Quick and Simple Statistics Peter Kasper. Basic Concepts Variables & Distributions Variables &...
17
Quick and Simple Quick and Simple Statistics Statistics Peter Kasper Peter Kasper
-
Upload
verity-allison -
Category
Documents
-
view
222 -
download
2
Transcript of Quick and Simple Statistics Peter Kasper. Basic Concepts Variables & Distributions Variables &...
Quick and Simple Quick and Simple StatisticsStatistics
Peter KasperPeter Kasper
Basic ConceptsBasic Concepts
Variables & DistributionsVariables & Distributions Mean & Standard DeviationMean & Standard Deviation Estimators & ErrorsEstimators & Errors Comparing Two Sample ResultsComparing Two Sample Results SignificanceSignificance t-Testst-Tests
Variables & Variables & DistributionsDistributions A Variable is anything that can A Variable is anything that can
be ..be ..– Measured (continuous variable)Measured (continuous variable)
E.g. The height of vegetation in an areaE.g. The height of vegetation in an area
– Counted (discreet variable)Counted (discreet variable) E.g. The number of birds in an areaE.g. The number of birds in an area
– Categorized (categorical variable)Categorized (categorical variable) E.g. Birds that are E.g. Birds that are grassland specialistsgrassland specialists
or or notnot grassland specialistsgrassland specialists
Variables & Variables & DistributionsDistributions Each variable has a “natural” Each variable has a “natural”
distribution i.e. the frequency at distribution i.e. the frequency at which particular values of the which particular values of the variable occurvariable occur
A common example is the “Bell A common example is the “Bell Curve”Curve”– In general can be arbitrary!In general can be arbitrary!
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
A “Bell Curve” distribution
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Another “Bell Curve” distribution
Different “average” value and different width
Mean & Standard Mean & Standard DeviationDeviation
The mean of a sample of The mean of a sample of nn measurements of some variable measurements of some variable is defined as ..is defined as ..
μμ = ∑= ∑ii vvii / / nn
The “standard deviation” or The “standard deviation” or width of the distribution is ..width of the distribution is ..
σσ 22 = ∑= ∑ii ( (vvii - - μμ ) )22 / ( / ( n n – 1 )– 1 )
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
For a Bell Curve distribution
68% of values are within 1 of the mean
95% of values are within 2 of the mean
Mean 2.82 Standard Deviation 0.385 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 2.70 Standard Deviation 0.4810 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 2.96 Standard Deviation 0.6520 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 2.81 Standard Deviation 0.6840 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 2.93 Standard Deviation 0.7180 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 2.93 Standard Deviation 0.73160 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 2.96 Standard Deviation 0.77320 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 3.00 Standard Deviation 0.73640 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 2.99 Standard Deviation 0.721000 Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Mean 3.00 Standard Deviation 0.71infinite Measurements
0
20
40
60
80
100
120
0 1 2 3 4 5
Estimators & ErrorsEstimators & Errors
The values for The values for μμ and and σσ from from finite samples vary even if the finite samples vary even if the samples are part of the same samples are part of the same distributiondistribution
They are Estimators of the They are Estimators of the “true” values.“true” values.
The “error” on the estimators The “error” on the estimators indicates how much variation is indicates how much variation is expectedexpected
Estimators & ErrorsEstimators & Errors
The The SStandard tandard EError on the mean of a rror on the mean of a sample of sample of nn measurements is defined measurements is defined as ..as ..
SESEμμ = = σσ / √ / √nn The error gets smaller as the sample The error gets smaller as the sample
size increasessize increases μμ has a “Bell Curve” distribution with has a “Bell Curve” distribution with
mean = “true mean” and SD = mean = “true mean” and SD = SESEμμ
Comparing ResultsComparing Results
Assume two different samples …Assume two different samples …– Are they from the same distribution?Are they from the same distribution?– Compare the two meansCompare the two means– The difference in the means will The difference in the means will
usually not be zerousually not be zero– How do we measure the significance How do we measure the significance
of an observed differenceof an observed difference It will clearly depend on the standard It will clearly depend on the standard
errorserrors
Comparing ResultsComparing Results
The standard error on the difference The standard error on the difference between to quantities is defined as ..between to quantities is defined as ..
SESEdiffdiff = √( = √( SESE1122 + + SESE22
22 ) )
If the two results are from the same If the two results are from the same distribution (“Null Hypothesis”)distribution (“Null Hypothesis”)– The difference will be a “Bell Curve” The difference will be a “Bell Curve”
distribution with mean = zero and distribution with mean = zero and Standard Deviation = Standard Deviation = SESEdiffdiff
SignificanceSignificance
Can use the difference of the Can use the difference of the means, means, ΔΔ, and its error to measure , and its error to measure the likelihood of an observed the likelihood of an observed difference.difference.
What is the probability of randomly What is the probability of randomly getting a bigger difference than we getting a bigger difference than we obtained?obtained?– Assuming a “Bell Curve” …Assuming a “Bell Curve” …– ProbabilityProbability = 32% if | = 32% if |ΔΔ| = | = SESEΔΔ
– ProbabilityProbability = 5% if | = 5% if |ΔΔ| = 2 x | = 2 x SESEΔΔ
SignificanceSignificance
This is NOT the same thing as the This is NOT the same thing as the “probability that the two “probability that the two distributions are the same”.distributions are the same”.– ProbabilityProbability = 1 if = 1 if ΔΔ = = 0 0 !!– But it is clearly not impossible to But it is clearly not impossible to
measure the same mean from two measure the same mean from two similar but different distributionssimilar but different distributions
t-Testst-Tests
When the size of a sample is small When the size of a sample is small (<30 measurements), we need to (<30 measurements), we need to take into account the uncertainty in take into account the uncertainty in the estimates of the Standard Errorsthe estimates of the Standard Errors
Instead of comparing Instead of comparing ΔΔ//SESEΔΔ with a with a “Normal” Bell curve distribution, “Normal” Bell curve distribution, Compare a variable Compare a variable tt with a with a tabulated t-distribution.tabulated t-distribution.
t-Testst-Tests
The t variable is defined as The t variable is defined as tt = = ΔΔ / √( / √( ss22//nn11 + + ss22//nn22 ))
wherewhere
ss22 = { ( = { (nn11-1)-1)SESE1122 + ( + (nn22-1) -1) SESE11
22} / } / ((nn11++nn22-2)-2)
The number of degrees of The number of degrees of freedom (needed by the tables) freedom (needed by the tables) are (are (nn11-1) and (-1) and (nn22-1)-1)
ConclusionConclusion
This talk has skipped a LOT of This talk has skipped a LOT of detailsdetails
It was designed to give you a feel It was designed to give you a feel for conceptsfor concepts
There are lots of resources on the There are lots of resources on the web and in the libraryweb and in the library
