given a sample from some population: what is a good “summary” value which well describes the...

• Given a sample from some population:• What is a good “summary” value which well

describes the sample?

• We will look at:

• Average (arithmetic mean)

• Median

• Mode

Measures of Location

For reference see (available on-line): “The Dynamic Character of Disguised Behaviour for Text-based, Mixed and Stylized Signatures”

LA Mohammed, B Found, M Caligiuri and D RogersJ Forensic Sci 56(1),S136-S141 (2011)

Histogram Points of Interest

• Velocity for the first segment of genuine signatures in (soon to be classic) Mohammed et al. study.

• What is a good summary number?

• How spread out is the data? (We will talk about this later)

• Arithmetic sample mean (average):• The sum of data divided by number of observations:


intuitive formula

fancy formula

• Example from LAM study: • Compute the average absolute size of segment 1 for

the genuine signature of subject 2:

Subj. 2; Gen; Seg. 1

Absolute Size (cm)

1 0.05482 0.29513 0.10264 0.10055 0.24916 0.12877 0.04968 0.22999 0.256

10 0.0538


• Example: • More useful: Consider again Absolute Average Velocity for

Genuine Signatures across all writers in the LAM study:

92 subjects × 10 measurements/subject = 920 velocity measurements

Average Absolute Average Velocity:


• Follow up question: • Is there a difference in the Abs. Avg. Veloc. for Genuine

signatures vs. Disguised signatures (DWM and DNM)??

Genuine DWM DNM

• We will learn how to answer this, but not yet.


• Sample median:• Ordering the n pieces of data from smallest value to

largest value, the median is the “middle value”:

• If n is odd, median is largest data point.

• If n is even, median is average of and largest data points.

th1

2

n

th

2

n th12

n


• Example: • Median of Average Absolute Velocity for Genuine Signatures,

LAM:

Avg


• Sample mode:• Needs careful definition but basically:

• The data value that occurs the most

Avg

mode = 9.2541

Med


• Some trivia:

Nice and symmetric:Mean = Median = Mode

Mean

Modes



Toss out the largest 5% and smallest 5% of the data

• Sample variance:• (Almost) the average of squared deviations from the

sample mean.

Measures of Data Spread

22

1

1

1

n

ii

s x xn

data point i

sample mean

there are n data points

2s s• Standard deviation is • The sample average and standard dev. are the most

common measures of central tendency and spread

• Sample average and standard dev have the same units

Measures of Data Spread• If you have “enough” data, you can fit a smooth

probability density function to the histogram


~ 68%±1s

~ 95%±2s

~ 99%±3s

• Trivia: The famous (standardized) “Bell Curve”

• Also called “normal” and “Gaussian”• Mean = 0

• Std Dev = 1

• Units are in

Std Devs

- - -

• Sample range:• The difference between the largest and smallest

value in the sample• Very sensitive to outliers (extreme observations)

• Percentiles:• The pth percentile data value, x, means that p-

percent of the data are less than or equal to x.• Median = 50th percentile


1st-%tile99th-%tile

1.520031.52008


Confidence Intervals

• A confidence interval (CI) gives a range in which a true population parameter may be found.

• Specifically, (1-α)×100% CIs for a parameter, constructed from a random sample (of a given sample size), will contain the true value of the parameter approximately (1-α)×100% of the time.

• α is called the “level of significance”

• Different from tolerance and prediction intervals


• Caution: IT IS NOT CORRECT to say that there a (1- α)×100% probability that the true value of a parameter is between the bounds of any given CI.

true valueof parameter

Here 90% of theCIs contain thetrue value of theparameter

Graphical representation of 90% CIs is for a parameter:

Take a sample.Compute a CI.

• Construction of a CI for a mean depends on:• Sample size n

• Standard error for means

• Level of confidence 1-α• α is significance level

• Use α to compute tc-value

• (1-α)×100% CI for population mean using a sample average and standard error is:


• Compute a 99% confidence interval for the mean using this sample set:


Fragment # Fragment nD1 1.520052 1.520033 1.520014 1.520045 1.520006 1.520017 1.520088 1.520119 1.52008

10 1.5200811 1.52008

Putting this together:[1.52005 - (3.17)(0.00001), 1.52005 + (3.17)(0.00001)]

99% CI for sample = [1.52002, 1.52009]

Hypothesis Testing• A hypothesis is an assumption about a statistic.

• Form a hypothesis about the statistic

• H0, the null hypothesis

• Identify the alternative hypothesis, Ha

• “Accept” H0 or “Reject” H0 in favour of Ha at a certain confidence level (1-α)×100%• Technically, “Accept” means “Do not Reject”

• The testing is done with respect to how sample values of the statistic are distributed• Student’s-t

• Gaussian

• Binomial

• Poisson

• Bootstrap, etc.

Hypothesis Testing• Hypothesis testing can go wrong:

• 1-β is called test’s power

• Do the thicknesses of float glass differ from non float glass?

• How can we use a computer to decide?

H0 is really true H0 is really false

Test rejects H0 Type I error. Probability is α

OK

Test accepts H0 OK Type II error. Probability is β

Importing External Data From a Spread Sheet

• Use R function read.csv:• Import (fake) float glass thickness data in file

glass_thickness_simulated.csv:

read.csv(“/Path/to/your/data/glass_thickness_simulated.csv", header=T)

Hypothesis Testing

Analysis of Variance

• Standard hypothesis testing is great for comparing two statistics.• What is we have more than two statistics to compare?

• Use analysis of variance (ANOVA)

• Note that the statistics to be compares must all be of the same type• Usually the statistic is an average “response” for

different experimental conditions or treatments.

Analysis of Variance• H0 for ANOVA

• The values being compared are not statistically different at the (1-a)×100% level of confidence

• Ha for ANOVA

• At least one of the values being compared is statically distinct.

• ANOVA computes an F-statistic from the data and compares to a critical Fc value for

• Level of confidence

• D.O.F. 1 = # of levels -1

• D.O.F. 2 = # of obs. - # of levels

Analysis of Variance• Levels are “categorical variables” and can be:

• Group names

• Experimental conditions

• Experimental treatments

Analysis of Variance

given a sample from some population: what is a good “summary” value which well describes the...

Documents

data slide

avg measures of location

med measures of location

histogram slide

spread sample average

sample median

later slide

sum of data