2. Basic_statistics 30 Sep 2013.pdf

8/14/2019 2. Basic_statistics 30 Sep 2013.pdf

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Basic Statistics - Concepts

and Examples


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Elementary Concepts

Variables: Variables are things that we measure, control, or manipulatein research. They differ in many respects, most notably in the role theyare given in our research and in the type of measures that can beapplied to them.

Observational vs. experimental research. Most empirical research

belongs clearly to one of those two general categories. In observationalresearch we do not (or at least try not to) influence any variables butonly measure them and look for relations (correlations) between someset of variables. In experimental research, we manipulate somevariables and then measure the effects of this manipulation on othervariables.

Dependent vs. independent variables. Independent variables are thosethat are manipulated whereas dependent variables are only measured orregistered.


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Systematic and Random Errors

Error:Defined as the difference between a

calculated or observed value and the true value

Blunders: Usually apparent either as obviously incorrect datapoints or results that are not reasonably close to the expected value.Easy to detect.

Systematic Errors:Errors that occur reproducibly from faultycalibration of equipment or observer bias. Statistical analysis ingenerally not useful, but rather corrections must be made based onexperimental conditions.

Random Errors:Errors that result from the fluctuations inobservations. Requires that experiments be repeated a sufficientnumber of time to establish the precision of measurement.


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Uncertainties

In most cases, cannot know what the true value is unless

there is an independent determination (i.e.different

measurement technique).

Only can consider estimatesof the error.

Discrepancyis the difference between two or more

observations. This gives rise to uncertainty.

Probable Error:Indicates the magnitude of the error we

estimate to have made in the measurements. Means that if

we make a measurement that we probably wont be

wrong by that amount.


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some univariate statistical terms:

mode: value that occurs most frequently in a distribution

(usually the highest point of curve)

may have more than one mode in a dataset

median: value midway in the frequency distribution

half the area of curve is to right and other to left

mean: arithmetic average

sum of all observations divided by # of observations

poor measure of central tendency in skewed distributions

range: measure of dispersion about mean(maximum minus minimum)

when max and min are unusual values, range may be

a misleading measure of dispersion


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Distribution vs. Sample Size


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histogramis a useful graphic representation of

information content of sample or parent population

many statistical tests assume

values are normally distributed

not always the case!

examine data prior

to processing

from: Jensen, 1996


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Deviations

The deviation, di, of any measurementxifrom the mean mof theparent distribution is defined as the difference betweenxiand m

Average deviation, a, is defined as the average of the magnitudes

of the deviations, which is given by the absolute value of the

deviations.

di xi m

a limN

1

N xi mi1

n


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variance: average squared deviation of all possible observations

from a sample mean (calculated from sum of squares)

standard deviation: positive square root of the variancesmall std dev: observations are clustered tightly

around a central value

large std dev: observations are scattered widely

about the mean

s2

i= lim [1/NS(xi- )2

]i=1

n

s2i= S(xi- )2i=1n

N- 1

where: is the mean,

xiis observed value, and

Nis the number of observations

N->

Number decreased fromNto

N - 1for the sample variance

as is used in the calculation


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Sample Mean and Standard Deviation

For a series of Nobservations, the most probable estimate of the

mean is the average x of the observations. We refer to this as

the sample mean x to distinguish it from the parent mean .

m x

1

N xi

s

2 1

N ix m

2

1

N i2

x m2

Sample Mean

Our best estimate of the standard deviation swould be from:

s2 s2

1

N1x

i

x

2

Sample Variance

But we cannot know the true parent mean so the best estimate

of the sample variance and standard deviation would be:


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Distributions

Binomial Distribution:Allows us to define theprobability,p, of observingxa specific combination of n

items, which is derived from the fundamental formulas for

the permutations and combinations.

Permutations:Enumerate the number of permutations,

Pm(n,x),of coin flips, when we pick up the coins one at

a time from a collection of n coins and putxof them

into the heads box.

Pm(n,x) n!

(n x)!

n!n(n1)(n2) (3)(2)(1)

1!1

0!1


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Distributions - cont.

Combinations: Relates to the number of ways we can

combine the various permutations enumerated above

from our coin flip experiment. Thus the number of

combinations is equal to the number of permutationsdivided by the degeneracy factorx! of the permutations.

C(n,x) Pm(n,x)x!

n!x!(nx)!

nx


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Probability and the Binomial Distribution

Coin Toss Experiment: If p is the probability of success (landing heads up)is not necessarily equal to the probability q = 1- pfor failure

(landing tails up) because the coins may be lopsided!

The probability for each of the combinations ofxcoins heads up and

n -xcoins tails up is equal topxqn-x. The binomial distributioncan be

used to calculate the probability:

PB(x,n,p) n

x

p

xqnx

n!

x!(nx)!px

(1 p)n x

The coefficientsPB(x,n,p)are closely related to the binomial theorem

for the expansion of a power of a sum:

p q n n

x

pxqnx

x0

n


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Mean and Variance: Binomial Distribution

m x n!

x! n x !p

x(1 p)n x

x0

n

np

The mean of the binomial distribution is evaluated by combining the

definition of with the function that defines the probability, yielding:

The average of the number of successes will approach a mean value

given by the probability for success of each item ptimes the number of

items. For the coin toss experimentp=1/2, half the coins should land

heads up on average.

s2 (x m) 2 n!

x! nx !px

(1 p)n x

x0

n

np(1 p)

If the the probability for a single successpis equal to the probability for

failurep=q=1/2, the final distribution is symmetric about the mean and

mode and median equal the mean. The variance, s2 m/2.


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Other Probability Distributions: Special Cases

Poisson Distribution:An approximation to the

binomial distribution for the special case when the averagenumber of successes is very much smaller than the

possible number i.e.


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Gaussian or Normal Error Distribution Details

Gaussian Distribution:Most important probabilitydistribution in the statistical analysis of experimental data.functional form is relatively simple and the resultantdistribution is reasonable. Again this is a special limitingcase to the binomial distribution where the number of

possible different observations, n, becomes infinitely largeyielding np >> 1. Most probable estimate of the mean from a random sample of

observations is the average of those observations!

G 2.354s

Tangent along the steepest portion

of the probability curve intersects

at e-1/2and intersects x axis at the

points x = 2s

Probable Error (P.E.)is defined as the

absolute value of the deviation such

thatPGof the deviation of any randomobservation is < 1/2

P.E. 0.6745s 0.2865 G


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For gaussian or normal error distributions:

Total area underneath curve is 1.00 (100%)

68.27% of observations lie within 1 std dev of mean

95% of observations lie within 2 std dev of mean

99% of observations lie within 3 std dev of mean

Variance, standard deviation, probable error, mean, and

weighted root mean square error are commonly used

statistical terms in geodesy.

compare (rather than attach significance to numerical value)


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2. Basic_statistics 30 Sep 2013.pdf