This poster is one of a series of three, designed by Stella Dudzic. The series includes: Hypothesis tests for one sample, Hypothesis tests for two samples, and Experimental Design and Hypothesis tests for several samples: ANOVA (Analysis of Variance)To view the other two posters and to place orders for these and for extra sets of all three posters, please visit the MEI website at www.mei.org.ukThe set of three posters is also available in a simplified format (on CD) to print in A4 size for student folders or for use on a whiteboard. Included on the CD are test statistics for one and two samples, and worked examples for analysis of variance (ANOVA).

Are the samples from populations with equal variance?

Use replication, i.e.get several values for each level of the factor

One-way analysis of variance (one between subjects factor)

Are the populations Normal (at least approximately)?

Testing whether all means are equal

Are there any “nuisance” factors?

How many factors of key interest are there?

Might the “nuisance” factors interact with each otherand/or the factor of interest?

NoHow many “nuisance” factors are there?

Use each level of the “nuisance” factor as a block

Is it possible to include each level of the factor of interest in each block?

Randomised block design. Possibly replication

Are the samples from populations with equal variance, which are at least approximately Normal?

Testing whether all means are equal, for each factor

Analysis of variance for randomised blocks

Beyond the scope of this poster, possibly Friedman's two-way analysis of variance by rank

Are the number of levels the same for all three factors?

Testing whether all means are equal, for each factor

Analysis of variance for Latin square

How many “nuisance” factors are there?

Are there any “nuisance” factors?

Are the samples from populations with equal variance and at least approximately Normal?

Have you used replication?

Might the factors interact with each other? Two-way analysis of

variance (two between subjects factors)

For each combination of factors, does the population have the same variance and is it at least approximately Normal?

Possibly balanced incomplete blocks or partially balanced incomplete blocks

Kruskal-Wallis one-way analysis of variance

One

Two

Yes

No

Yes

No

Yes

No

Yes

No

One

Two

Yes

No

Yes

No

Yes

No

Yes

No

One

Yes

No

Yes

No

YesYes

No

YesNo

No

More advanced techniques needed (e.g. transformations or General Linear Model). Beyond the scope of this poster

Analysis beyond the scope of this poster

Latin square design

Specialised design (possibly Graeco Latin square)

Specialised design beyond the scope of this poster

Analysis beyond the scope of this poster

No suitable common testMore than two

Use a two way factorial design with randomisation and, possibly, replication

Analysis similar to that for two factors of key interest

Possibly factorial design Analysis beyond the scope of this poster

Two or more

More than two

Beyond the scope of this poster

Use a two way factorial design with randomisation and, possibly, replication

Beyond the scope of this poster Beyond the scope of this poster

Are you prepared to assume that the factors do not interact?

Testing whether all means are equal, for each factor

Testing whether all means are equal, for each factor, and whether interactions between factors exist

No simple general procedure - beyond the scope of this poster

Two-way analysis of variance(no interaction)

Two-way analysis of variance, with interaction interpreted as residual

See

www.winterolympics.external.bbc.co.uk/event-results-schedules/index.htmlfor results from theWinter Olympics

This poster is also available as a download to print in A4 size for student folders or for use on a whiteboard. Please visit www.mei.org.uk for more information.This poster is one of a series of three, produced by Stella Dudzic for MEI.

This poster is one of a series of three, designed by Stella Dudzic. The series includes: Hypothesis tests for one sample, Hypothesis tests for two samples, and Experimental Design and Hypothesis tests for several samples: ANOVA (Analysis of Variance)To view the other two posters and to place orders for these and for extra sets of all three posters, please visit the MEI website at www.mei.org.ukThe set of three posters is also available in a simplified format (on CD) to print in A4 size for student folders or for use on a whiteboard. Included on the CD are test statistics for one and two samples, and worked examples for analysis of variance (ANOVA).

Are the data single variable or bivariate?

Single variable

Test on variance

Test of proportion Binomial test or Normal approximation

For Normal population 2 test for variance�

Are the data in a frequency table with expected frequency of at least 5 for each group?

Goodness of �t test

Yes test or Kolmogorov-Smirnov 2�

No Kolmogorov-Smirnov test

Categories in a contingency table

Bivariate data

Spearman’s rank correlation test or Kendall’s rank correlation test

No

Yes Pearson’s product moment correlation test

Are the variables categories or numbers? Number pairs Are the data from a bivariate

Normal distribution?

2 test�

Test on mean/median

Do you have a large sample?

Symmetrical distribution

Poisson

What distribution are the data from?

Yes Do you know the variance?

Yes Normal test

No 2Estimate variance as

and use Normal tests

No

No

Are the data from a Normal distribution?

Yes Yes

Do you know the variance?

Other Sign test

Wilcoxon single sample test

Poisson test

Normal test

2Estimate variance as and use test

st

Hypothesis tests for one sample

Contingency table male female

right handed 32 28

left handed 7 5

Test statistic for Kolmogorov-Smirnov test

N(0, 1) probability density

No

With a large set of data, the scatter diagram for a bivariate Normal distribution is approximately elliptical.

Test statistics for all these tests can be found at www.mei.org.uk/teachersupport There are some specialised tests which are not included on this poster. © MEI 2008

Use replication, i.e.get several values for each level of the factor.

No

Test on mean/median

Do you have a large sample?

Do you know the variance?

Normal test

Estimate variance as s²and use Normal test

Normal test

Estimate variance as s²and use t test

Are the data from a Normal distribution?

Do you know the variance?

What distribution are the data from?Are the

data single variable or bivariate? Test on

varianceFor Normal population

Test of proportion

Goodness of fit test

Poisson

Symmetrical Distribution

Other

Poisson test

Wilcoxon singlesample test

Sign test

Binomial test or Normal approximation

Pearson’s productmoment correlation test

Spearman’s rankcorrelation test or Kendall’srank correlation test

Test statistics for all these tests can be found at www.mei.org.uk/teachersupportThere are some specialised tests which are not included on this poster.

Are the datafrom a bivariate Normal distribution?(see fig 2)

Number pairs

Are thevariablescategories or numbers?

Bivariatedata

Singlevariable

Categories in a contingency table (see fig 1)

With a large set of data, the scatter diagram for a bivariate Normal distribution is approximately elliptical

Test statistic for Kolmogorov-Smirnov test

x0

0.25

0.5

0.75

1

cum

ulat

ive

pro

bab

ility

observed

EXPECTED

D

N(0,1) probability density

0

No

Yes

Yes

No

Yes

No

Yes

No

Yes

Contingency table

Male female

righthanded

lefthanded

32 28

7 5

fig 2 fig 3

fig 4.

fig 1

38 has come up 213 times to end March 2010 but 20 has only come up 148 times. See www.lottery.co.uk/statistics/ for data. You could use a goodness of fit test to check if there is evidence that the lottery is not fair.

38 has come up 213 times to end March 2010 but 20 has only come

up 148 times

See www.lottery.co.uk/statistics/

for data

You could use a goodness of fit test to check if there is evidence that the lottery is not fair

test for variance

test or Kolmogorov-Smirnov (see fig 3)

test

Are the data in a frequency table with expected frequency of at least 5 for each group? Kolmogorov-Smirnov testNo

Yes test or Kolmogorov-Smirnov (see fig 3.)

In the National lottery any one ticket has exactly the same odds of winning as any other ticket right?....

If the national lottery was fair, we wouldexpect each number to be chosenthe same number of times, right?

This poster is also available as a download to print in A4 size for student folders or for use on a whiteboard. Please visit www.mei.org.uk for more information.This poster is one of a series of three, produced by Stella Dudzic for MEI.

This poster is one of a series of three, designed by Stella Dudzic. The series includes: Hypothesis tests for one sample, Hypothesis tests for two samples, and Experimental Design and Hypothesis tests for several samples: ANOVA (Analysis of Variance)To view the other two posters and to place orders for these and for extra sets of all three posters, please visit the MEI website at www.mei.org.ukThe set of three posters is also available in a simplified format (on CD) to print in A4 size for student folders or for use on a whiteboard. Included on the CD are test statistics for one and two samples, and worked examples for analysis of variance (ANOVA).

Are yoursamplesmatched?

Unpairedsamples

Matched (paired)samples

Test on difference ofmeans/medians

Do youhave largesamples?

To do a test on paired samples, first find the differences between paired data values and then proceed as for

a single sample test

Are thedifferencesNormallydistributed?

Do you knowthe variance of the differences?

Do you knowthe variance of the differences?

Are thedifferencessymmetricallydistributed?

Are thevariancesequal?

Are the data from distributions with the same shape?

Do you know thevariances?

Are the datafrom Normaldistributions?

Do youhave largesamples?

Test on difference of means/medians

Testing whether they are from the same distribution

Test ondifference of variances

Do you know thevariances?

No

Yes

No

Yes

No

Yes

No

Yes

No

Yes

No

Yes

No

Yes

No

YesNo

YesNo

Yes

N(0,1) probability density

0

Normal test

Normal test

Estimate variance of differencesusing s² and use Normal test

Estimate variance of differencesusing s² and use t test

Wilcoxon pairedsample test

Sign test

Kolmogorov-Smirnov 2-sample test

Normal test

Estimate variances usings²,s² and use Normal test

Normal test

t test with pooledestimate of variance

No suitable simple test

Wilcoxon rank sum testor Mann Whitney U test

No suitable simple test

F test

No

Yes

In the National lottery any one ticket has exactly the same odds of winning as any other ticket right?....

If the national lottery was fair, we wouldexpect each number to be chosenthe same number of times, right?

Explananation of how it works etc etc, Explananation of how it works etc etc, Explananation of how it works etc etcExplananation of how it works etc etcExplananation of how it works etc etc

A survey of TV watching habits is conducted with the following results

Number of hours of TV watched per week Sample size Sample mean Sample varianceWomen 50 11.2 135.2Men 60 9.6 66.9

Does this provide evidence thatthere is a difference in the mean number of hours of TVwatched by men and women?