Descriptive Statistics: Variability Lesson 5

Theories & Statistical Models n Theories l Describe, explain, & predict real- world events/objects n Models l Replicas of real-world events/objects l Can test predictions ~

Models & Fit n Model not exact replica l Smaller, simulated n Sample l Model of population l Introduces error n Fit l How well does model represent population? l Amount of error in model l Good fit  more useful ~

Models in Psychology n My research model l Domestic chicks l Effects of pre-/postnatal drug use l Addiction & its consequences n Who/What do most psychologists study? l Rats, pigeons, intro. psych. students n External validity l Good fit with real-world populations? ~

The General Linear Model n Relationship b/n predictor & outcome variables form straight line l Correlation, regression, analysis of variance l Other more complex models ~

The Mean as a Statistical Model n Very simple model l 1 number represents all the observations l Often hypothetical value u e.g., mean # friends = 2.6 n Error introduced l Actual # friends = mean + error n Deviation (deviance) l ~

Distributions: 3 useful features n Summarizes important characteristics of data n 1. What is shape of the distribution? n n 2. Where is middle of distribution? n 3. How wide is distribution?

Assessing the Fit of the Mean n How well does it represent all observations? l On average near or far from mean? u Distance from mean l Or width of distribution n Variability l How much do scores vary from the mean? ~

  For which group is the mean a better fit for the data? Mean Daily Temperature

Measures of Variability n Deviation: for a single score n Range l Highest value – lowest value + 1 n Standard deviation l Conceptually: mean of all deviation scores l average distance of scores from mean n Variance l Used to calculate standard deviation l Also used in analysis of variance ~

From the Dictionary n Deviation: departure from a standard or norm. n Variance: the state, quality, or fact of being variable, divergent, different, or anomalous. n Error: a deviation from accuracy or correctness n Variability: something that may or does vary; a variable feature or factor n Variation: something that may or does vary; a variable feature or factor ~

Calculating the Standard Deviation n Why only conceptually mean of deviation scores? n If n What is mean deviation?  (X i –  ) = 0 ~ XiXi X i - 

4 Steps to Standard Deviation n 1. Calculate deviation scores n 2. Sums of squared deviations l Or Sums of squares (SS) n 3. Variance l mean of squared deviations (MS) n 4. Standard deviation l square root of variance ~

Standard Deviation (SD) n Conceptually mean deviation score for all data l Gives width (dispersion) of distribution n Describing a distribution l Report mean & standard deviation  ~

Samples & Variability n Usually study samples u to learn about populations l Sampling introduces error l Change symbols & formula

Samples: Degrees of Freedom (df) n df = N – 1 l For a single sample (or group) n s tends to underestimate s l Fewer Xi used to calculate l Dividing by N-1 boosts value of s n Also used for l Confidence intervals for sample means l Critical values in hypothesis testing ~

Level Of Measurement & Variability n Which can be used? n nominal l none n ordinal l range only n interval/ratio l all 3 OK l range, standard deviation, & variance ~

Statistical Models n Representation of the population l We will focus on linear models n Mean is a simple model l One number represents all data l Both n Standard deviation l measures fit of model l Better fit  more useful l Smaller ~