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Statistics of samples
Populations and samples
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POPULATION
All individuals
All objects
by
Gilberto E. Urroz,
March 2006
All measurements
Populations, samples and
statistical inference
Population size
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Finite population
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Infinite population
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e.g., students in this class
e.g., possible values of a length
Extremely large populations
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e.g., the population of the U.S.
treat as infinite
Samples
Should be random or unbiased
Numeric sample
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Example: monthly precipitation data (in)
[0.05, 0.07, 0.10, 0.12, 0.22, 0.50, ..., 0.10]
Each element equally likely to be chosen
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Biased sample = not representative
[x1, x2, ..., xn]
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Described by sample statistics
Represent it as the list:
n = sample size
Sample statistics
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Measures of central tendency
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Mean deviation, median deviation, variance,
standard deviation, range, interquartile range
Coefficient of variation
Measures that split the data
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Extracting sample statistics out
of a numerical sample
Mean, geometric mean, harmonic mean,
median, mode(s)
Measures of spread or variation
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Data Reduction
Quartiles, percentiles, deciles
Moments
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Skewness, kurtosis
Maple context-menu for statistics
Entering data into Maple
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Click on list, right-click, choose Statistics
Type the data
Generate random data
– LinearAlgebra[RandomMatrix](1,n)
or
– with(Statistics);
– X := RandomVariable(Normal(μ,σ))
– Sample(X,n)
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Read data from a file
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Tools>Assistants>Import Data...
Stored as a matrix
LinearAlgebra[Column] – extract columns
Convert to a list
Measures of central tendency
Measures of central tendency
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Mean
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Quadratic mean
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Geometric mean
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Median
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Harmonic mean
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Mode(s) = value(s) that repeat the most in
sample
Deviations from the mean
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Differences between each data value and the
mean
Measures of spread
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Mean Deviation, or mean absolute deviation
Variance
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Sum of deviations equals zero
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Uses
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Quartiles
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Quartiles, Inter-quartile range
Q1 = first quartile
25% of data below Q1, 75% above
Q2 = median (second quartile)
50% of data below Q2, 50% above
Q3 = third quartile
75% of data below Q3, 25% above
IQR = Q3-Q1
Contains 50% of the data
Coefficient of variation
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Standard deviation, s = square root of
variance
Maple definition
Skewness & Kurtosis
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Skewness
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Kurtosis
Population standard deviation (for finite
populations)
Alternative definition:
Five point summary
Contains
(1) Minimum, (2) Lower hinge, (3) Median,
(4) Upper hinge, (5) Maximum
Data Summary
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Mean
Standard deviation
Skewness
Kurtosis
Minimum value
Maximum value
Cumulative weight = n (for a sample
without weights)