Stats 3xE3

correlational studies

measure correlations between predictor and criterion variables

subjects come with their own set of variables

experimental studies

studies in which the independent variables are directly manipulated and the effects on the dependent variable are examined

random assignment of subjects to experimental condition

descriptive statistics

numerical data used to measure and describe characteristics of groups.

Includes measures of central tendency (mean) and measures of variation (variance)

inferential statistics

using samples to make claims about the populations

population

all events, people, scores of interest

sample

a subset of the population

random sample

each member of the population has an equal chance of being selected

convenience sample

only members of the population who are easily accessible are selected

cons of correlational study

cannot establish causation, hard to know if the value of criterio is caused be a predictor, a lot of other variables may come into play

random assignment

assigning participants to experimental and control conditions by chance, thus minimizing preexisting differences between those assigned to the different groups

Causality

when we change the value of x, the probability of y occuring also changes

IVs

variables that are manipulated by the experimenter

DVs

key variables of interest that we measure and analyze

between-subjects design

A research design in which different groups of participants are randomly assigned to experimental conditions or to control conditions.

within-subjects design

participants are exposed to all levels of the independent variable

pros of within-subjects design

better control on the individual differences

no differences between two people

cons of within-subject design

practice effect, and boredom

2 group counter balance within subject design

divides your participants into two distinct groups to experience all experimental conditions in opposing orders, which controls for order and practice effects

categorical/ qualitative variables

lack numeical properites

can be nominal (no order)

can be ordinal (meaningful order)

numerical (quantitative) variables

have values that represent a counted or measured quantity

differences between levels are consistent and meaningful

zero point may be arbitrary or meaninful

line graph

used to show relation between quantitative measures

each value on x-axis has one data point

often, variable on x-axis is continuous

scatter plot

useful for visualising the association between two quantitative variables

each value on x-axis can have multiple data points

each dot represents one particpants

pie charts

bad because:

no common reference point for each slice

hard to trach change over time

requires clunky labels or legends

hated by colour blind people

skewness

Measure of asymmetry in data distribution.

kurtosis

the frequency/propability of scores that are far from the centre of distribution

high kurtosis

more outlier scores -> fatter tails

low kurtosis

light tails, lack of outliers

mode

the most frequently occurring score(s) in a distribution

if two adjacent scores occur with equal frequency- average of those two scores

if two non-adjacent scores occur with equal frequency distribution is bi-modal

cons of mode

- only gives info about a single score(s)

- sensitive to frequent extreme scores

- doesn't account for variability

- changing just one observation can change which value is the mode, even though the overall data barely changed.

pros of mode

- value will appear in the data set

- easy to understand

- robust to extreme scores

median

the middle score in a distribution; half the scores are above it and half are below it

- tells you how a score ranks within a distribution of scores

percentiles

the proportion of values in a sample that fall below a given value

median > mode

positive skew

mode < median

negative skew

pros of median

Not affected by extreme scores

stable even when mode is undefined

easy to define as the middle score

cons of median

difficult to use in statistical theorems and calculations

no simple formula

mean

average

pros of mean

-summarizes data in a way that is easy to understand

-uses all the data and is the most--used measure of central tendency

minimises the distances (deviations) from ach score (balance point)

every observation contributes to the mean

-best at minimizing squared errors

cons of mean

-affected by outliers

-values may not actually exist in data

trimmed means

means calculated on data for which we have discarded a certain percentage of the data at each end of the distribution

range

the difference between the highest and lowest scores in a distribution

cons of range

distorted by outliers

pros of the interquartile range

not sensitive to extreme scores

pearson r

A method of computing correlation when both variables are linearly related and continuous

- index of goodness of fit

-how close points are to line

linear relationship

A relationship that has a straight line graph

curvilinear

best fit line is characterized by curved lines

monotonic trend

as x increases y increases, might not be a straight line

non monotonic trend

As X increases, Y changes direction at least once

covariance

A measure of linear association between two variables. Positive values indicate a positive relationship; negative values indicate a negative relationship

covariance depends on

the sum on the products of deviation scores

cons of covariance

sensitive to the spread of x and y

depends on our choice of units

weak r

0.1 - 0.3 (-0.1 to -0.3)

moderate r

0.3-0.5 (-0.3 to -0.5)

strong r

0.5 - 1 (-0.5 t -1)

factors that affect correlation (r)

non-linearity

extreme scores

restricted range

heterogenous sub groups

simpson's paradox

when averages are taken across different groups, they can appear to contradict the overall averages

a trend that exists in several groups can disappear or reverse when groups are combined

pearson product moment correlation coefficient (r)

for two continuous variables

spearman's correlation coefficient for ranked data (rho, p, or ra)

two ranked/ ordinal variables

sort into ranks then you calculate r

point-biserial correlation (rpb)

1 dichotomous and 1 continuous variable

- correct/incorrect on a single MCQ vs total exam score)

calculate r

Phi Correlation (rφ)

two dichotomous variables

yes/no vs yes/no

calculate r

rs

measures monotonicity of X,Y association

rs is sensitive to

extreme scores affecting monotonicity

when to use rs

if your data is ranked and intervals are meaningless

point estimate

a summary statistic from a sample that is just one number used as an estimate of the population parameter

bootstrapping

using one sample to estimate how much your result might vary across repeated samples

r doesn't tell you anything about

slope or best fit line

Confidence Interval

the range of values within which a population parameter is estimated to lie

--% of the calculated intervals would be expected to contain the true parameter value from our population

p value

how unusual your observed result would be if the null hypothesis were true

describes how unusual your sample r is, assuming H0 is true.

type 1 error

Rejecting null hypothesis when it is true

Concluding there's evidence of a correlation when the population correlation is actually zero.

linear regression

a type of regression that models that relationship with a straight line when there's one predictor

R^2

the proportion (percent) of the variation in the values of y that can be accounted for by the least squares regression line

residual standard error

the standard deviation of residuals.

coefficients

the parameters of the regression line

multiple r-squared

Proportion of variance explained by predictors.

Multiple Regression

a statistical technique that includes two or more predictor variables in a prediction equation

normal distribution is defined by

standard deviation and mean

population standard deviation

how spread out individual values are around the population mean.

t-test

a statistical test used to evaluate the size and significance of the difference between two means

sampling distribution of the mean

the distribution of sample means over repeated sampling from one population

standard error of mean

the standard deviation of the sampling distribution of sample means

sampling distribution

probability/ frequency distribution of a sample statistic

sampling error

the difference between a sample statistic and the true population value, caused by which individuals happen to be sampled.

Central Limit Theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution.

type 2 error

we fail to reject the H0 when it is not true

there was actually a change

reject H0 (Type 1 error)

p = a

Do not reject Ho (H0 not true)

p = 1 - a

Reject H0 (H0 is false)

correct decision

Do no reject H0 (H0 is false)

p = b

Effect size

how large the real effect or difference is.

Larger effect size

Easier to detect; increases power and decreases β.

Statistical power

Probability of correctly rejecting a false null hypothesis.

1 - b

Alpha (α)

the probability of making a type I error

Beta (β)

probability of making a type II error

Lower alpha

Stricter rejection rule → more missed effects → higher b

Effect size can be measured

Difference between the true mean and the null mean

Larger sample size

More precise sample means → easier detection → lower b

Sample size and SEM

As n increases, SEM decreases

Smaller SEM

Sample means cluster more closely around the true population mean

factors affecting the probability of making a type 2 error

alpha level, effect size, and sample size