Stats OHPs 3

8/6/2019 Stats OHPs 3

http://slidepdf.com/reader/full/stats-ohps-3 1/22

Avoiding Damned Lies – Understanding Statistical Ideas, Alan Dix www.meandeviation.com 26

measures of averages

and variation

• lies, damn lies and ...mode(s), med ian and mean

• square peoplevariance and stand ard

deviation




average?three typ ical measures:

• mode(s):“more people use dogo than any other dog food”

• med ian“half of all salaries are greater than £15000 p .a.”

• mean“if salaries were d ivided evenly . . .”




income £

nos of peoplethree modes

mode(s)

• not w id ely u sed

• may have morethan one mod e

• the bump maybe anyw here!

• sensitive




J. Bloggs 3500F. Mole 5600K. Giles 8000J. Smith 8300B. Roberts 8450S. Claus 8450A. Jones 8680H. Lee 15750M. Warren 17500

T. Smyth-Boule 20000028423

mediansalary£8450

meansalary!

sensitivity of mean

• one big value ...• union qu otes

median• employer the

mean

• lies, d amn lies ...




why use the mean?

• median is more robust• mean is more manipulable

numberof people

meansalary

mediansalary

group 1 10 15000 12500

group 2 10 23000 16000grp 1 & grp 2 19000 ?




measures of variation

inter-quartilerange = 14–9

27

81 01 11 21 21 3

1 31 51 82 31 2 mean

2 1 0 1 0 07 5 2 58 4 1 6

1 0 2 4

1 1 1 11 2 0 01 2 0 01 3 1 11 3 1 11 5 3 91 8 6 3 62 3 1 1 1 2 1

4 2 6

differencefrom m ean

square of difference

varianceaverage

difference

stand ard d eviationσ = √ variance




which is best?

a bit like averages . . .• inter-quartile range is robust

• variances ad d u p

• standard deviations meaningful




square people

if data is people buying ‘d ogo’

variance is 26 square people!

stand ard deviation

σ = √ variance= 5.1 p eop le




the ‘real’ world

• the sample – actual measured d ata• the population

– large set from w hich the data is draw n

– especially for su rveys etc.

• the id eal

– the ‘typical’ user, the fair coin– unrepeatable events – the fall of a raind rop– a theoretical d istribution




the job of statisticsreal world

sample data

measurement

statistics!




different means

x average of the measured d ata~ sample mean

y average of the ‘real’ world~ popu lation mean

z theoretical mean of the ‘d istribu tion’e.g. mean d ie score = 3.5




real mean

sample mean

estimator

µ

µ

estimating the mean

sample mean estimatesreal (popu lation) mean




real mean

sample mean

estimator

µ

µ

strange but true

the meanof the mean

is the mean

i.e. theoretical meanof sample mean

is real mean!!!!!




real mean

sample mean

estimator

µ

µ

law of large numbers

if samples are ind epend ent(or nearly so)

bigger sample ⇒ better estimate




how good an estimate

• each d ata item has some variabilityhead=1/ tail=0: 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 1 1

• sum s of data items have variabilitynos of head s: 12 11 9 13 8 8 8 11 8 11

• means of data item have variabilityaverages: 0.6 0.65 0.45 0.65 0.4 0.4 0.4 0.55 0.4 0.55

better = less variability




variability of sumsvariances ad d up*:

variance(sum of 100 items)= 100 × variance(each item)

standard deviation = √ variances.d . of sum of 100 items= 10 × s.d . each item

square root ru le: σ(n items) = √ n σ(eachitem)

i.e. bigger, bu t prop ortionately less

* only if items are ind epend ent (actually closely related to Pythagoras' theorem!)




variability of mean

mean is sum/ nos. of items:σ(mean of 100 items)= σ(sum each item)/ 100= σ(each item) / 10

square root ru le for means:σ(mean of n item s)* =

1

√ n σ(each item)

* called stand ard error (s.e.) of mean




so what?

experiments, d ata collection etc....

to halve the variationneed 4 times as many subjects




solved it?x seeing through rand omness

use sample mean as estimator

y know ing w hen you haveσ(mean) = σ(item)/ √ n

? w hat is σ(item)estimate it from sample!




real variance

sample data

estimator

σ2

Σ µ(x- ) 2

n–1

estimating σ(item)

use sample variance/ s.d .as estimateof real variance

N.B. only an estimate

OK . . . bu t a tid bit small on average(biased estimator)

✰ that’s why stats. form ulae are fu ll of √ n-1






drunkard's walk

• a d ru nk w and ers hom e❖"""" sometimes he takes one step forw ard s

sometimes one step back ❙

? after n stepshow far is he from w here he started

! another example of √ n behaviour

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Stats OHPs 3

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