the sample mean.

Confidence Intervals

•They are used to help us estimate or feel confident that the actual u (mean) is within a certain range of the mean of means from many samples. So that to be 68% confident that the confidence interval contains the parameter u (mean): •.68CI = X +/- alpha(x) (with line over the X and x) •The parameter would lie within 68% of this interval. But in reality we only take 1 sample. Thus we need to increase the confidence intervals. •For 90% confidence intervals: .90CI = X +/-1.645(alpha(x). (with lines over X and x again). •But 95% confidence interval most common! 95CI = X +/-1.96(alpha) (x) (lines over X and x again).