Grouped data – Weighted data
Grouped data
A lecture by
Gilberto E. Urroz
March 2006
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Reference worksheet:
Data summarized through a frequency
distribution
Basic information needed:
–
BasicStatisticsMaple10_WEIGHTED_Part6.mw
–
Remember to use
> with(Statistics):
Variance
Count and mean value
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Class marks (xi ) and frequencies (fi ), i = 1, 2, ...,
k
Class boundaries (xBi ) and frequencies (fi ), thus
xi= (xBi+xBi+1)/2
The variance is calculated as
The total number of data points (count) in the
original sample is
where
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The mean value is calculated as:
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Note: if fi = 1, Sff = n, ⇒
(variance of a sample)
Other measures of central
tendency
Calculations using Maple
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Enter (x,f) as lists, e.g.,
> X:=[3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, 10.5 ]
> f :=[ 2, 6, 12, 18, 11, 5, 3, 1 ]
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Calculations:
> xbar := Mean(X,weights=f)
> s2 := Variance(X,weights=f)
> s := StandardDeviation(X,weights=f)
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Try the following commands:
> xgeom := GeometricMean(X,weights=f)
> xharm := HarmonicMean(X,weights=f)
> qmean := QuadraticMean(X,weights=f)
> median := Median(X,weights=f)
> mode := Mode(X,weights=f)
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Other measures of spread or
variation
Data summary on grouped data
Try the following commands:
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> meandeviation := MeanDeviation(X,weights = f)
> mediandeviation := MedianDeviation(X,weights = f)
> xrange := Range(X,weights = f)
> IQR := interquartileRange(X,weights = f)
> CoeffOfVariation := Variation(X,weights = f)
> Q1 := Quartile(X, 1, weights = f)
> Q3 := Quartile(X, 3, weights = f)
> D6 := Percentile(X, 60, weights = f)
> P35 := Percentile(X, 35, weighte = f)
> Xv := convert(X,vector)
> fv := convert(f,vector)
> DataSummary(Xv, weights = fv)
Finding class marks given
classes
Five-point summary on grouped data
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Needs data in vector format
> Xv := convert(X,vector)
> fv := convert(f,vector)
> FivePointSummary(Xv, weights = fv)
Needs data in vector format
Enter classes as ranges:
>classes:=[2..2.95,3..4.95, ..., 8..9.95]
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Class marks are calculated as follows:
> k := nops(classes)
>
Finding classes given class marks - 1
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Simplest approach: given class marks:
X := [X1, ..., Xk]
Finding classes given class marks - 2
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Use the following user-defined function:
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XLimits:=GetClassLimits(X)
Assume lowest class limit is:
XL1 = X1-(X2-X1)/2
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Remaining class limits are calculated as:
Xlj+1 = 2Xj – Xlj, for j = 1, 2, ..., k
Frequency Table for Grouped Data - 1
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Given class limits, XL, and frequencies, f, as
lists, form a tally list, e.g.,
Frequency Table for Grouped Data - 2
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Function TallyFrequencyTable:
>Xfreq:=[seq(XLj ..Xlj+1 = fj , j = 1..k) ]
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Use function TallyFrequencyTable to
generate the frequency table:
>TallyFrequencyTable(Xfreq)
Frequency Table for Grouped Data - 2
Application of TallyFrequencyTable:
>TallyFrequencyTable(Xfreq)
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Histogram for Grouped Data - 1
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Use function TallyHistogram:
Histogram for Grouped Data - 1
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Use function TallyHistogram:
Histogram for Grouped Data - 2
>TallyHistogram(Xfreq, magenta)
Weighted data
Weighted data grouping
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Sample with repeating numbers:
Data can be grouped into data points (XV)
and frequency counts (Xf)
Data summary for weighted data
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Tally and separate values and frequencies:
Convert lists XV and Xf to vectors, then use
function DataSummary
Five point summary for weighted data
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Use function FivePointSummaryI(XVV):
Note: re-defining 'median' before summarizing
data.
Frequency table for weigthed data
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Use function TallyFrequencyTable(XL):
Bar chart for weighted data