**Dr Michelle E. Kelly**

**PS104 Statistics Revision**

**Descriptive**

**Inferential**

**Central Tendency**

Mean

**Dispersion/ Variability**

Variance

Characterises the spread or dispersion of scores

i.e. how spread out they are and how similar they are to the center of the data.

Usually the variance is reported in tables which summarise a variable along with other statistics such as the mean and range.

There is one problem with our variance formula in estimating the population variance

The sample variance tends to underestimate the population variance .... to correct for this, we calculate the variance estimate

Variance formula using N - 1

Standard Deviation

Normal Distribution

z

- scores

The most common measure of variability..

Standard deviation (SD) measures the amount of variation from the average.

When reporting mean – also report SD

Mean = 4 (

SD

= 2.58)

In everyday life many variables are normally distributed.

We use this information to make assumptions about the way our populations are distributed.

Many statistical tests are based on the assumption that data are normally distributed.

A z-score tells us the number of standard deviations which a particular score is above or below the mean of the set of scores .

Standardizing scores so they can be compared meaningfully.

Z-scores allow us to determine how each score compares to the other scores in a data set...

How well did one student perform in an English test compared to the other 50 students?

Which students came in the top 10% of the class?

x - mean

z =

_________

standard deviation

Standard Error

Sampling distribution

The standard deviation of the means of samples is the standard error of the mean:

The degree to which sample means deviate from the mean of your sample means

• Standard error of the mean = estimated standard deviation

---------------------------------

square root of sample size

**Describe data**

Make inferences about population based on sample

**Parametric**

**Non - Parametric**

**Pearson's r**

**Independent t-test**

**Dependent t-test**

**Spearman's Rho**

**Mann Whitney U-test**

**Wilcoxon Matched Pairs Test**

**Chi Square Test**

Measure of association between nominal variables

SCORE

DATA

1. Ranked

2. Category

Violate assumptions

Correlation

Relationship

Difference

Spearman’s Rho

- association between two sets of ordinally ranked data.

Categorical data are analysed using

Chi-square

(or

Fishers Exact Probability, Yates Correction, Odds Ratio)

.

Mann Whitney, Wilcoxon Matched Pairs

examine differences between ranked data.

Non-parametric statistics generally test hypotheses involving ordinal rankings of data or frequencies...

They make no assumptions about the populations or the shape of their distributions (unlike parametric tests).

‘Distribution free’

Non-Parametric Tests

Assumptions of parametric test violated...

The formulae for parametric tests involve calculations of means and standard deviations.

Parametric tests make assumptions about the characteristics of the populations from which samples are drawn.

'Powerful'

Parametric Tests

Pearson's

Product Moment Correlation Coefficient - relationship between variables

Independent t-test

- difference between two groups (sets of means)

Dependent t-test

- difference between related sets of means (time point 1 and 2)

Correlation

‘What is the relationship between stress and heart disease?’

Correlational studies

Want to know whether the variables vary together i.e. is there an association or correlation between them?

'Alcohol consumption and reaction time'

'Maths scores and music ability'

Pairs of values - want to determine if there is a positive or negative correlation between the two.

Correlation Coefficient

Pearson’s r

tells us two major pieces of information:

(1) How close the points on a scatterplot fit the best-fitting

straight line.

(2) Whether the slope of the scatterplot is positive or negative.

'Single numerical index'

'How strong is the relationship?'

Draw Scatterplot!

Interpreting and reporting results

-0.90 indicates a very strong negative relationship

Mathematical scores were significantly negatively correlated with musical scores, r(8) = -0.90, p<0.05.

Calculate Pearson’s r, note the sample size and consult the table to obtain the corresponding critical value.

Ignoring the sign, your value must be equal to or greater than the critical value to be statistically significant...

Identical to Pearson’s r except that instead of taking the scores directly from your data, the scores are ranked from smallest to largest.

Spearman’s rho is Pearson’s r calculated on ranked scores (it is the non-parametric counterpart of Pearson’s r)

Apply the same Pearson’s

r

formula!

Report and interpret the results the same way too (using Spearman's significance tables)!

Chi-square

measures the relationship/association or differences between two categorical variables.

Correlation coefficient - used to assess association between two variables measured on an ordinal or interval scale.

'Differences between observed and expected frequencies...'

Calculating expected frequencies

“ The chi-square value of 53.6 (

df

=3) was found to have an associated probability value of <0.05. Thus, we can accept that there is a significant difference between the observed and expected frequencies and we can conclude, that the four brands of chocolate are not equally popular. The table shows that more people (n=60) prefer Snickers to the other brands.

Alternatives to Chi-square

Combine categories

The Fisher Exact Probability Test (!)

Yates’ correction

The expected cell frequencies rule whereby no expected frequency should fall below 5 - can be addressed in three ways:

Phi Coefficient - standard 2x2

Cramers V - larger than 2x2

'Strength of association?'

Independent t-test

Comparisons between two groups of scores:

Each group of scores is obtained from two separate groups of individuals

‘unrelated’ or ‘between-participants’

A difference is statistically significant at a certain level (of

df

) only if the observed value of

t

= or e

xceeds the table value.

"There was no significant difference between times taken to sort into two as opposed to four piles, t (17) = 1.241, p > 0.05."

Random allocation of participants into experimental vs control.

Dependent

t

-test

'Related or within participants'

Scores are obtained from the same individuals but on two separate occasions / Matched pairs.

Compare scores at time 1 and 2

Non-parametric alternative to the independent t-test.

Assess whether a statistically significant difference exists between two independent samples of rank-ordered data.

It can be applied when: the data are ordinal; were randomly selected; and tied ranks are dealt with appropriately.

The Mann-Whitney U-test

Wilcoxon matched-pairs test

Examines whether a difference between 2 dependent samples of ordinal rankings is significant.

(1) The difference scores show a very asymmetrical distribution

(2) There are outliers in the difference scores (i.e. a small number of difference scores are very different from the majority).

T-Test Notes

Robust - guards against Type I Errors

Smaller standard error values tend to occur when sample sizes are large, two conditions that lead to larger

t

statistics.

The power of a t-test is influenced by: (a) the selected significance level (eg. p< 0.01 or p<0.05); (b) variability within the sample data; (c) the size of a sample(s); and (d) the magnitude of the difference between means.

Related designs have the distinct statistical advantage of reducing error variance, also need fewer participants.

Dependent

t

-test

Counterbalance to reduce risk of carryover effects.

In contrast to between-grps designs, related designs draw error variance from one source (e.g. one grp of participants) rather than two (e.g. two independent samples of participants)…..

This leads to a smaller standard error in the t-test which means there is a greater likelihood of rejecting the null hypothesis (i.e. more likely to obtain statistically significant result).

Error variance - differential behaviour of participants within the samples as well as experimental error.... Dependent designs reduce this!