Parametric Statistics

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Transcript Parametric Statistics

Parametric Statistics
Descriptive statistics
Hypothesis testing
Definitions:
• Population: the entire set about which info is
needed, Greek letters are used
• Sample: a subset studied; random samples
• Parameter – numerical characteristic
• Inferential statistics
• Discrete and continuous
• Distribution: the pattern of variation of a variable
• One sided probability: comparing two data sets
(ie. a > b); two-sided probability: a not equal to b
Detection Limit
Action Limit; 2s, 97.7% certain that signal observed is
not random noise.
Detection Limit; 3s, 93.3% certain to detect signal
above the 2s limit when the analyte is at this
concentration.
• Quantitation Limit; 10s signal required for 10% RSD
• Type I Error: identification of random noise as signal
• Type II Error: not identifying signal that is present.
Numerical Descriptive Statistics
Types of numerical summary statistics:
• Measures of Location
–
–
•
Measures of Center
Other Measures
Measures of Variability
Probability Density Function
• Probability density function – f(x) –
probability of obtaining result x for the
variable in your experiment (relative
frequency for discreet measurements)
• The total Sf(xi)=1 the total sum of all relative
frequencies
• Distribution function – probability that x is
less than or equal to xi:
– F(xi)= = Sf(xj) over all j such that xj xi
Discreet Data PDFs
• Binomial distribution:
f(x) = (n!/(x!(n-x)!))px(1-p)n-x
• The probability of getting the result of
interest (success) x times out of n, if the
overall probability of the result is p
• Note that here, x is a discrete variable
– Integer values only
Uses of the Binomial Distribution
• Quality assurance
• Genetics
• Experimental design
Binomial PDF: Example
•
•
•
•
n = 6; number of dice rolled
pi = 1/6; probability of rolling a 2 on any die
x = [0 1 2 3 4 5 6]; sample results # of 2s out of 6
Graph of f(x) versus x for rolling a “2”
Binomial PDF: Example 2
• n = 8; number of puppies in litter
• pi = 1/2; probability of any pup being male
• x = [0 1 2 3 4 5 6 7 8] example data for the # of
males out of 8
• Graph of f(x) versus x
Binomial PDF: Characteristics
• Shape is determined by values of n and p
(parameters of the distribution function)
– Only truly symmetric if p = 0.5
– Approaches Poisson’s distribution if n is very
large and p is very small,
– Approaches the normal distribution if n is large,
and p is not small
• Mean number of “successes” or the
expectation value X = np
• Variance is np(1- p)
Poisson Distribution
• Can be derived as a limiting form of Binomial
Distribution
– when n∞ as the mean l=np remains constant
– this means conducting a large number of trials with
p very small
• Can be derived directly from basic assumptions
• Assumptions determine the real situations where
Poisson’s distribution is useful
Simeon D. Poisson (1781-1840)
Poisson’s Assumptions
• Time or other interval type study
• The time interval is small
• The probability of one success is proportional
to the time interval
• The number of successes in one time interval
is independent of the number of successes in
another time interval
Derive Poisson from basic
assumptions
• Derivation by Induction
– To find an expression for p(x), first find p(0),
then p(k) and p(k+1) then generalize for p(x).
• Basic properties used:
Poisson’s Assumptions: Example
• The probability of one photon arriving in
the time interval Dt is proportional to Dt
when Dt is small
• The probability that more than one photon
arrives in Dt is negligible for small Dt
• The number of photons that arrive in one
time interval is independent of the number
of photons that arrive in any other nonoverlatping interval
Normal Approximation to Poisson’s
Distribution
• http://www.stat.ucla.edu/~dinov/courses_stu
dents.dir/Applets.dir/NormalApprox2Poisso
nApplet.html
Measures of Center
• Mode
• Median
• Population Mean (μ) and Sample Average (x)
Measures of Spread
• Variance – square of standard
deviation
• Standard deviation:
– Population standard deviation s:
large sample sets, the population
mean (μ) is known.
– Sample standard deviation (s):
small sample sets, sample average
(x) is used.
– Pooled standard deviation (s ).
When several small sets have the
same sources of indeterminate
error (ie: the same type of
measurement but different
samples)
Standard Error of the Mean
• uncertainty in the average(sm);
different from the standard deviation
s (variation for each measurement);
if n=1, sm= s
• i. If s is known, the uncertainty in the
mean is:
• ii. If s is unknown, use the t-score to
compensate for the uncertainty in s.
• t - from a table for % confidence
level and n-1 degrees of freedom.
(one degree of freedom is used to
calculate the mean.)
Chebychev and Empirical
Rules
• 's Rule The proportion of observations within k
standard deviations of the mean, where , is at
least , i.e., at least 75%, 89%, and 94% of the
data are within 2, 3, and 4 standard deviations of
the mean, respectively.
• Empirical Rule If data follow a bell-shaped curve,
then approximately 68%, 95%, and 99.7% of the
data are within 1, 2, and 3 standard deviations of
the mean, respectively.
Z-score
• -scores are a means of answering the question
``how many standard deviations away from the
mean is this observation?''
z
0
1
2
3
P 1sided
0.5
.84
.98
.999
P 2sided
0
.68
.95
.99
Confidence Interval
• The range of uncertainty
in a value at a stated
percent confidence
• Percent confidence… that
the value is within the
stated range
 s is known
 s is unknown
Look up the appropriate
z or t values to use:
x +/- t*s/sqrt(N)
http://math.uc.edu/~brycw/classes/148/tables.htm
Inferential Statistics
• Comparing two
sample means
T-test (Student's t)
• Used to calculate the confidence intervals of
a measurement when the population
standard deviation s is not known
• Used to compare two averages
• corrects for the uncertainty of the sample
standard deviation (s) caused by taking a
small number of samples.
Comparison Tests
Comparing the sample to the true value.
Comparing two experimental averages
Significance Testing
• Confidence interval
• Statistical Hypotheses
• Ho and H1
Comparison Test: Comparing the
sample to the true value
Method #1.
• If the difference between the measured value
and the true value (μ) is greater than the
uncertainty in the measurement, then there is a
significant difference between the two values at
that confidence level.
Method #2.
• experimental t-score (t ) is compared to t-critical
(found in a table)
• There is a significant difference if experimental
t is greater than critical t .
• t is chosen for N-1 degrees of freedom at the
desired percent confidence interval.
• If the experimental value may be greater or less
than the true value, use a two sided t-score. If …
Comparison Tests: Comparing two
experimental averages.
t-test:
• use the pooled standard deviation and calculate t as:
experimental
• If experimental t is greater than critical t then there is
a significant difference between the two means.
• t is determined at the appropriate confidence level
from a table
• the t-statistic for N + N - 2 degrees of freedom.
T-table