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Author's title

Author*Unverified author*
R Software Modulerwasp_centraltendency.wasp
Title produced by softwareCentral Tendency
Date of computationSun, 16 Dec 2007 04:44:04 -0700
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2007/Dec/16/t11978047542k7bi0xleqd5kvj.htm/, Retrieved Tue, 26 May 2026 09:14:46 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=4140, Retrieved Tue, 26 May 2026 09:14:46 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact476
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Central Tendency] [Central tendency ...] [2007-12-16 11:44:04] [908a52009ae83f013a8231794220d3ac] [Current]
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Dataseries X:
93,9
89,8
93,4
101,5
110,4
105,9
108,4
113,9
86,1
69,4
101,2
100,5
98,0
106,6
90,1
96,9
125,9
112,0
100,0
123,9
79,8
83,4
113,6
112,9
104,0
109,9
99,0
106,3
128,9
111,1
102,9
130,0
87,0
87,5
117,6
103,4
110,8
112,6
102,5
112,4
135,6
105,1
127,7
137,0
91,0
90,5
122,4
123,3
124,3
120,0
118,1
119,0
142,7
123,6
129,6
151,6
110,4
99,3
129,1
134,1




Summary of compuational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time1 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135

\begin{tabular}{lllllllll}
\hline
Summary of compuational transaction \tabularnewline
Raw Input & view raw input (R code)  \tabularnewline
Raw Output & view raw output of R engine  \tabularnewline
Computing time & 1 seconds \tabularnewline
R Server & 'Gwilym Jenkins' @ 72.249.127.135 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=4140&T=0

[TABLE]
[ROW][C]Summary of compuational transaction[/C][/ROW]
[ROW][C]Raw Input[/C][C]view raw input (R code) [/C][/ROW]
[ROW][C]Raw Output[/C][C]view raw output of R engine [/C][/ROW]
[ROW][C]Computing time[/C][C]1 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Gwilym Jenkins' @ 72.249.127.135[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=4140&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=4140&T=0

As an alternative you can also use a QR Code:  

The GUIDs for individual cells are displayed in the table below:

Summary of compuational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time1 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135







Central Tendency - Ungrouped Data
MeasureValueS.E.Value/S.E.
Arithmetic Mean109.632.1393822699964451.2437639301284
Geometric Mean108.378950429418
Harmonic Mean107.102344101729
Quadratic Mean110.854757829032
Winsorized Mean ( 1 / 20 )109.6552.0451063611090253.6182381930181
Winsorized Mean ( 2 / 20 )109.5851.9681917265135555.6780106956951
Winsorized Mean ( 3 / 20 )109.651.9226710175711157.030037379208
Winsorized Mean ( 4 / 20 )109.611.8879168264261158.0587017742172
Winsorized Mean ( 5 / 20 )109.311.8086526183281660.4372552762738
Winsorized Mean ( 6 / 20 )109.51.7554917864887362.3756834653258
Winsorized Mean ( 7 / 20 )109.4766666666671.7376121134688963.0040880919692
Winsorized Mean ( 8 / 20 )109.5033333333331.7224882912144963.5727591832425
Winsorized Mean ( 9 / 20 )109.3983333333331.6746371420217665.326589616458
Winsorized Mean ( 10 / 20 )109.4983333333331.5472802668248170.7682607224326
Winsorized Mean ( 11 / 20 )109.2966666666671.4795287970815673.872618689315
Winsorized Mean ( 12 / 20 )109.8166666666671.364416985375580.4861474488637
Winsorized Mean ( 13 / 20 )109.991.3154894257950383.6114664574568
Winsorized Mean ( 14 / 20 )110.1533333333331.2677696654638986.8874972592334
Winsorized Mean ( 15 / 20 )110.0033333333331.2173927096119090.3597766479169
Winsorized Mean ( 16 / 20 )109.551.08172936452659101.273020399094
Winsorized Mean ( 17 / 20 )109.4083333333331.01467840683795107.825624942866
Winsorized Mean ( 18 / 20 )109.3483333333330.940700819610074116.241350123049
Winsorized Mean ( 19 / 20 )109.2850.901921442330535121.169089535793
Winsorized Mean ( 20 / 20 )108.3850.668677018299762162.088717024535
Trimmed Mean ( 1 / 20 )109.61.9693999073728855.6514700694807
Trimmed Mean ( 2 / 20 )109.5410714285711.8759063208440258.3936789440988
Trimmed Mean ( 3 / 20 )109.5166666666671.8124130379870860.4258876819267
Trimmed Mean ( 4 / 20 )109.4653846153851.7554480690675462.3575180287336
Trimmed Mean ( 5 / 20 )109.4221.6976268910524764.4558592802226
Trimmed Mean ( 6 / 20 )109.451.6519760357508366.2539877282508
Trimmed Mean ( 7 / 20 )109.4391304347831.6098323442003367.9816943851663
Trimmed Mean ( 8 / 20 )109.4318181818181.5599329576017670.151616227186
Trimmed Mean ( 9 / 20 )109.4190476190481.4984592865320673.0210347404768
Trimmed Mean ( 10 / 20 )109.42251.4306824958201476.482727872667
Trimmed Mean ( 11 / 20 )109.4105263157891.3767154795805579.4721407135802
Trimmed Mean ( 12 / 20 )109.4277777777781.3221471900812982.7652008783149
Trimmed Mean ( 13 / 20 )109.3705882352941.2792288885220385.4972782561659
Trimmed Mean ( 14 / 20 )109.281251.2313318705382588.7504438200159
Trimmed Mean ( 15 / 20 )109.1566666666671.1751026693548592.8911741189347
Trimmed Mean ( 16 / 20 )109.0357142857141.1081732702860998.392297675223
Trimmed Mean ( 17 / 20 )108.9615384615381.05713803658858103.072195579265
Trimmed Mean ( 18 / 20 )108.8958333333331.00174628623254108.706001539451
Trimmed Mean ( 19 / 20 )108.8272727272730.941909771742022115.538957119004
Trimmed Mean ( 20 / 20 )108.7550.857059692080705126.893145255697
Median110.15
Midrange110.5
Midmean - Weighted Average at Xnp108.829032258065
Midmean - Weighted Average at X(n+1)p109.156666666667
Midmean - Empirical Distribution Function108.829032258065
Midmean - Empirical Distribution Function - Averaging109.156666666667
Midmean - Empirical Distribution Function - Interpolation109.156666666667
Midmean - Closest Observation108.829032258065
Midmean - True Basic - Statistics Graphics Toolkit109.156666666667
Midmean - MS Excel (old versions)109.28125
Number of observations60

\begin{tabular}{lllllllll}
\hline
Central Tendency - Ungrouped Data \tabularnewline
Measure & Value & S.E. & Value/S.E. \tabularnewline
Arithmetic Mean & 109.63 & 2.13938226999644 & 51.2437639301284 \tabularnewline
Geometric Mean & 108.378950429418 &  &  \tabularnewline
Harmonic Mean & 107.102344101729 &  &  \tabularnewline
Quadratic Mean & 110.854757829032 &  &  \tabularnewline
Winsorized Mean ( 1 / 20 ) & 109.655 & 2.04510636110902 & 53.6182381930181 \tabularnewline
Winsorized Mean ( 2 / 20 ) & 109.585 & 1.96819172651355 & 55.6780106956951 \tabularnewline
Winsorized Mean ( 3 / 20 ) & 109.65 & 1.92267101757111 & 57.030037379208 \tabularnewline
Winsorized Mean ( 4 / 20 ) & 109.61 & 1.88791682642611 & 58.0587017742172 \tabularnewline
Winsorized Mean ( 5 / 20 ) & 109.31 & 1.80865261832816 & 60.4372552762738 \tabularnewline
Winsorized Mean ( 6 / 20 ) & 109.5 & 1.75549178648873 & 62.3756834653258 \tabularnewline
Winsorized Mean ( 7 / 20 ) & 109.476666666667 & 1.73761211346889 & 63.0040880919692 \tabularnewline
Winsorized Mean ( 8 / 20 ) & 109.503333333333 & 1.72248829121449 & 63.5727591832425 \tabularnewline
Winsorized Mean ( 9 / 20 ) & 109.398333333333 & 1.67463714202176 & 65.326589616458 \tabularnewline
Winsorized Mean ( 10 / 20 ) & 109.498333333333 & 1.54728026682481 & 70.7682607224326 \tabularnewline
Winsorized Mean ( 11 / 20 ) & 109.296666666667 & 1.47952879708156 & 73.872618689315 \tabularnewline
Winsorized Mean ( 12 / 20 ) & 109.816666666667 & 1.3644169853755 & 80.4861474488637 \tabularnewline
Winsorized Mean ( 13 / 20 ) & 109.99 & 1.31548942579503 & 83.6114664574568 \tabularnewline
Winsorized Mean ( 14 / 20 ) & 110.153333333333 & 1.26776966546389 & 86.8874972592334 \tabularnewline
Winsorized Mean ( 15 / 20 ) & 110.003333333333 & 1.21739270961190 & 90.3597766479169 \tabularnewline
Winsorized Mean ( 16 / 20 ) & 109.55 & 1.08172936452659 & 101.273020399094 \tabularnewline
Winsorized Mean ( 17 / 20 ) & 109.408333333333 & 1.01467840683795 & 107.825624942866 \tabularnewline
Winsorized Mean ( 18 / 20 ) & 109.348333333333 & 0.940700819610074 & 116.241350123049 \tabularnewline
Winsorized Mean ( 19 / 20 ) & 109.285 & 0.901921442330535 & 121.169089535793 \tabularnewline
Winsorized Mean ( 20 / 20 ) & 108.385 & 0.668677018299762 & 162.088717024535 \tabularnewline
Trimmed Mean ( 1 / 20 ) & 109.6 & 1.96939990737288 & 55.6514700694807 \tabularnewline
Trimmed Mean ( 2 / 20 ) & 109.541071428571 & 1.87590632084402 & 58.3936789440988 \tabularnewline
Trimmed Mean ( 3 / 20 ) & 109.516666666667 & 1.81241303798708 & 60.4258876819267 \tabularnewline
Trimmed Mean ( 4 / 20 ) & 109.465384615385 & 1.75544806906754 & 62.3575180287336 \tabularnewline
Trimmed Mean ( 5 / 20 ) & 109.422 & 1.69762689105247 & 64.4558592802226 \tabularnewline
Trimmed Mean ( 6 / 20 ) & 109.45 & 1.65197603575083 & 66.2539877282508 \tabularnewline
Trimmed Mean ( 7 / 20 ) & 109.439130434783 & 1.60983234420033 & 67.9816943851663 \tabularnewline
Trimmed Mean ( 8 / 20 ) & 109.431818181818 & 1.55993295760176 & 70.151616227186 \tabularnewline
Trimmed Mean ( 9 / 20 ) & 109.419047619048 & 1.49845928653206 & 73.0210347404768 \tabularnewline
Trimmed Mean ( 10 / 20 ) & 109.4225 & 1.43068249582014 & 76.482727872667 \tabularnewline
Trimmed Mean ( 11 / 20 ) & 109.410526315789 & 1.37671547958055 & 79.4721407135802 \tabularnewline
Trimmed Mean ( 12 / 20 ) & 109.427777777778 & 1.32214719008129 & 82.7652008783149 \tabularnewline
Trimmed Mean ( 13 / 20 ) & 109.370588235294 & 1.27922888852203 & 85.4972782561659 \tabularnewline
Trimmed Mean ( 14 / 20 ) & 109.28125 & 1.23133187053825 & 88.7504438200159 \tabularnewline
Trimmed Mean ( 15 / 20 ) & 109.156666666667 & 1.17510266935485 & 92.8911741189347 \tabularnewline
Trimmed Mean ( 16 / 20 ) & 109.035714285714 & 1.10817327028609 & 98.392297675223 \tabularnewline
Trimmed Mean ( 17 / 20 ) & 108.961538461538 & 1.05713803658858 & 103.072195579265 \tabularnewline
Trimmed Mean ( 18 / 20 ) & 108.895833333333 & 1.00174628623254 & 108.706001539451 \tabularnewline
Trimmed Mean ( 19 / 20 ) & 108.827272727273 & 0.941909771742022 & 115.538957119004 \tabularnewline
Trimmed Mean ( 20 / 20 ) & 108.755 & 0.857059692080705 & 126.893145255697 \tabularnewline
Median & 110.15 &  &  \tabularnewline
Midrange & 110.5 &  &  \tabularnewline
Midmean - Weighted Average at Xnp & 108.829032258065 &  &  \tabularnewline
Midmean - Weighted Average at X(n+1)p & 109.156666666667 &  &  \tabularnewline
Midmean - Empirical Distribution Function & 108.829032258065 &  &  \tabularnewline
Midmean - Empirical Distribution Function - Averaging & 109.156666666667 &  &  \tabularnewline
Midmean - Empirical Distribution Function - Interpolation & 109.156666666667 &  &  \tabularnewline
Midmean - Closest Observation & 108.829032258065 &  &  \tabularnewline
Midmean - True Basic - Statistics Graphics Toolkit & 109.156666666667 &  &  \tabularnewline
Midmean - MS Excel (old versions) & 109.28125 &  &  \tabularnewline
Number of observations & 60 &  &  \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=4140&T=1

[TABLE]
[ROW][C]Central Tendency - Ungrouped Data[/C][/ROW]
[ROW][C]Measure[/C][C]Value[/C][C]S.E.[/C][C]Value/S.E.[/C][/ROW]
[ROW][C]Arithmetic Mean[/C][C]109.63[/C][C]2.13938226999644[/C][C]51.2437639301284[/C][/ROW]
[ROW][C]Geometric Mean[/C][C]108.378950429418[/C][C][/C][C][/C][/ROW]
[ROW][C]Harmonic Mean[/C][C]107.102344101729[/C][C][/C][C][/C][/ROW]
[ROW][C]Quadratic Mean[/C][C]110.854757829032[/C][C][/C][C][/C][/ROW]
[ROW][C]Winsorized Mean ( 1 / 20 )[/C][C]109.655[/C][C]2.04510636110902[/C][C]53.6182381930181[/C][/ROW]
[ROW][C]Winsorized Mean ( 2 / 20 )[/C][C]109.585[/C][C]1.96819172651355[/C][C]55.6780106956951[/C][/ROW]
[ROW][C]Winsorized Mean ( 3 / 20 )[/C][C]109.65[/C][C]1.92267101757111[/C][C]57.030037379208[/C][/ROW]
[ROW][C]Winsorized Mean ( 4 / 20 )[/C][C]109.61[/C][C]1.88791682642611[/C][C]58.0587017742172[/C][/ROW]
[ROW][C]Winsorized Mean ( 5 / 20 )[/C][C]109.31[/C][C]1.80865261832816[/C][C]60.4372552762738[/C][/ROW]
[ROW][C]Winsorized Mean ( 6 / 20 )[/C][C]109.5[/C][C]1.75549178648873[/C][C]62.3756834653258[/C][/ROW]
[ROW][C]Winsorized Mean ( 7 / 20 )[/C][C]109.476666666667[/C][C]1.73761211346889[/C][C]63.0040880919692[/C][/ROW]
[ROW][C]Winsorized Mean ( 8 / 20 )[/C][C]109.503333333333[/C][C]1.72248829121449[/C][C]63.5727591832425[/C][/ROW]
[ROW][C]Winsorized Mean ( 9 / 20 )[/C][C]109.398333333333[/C][C]1.67463714202176[/C][C]65.326589616458[/C][/ROW]
[ROW][C]Winsorized Mean ( 10 / 20 )[/C][C]109.498333333333[/C][C]1.54728026682481[/C][C]70.7682607224326[/C][/ROW]
[ROW][C]Winsorized Mean ( 11 / 20 )[/C][C]109.296666666667[/C][C]1.47952879708156[/C][C]73.872618689315[/C][/ROW]
[ROW][C]Winsorized Mean ( 12 / 20 )[/C][C]109.816666666667[/C][C]1.3644169853755[/C][C]80.4861474488637[/C][/ROW]
[ROW][C]Winsorized Mean ( 13 / 20 )[/C][C]109.99[/C][C]1.31548942579503[/C][C]83.6114664574568[/C][/ROW]
[ROW][C]Winsorized Mean ( 14 / 20 )[/C][C]110.153333333333[/C][C]1.26776966546389[/C][C]86.8874972592334[/C][/ROW]
[ROW][C]Winsorized Mean ( 15 / 20 )[/C][C]110.003333333333[/C][C]1.21739270961190[/C][C]90.3597766479169[/C][/ROW]
[ROW][C]Winsorized Mean ( 16 / 20 )[/C][C]109.55[/C][C]1.08172936452659[/C][C]101.273020399094[/C][/ROW]
[ROW][C]Winsorized Mean ( 17 / 20 )[/C][C]109.408333333333[/C][C]1.01467840683795[/C][C]107.825624942866[/C][/ROW]
[ROW][C]Winsorized Mean ( 18 / 20 )[/C][C]109.348333333333[/C][C]0.940700819610074[/C][C]116.241350123049[/C][/ROW]
[ROW][C]Winsorized Mean ( 19 / 20 )[/C][C]109.285[/C][C]0.901921442330535[/C][C]121.169089535793[/C][/ROW]
[ROW][C]Winsorized Mean ( 20 / 20 )[/C][C]108.385[/C][C]0.668677018299762[/C][C]162.088717024535[/C][/ROW]
[ROW][C]Trimmed Mean ( 1 / 20 )[/C][C]109.6[/C][C]1.96939990737288[/C][C]55.6514700694807[/C][/ROW]
[ROW][C]Trimmed Mean ( 2 / 20 )[/C][C]109.541071428571[/C][C]1.87590632084402[/C][C]58.3936789440988[/C][/ROW]
[ROW][C]Trimmed Mean ( 3 / 20 )[/C][C]109.516666666667[/C][C]1.81241303798708[/C][C]60.4258876819267[/C][/ROW]
[ROW][C]Trimmed Mean ( 4 / 20 )[/C][C]109.465384615385[/C][C]1.75544806906754[/C][C]62.3575180287336[/C][/ROW]
[ROW][C]Trimmed Mean ( 5 / 20 )[/C][C]109.422[/C][C]1.69762689105247[/C][C]64.4558592802226[/C][/ROW]
[ROW][C]Trimmed Mean ( 6 / 20 )[/C][C]109.45[/C][C]1.65197603575083[/C][C]66.2539877282508[/C][/ROW]
[ROW][C]Trimmed Mean ( 7 / 20 )[/C][C]109.439130434783[/C][C]1.60983234420033[/C][C]67.9816943851663[/C][/ROW]
[ROW][C]Trimmed Mean ( 8 / 20 )[/C][C]109.431818181818[/C][C]1.55993295760176[/C][C]70.151616227186[/C][/ROW]
[ROW][C]Trimmed Mean ( 9 / 20 )[/C][C]109.419047619048[/C][C]1.49845928653206[/C][C]73.0210347404768[/C][/ROW]
[ROW][C]Trimmed Mean ( 10 / 20 )[/C][C]109.4225[/C][C]1.43068249582014[/C][C]76.482727872667[/C][/ROW]
[ROW][C]Trimmed Mean ( 11 / 20 )[/C][C]109.410526315789[/C][C]1.37671547958055[/C][C]79.4721407135802[/C][/ROW]
[ROW][C]Trimmed Mean ( 12 / 20 )[/C][C]109.427777777778[/C][C]1.32214719008129[/C][C]82.7652008783149[/C][/ROW]
[ROW][C]Trimmed Mean ( 13 / 20 )[/C][C]109.370588235294[/C][C]1.27922888852203[/C][C]85.4972782561659[/C][/ROW]
[ROW][C]Trimmed Mean ( 14 / 20 )[/C][C]109.28125[/C][C]1.23133187053825[/C][C]88.7504438200159[/C][/ROW]
[ROW][C]Trimmed Mean ( 15 / 20 )[/C][C]109.156666666667[/C][C]1.17510266935485[/C][C]92.8911741189347[/C][/ROW]
[ROW][C]Trimmed Mean ( 16 / 20 )[/C][C]109.035714285714[/C][C]1.10817327028609[/C][C]98.392297675223[/C][/ROW]
[ROW][C]Trimmed Mean ( 17 / 20 )[/C][C]108.961538461538[/C][C]1.05713803658858[/C][C]103.072195579265[/C][/ROW]
[ROW][C]Trimmed Mean ( 18 / 20 )[/C][C]108.895833333333[/C][C]1.00174628623254[/C][C]108.706001539451[/C][/ROW]
[ROW][C]Trimmed Mean ( 19 / 20 )[/C][C]108.827272727273[/C][C]0.941909771742022[/C][C]115.538957119004[/C][/ROW]
[ROW][C]Trimmed Mean ( 20 / 20 )[/C][C]108.755[/C][C]0.857059692080705[/C][C]126.893145255697[/C][/ROW]
[ROW][C]Median[/C][C]110.15[/C][C][/C][C][/C][/ROW]
[ROW][C]Midrange[/C][C]110.5[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - Weighted Average at Xnp[/C][C]108.829032258065[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - Weighted Average at X(n+1)p[/C][C]109.156666666667[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - Empirical Distribution Function[/C][C]108.829032258065[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - Empirical Distribution Function - Averaging[/C][C]109.156666666667[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - Empirical Distribution Function - Interpolation[/C][C]109.156666666667[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - Closest Observation[/C][C]108.829032258065[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - True Basic - Statistics Graphics Toolkit[/C][C]109.156666666667[/C][C][/C][C][/C][/ROW]
[ROW][C]Midmean - MS Excel (old versions)[/C][C]109.28125[/C][C][/C][C][/C][/ROW]
[ROW][C]Number of observations[/C][C]60[/C][C][/C][C][/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=4140&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=4140&T=1

As an alternative you can also use a QR Code:  

The GUIDs for individual cells are displayed in the table below:

Central Tendency - Ungrouped Data
MeasureValueS.E.Value/S.E.
Arithmetic Mean109.632.1393822699964451.2437639301284
Geometric Mean108.378950429418
Harmonic Mean107.102344101729
Quadratic Mean110.854757829032
Winsorized Mean ( 1 / 20 )109.6552.0451063611090253.6182381930181
Winsorized Mean ( 2 / 20 )109.5851.9681917265135555.6780106956951
Winsorized Mean ( 3 / 20 )109.651.9226710175711157.030037379208
Winsorized Mean ( 4 / 20 )109.611.8879168264261158.0587017742172
Winsorized Mean ( 5 / 20 )109.311.8086526183281660.4372552762738
Winsorized Mean ( 6 / 20 )109.51.7554917864887362.3756834653258
Winsorized Mean ( 7 / 20 )109.4766666666671.7376121134688963.0040880919692
Winsorized Mean ( 8 / 20 )109.5033333333331.7224882912144963.5727591832425
Winsorized Mean ( 9 / 20 )109.3983333333331.6746371420217665.326589616458
Winsorized Mean ( 10 / 20 )109.4983333333331.5472802668248170.7682607224326
Winsorized Mean ( 11 / 20 )109.2966666666671.4795287970815673.872618689315
Winsorized Mean ( 12 / 20 )109.8166666666671.364416985375580.4861474488637
Winsorized Mean ( 13 / 20 )109.991.3154894257950383.6114664574568
Winsorized Mean ( 14 / 20 )110.1533333333331.2677696654638986.8874972592334
Winsorized Mean ( 15 / 20 )110.0033333333331.2173927096119090.3597766479169
Winsorized Mean ( 16 / 20 )109.551.08172936452659101.273020399094
Winsorized Mean ( 17 / 20 )109.4083333333331.01467840683795107.825624942866
Winsorized Mean ( 18 / 20 )109.3483333333330.940700819610074116.241350123049
Winsorized Mean ( 19 / 20 )109.2850.901921442330535121.169089535793
Winsorized Mean ( 20 / 20 )108.3850.668677018299762162.088717024535
Trimmed Mean ( 1 / 20 )109.61.9693999073728855.6514700694807
Trimmed Mean ( 2 / 20 )109.5410714285711.8759063208440258.3936789440988
Trimmed Mean ( 3 / 20 )109.5166666666671.8124130379870860.4258876819267
Trimmed Mean ( 4 / 20 )109.4653846153851.7554480690675462.3575180287336
Trimmed Mean ( 5 / 20 )109.4221.6976268910524764.4558592802226
Trimmed Mean ( 6 / 20 )109.451.6519760357508366.2539877282508
Trimmed Mean ( 7 / 20 )109.4391304347831.6098323442003367.9816943851663
Trimmed Mean ( 8 / 20 )109.4318181818181.5599329576017670.151616227186
Trimmed Mean ( 9 / 20 )109.4190476190481.4984592865320673.0210347404768
Trimmed Mean ( 10 / 20 )109.42251.4306824958201476.482727872667
Trimmed Mean ( 11 / 20 )109.4105263157891.3767154795805579.4721407135802
Trimmed Mean ( 12 / 20 )109.4277777777781.3221471900812982.7652008783149
Trimmed Mean ( 13 / 20 )109.3705882352941.2792288885220385.4972782561659
Trimmed Mean ( 14 / 20 )109.281251.2313318705382588.7504438200159
Trimmed Mean ( 15 / 20 )109.1566666666671.1751026693548592.8911741189347
Trimmed Mean ( 16 / 20 )109.0357142857141.1081732702860998.392297675223
Trimmed Mean ( 17 / 20 )108.9615384615381.05713803658858103.072195579265
Trimmed Mean ( 18 / 20 )108.8958333333331.00174628623254108.706001539451
Trimmed Mean ( 19 / 20 )108.8272727272730.941909771742022115.538957119004
Trimmed Mean ( 20 / 20 )108.7550.857059692080705126.893145255697
Median110.15
Midrange110.5
Midmean - Weighted Average at Xnp108.829032258065
Midmean - Weighted Average at X(n+1)p109.156666666667
Midmean - Empirical Distribution Function108.829032258065
Midmean - Empirical Distribution Function - Averaging109.156666666667
Midmean - Empirical Distribution Function - Interpolation109.156666666667
Midmean - Closest Observation108.829032258065
Midmean - True Basic - Statistics Graphics Toolkit109.156666666667
Midmean - MS Excel (old versions)109.28125
Number of observations60



Parameters (Session):
Parameters (R input):
R code (references can be found in the software module):
geomean <- function(x) {
return(exp(mean(log(x))))
}
harmean <- function(x) {
return(1/mean(1/x))
}
quamean <- function(x) {
return(sqrt(mean(x*x)))
}
winmean <- function(x) {
x <-sort(x[!is.na(x)])
n<-length(x)
denom <- 3
nodenom <- n/denom
if (nodenom>40) denom <- n/40
sqrtn = sqrt(n)
roundnodenom = floor(nodenom)
win <- array(NA,dim=c(roundnodenom,2))
for (j in 1:roundnodenom) {
win[j,1] <- (j*x[j+1]+sum(x[(j+1):(n-j)])+j*x[n-j])/n
win[j,2] <- sd(c(rep(x[j+1],j),x[(j+1):(n-j)],rep(x[n-j],j)))/sqrtn
}
return(win)
}
trimean <- function(x) {
x <-sort(x[!is.na(x)])
n<-length(x)
denom <- 3
nodenom <- n/denom
if (nodenom>40) denom <- n/40
sqrtn = sqrt(n)
roundnodenom = floor(nodenom)
tri <- array(NA,dim=c(roundnodenom,2))
for (j in 1:roundnodenom) {
tri[j,1] <- mean(x,trim=j/n)
tri[j,2] <- sd(x[(j+1):(n-j)]) / sqrt(n-j*2)
}
return(tri)
}
midrange <- function(x) {
return((max(x)+min(x))/2)
}
q1 <- function(data,n,p,i,f) {
np <- n*p;
i <<- floor(np)
f <<- np - i
qvalue <- (1-f)*data[i] + f*data[i+1]
}
q2 <- function(data,n,p,i,f) {
np <- (n+1)*p
i <<- floor(np)
f <<- np - i
qvalue <- (1-f)*data[i] + f*data[i+1]
}
q3 <- function(data,n,p,i,f) {
np <- n*p
i <<- floor(np)
f <<- np - i
if (f==0) {
qvalue <- data[i]
} else {
qvalue <- data[i+1]
}
}
q4 <- function(data,n,p,i,f) {
np <- n*p
i <<- floor(np)
f <<- np - i
if (f==0) {
qvalue <- (data[i]+data[i+1])/2
} else {
qvalue <- data[i+1]
}
}
q5 <- function(data,n,p,i,f) {
np <- (n-1)*p
i <<- floor(np)
f <<- np - i
if (f==0) {
qvalue <- data[i+1]
} else {
qvalue <- data[i+1] + f*(data[i+2]-data[i+1])
}
}
q6 <- function(data,n,p,i,f) {
np <- n*p+0.5
i <<- floor(np)
f <<- np - i
qvalue <- data[i]
}
q7 <- function(data,n,p,i,f) {
np <- (n+1)*p
i <<- floor(np)
f <<- np - i
if (f==0) {
qvalue <- data[i]
} else {
qvalue <- f*data[i] + (1-f)*data[i+1]
}
}
q8 <- function(data,n,p,i,f) {
np <- (n+1)*p
i <<- floor(np)
f <<- np - i
if (f==0) {
qvalue <- data[i]
} else {
if (f == 0.5) {
qvalue <- (data[i]+data[i+1])/2
} else {
if (f < 0.5) {
qvalue <- data[i]
} else {
qvalue <- data[i+1]
}
}
}
}
midmean <- function(x,def) {
x <-sort(x[!is.na(x)])
n<-length(x)
if (def==1) {
qvalue1 <- q1(x,n,0.25,i,f)
qvalue3 <- q1(x,n,0.75,i,f)
}
if (def==2) {
qvalue1 <- q2(x,n,0.25,i,f)
qvalue3 <- q2(x,n,0.75,i,f)
}
if (def==3) {
qvalue1 <- q3(x,n,0.25,i,f)
qvalue3 <- q3(x,n,0.75,i,f)
}
if (def==4) {
qvalue1 <- q4(x,n,0.25,i,f)
qvalue3 <- q4(x,n,0.75,i,f)
}
if (def==5) {
qvalue1 <- q5(x,n,0.25,i,f)
qvalue3 <- q5(x,n,0.75,i,f)
}
if (def==6) {
qvalue1 <- q6(x,n,0.25,i,f)
qvalue3 <- q6(x,n,0.75,i,f)
}
if (def==7) {
qvalue1 <- q7(x,n,0.25,i,f)
qvalue3 <- q7(x,n,0.75,i,f)
}
if (def==8) {
qvalue1 <- q8(x,n,0.25,i,f)
qvalue3 <- q8(x,n,0.75,i,f)
}
midm <- 0
myn <- 0
roundno4 <- round(n/4)
round3no4 <- round(3*n/4)
for (i in 1:n) {
if ((x[i]>=qvalue1) & (x[i]<=qvalue3)){
midm = midm + x[i]
myn = myn + 1
}
}
midm = midm / myn
return(midm)
}
(arm <- mean(x))
sqrtn <- sqrt(length(x))
(armse <- sd(x) / sqrtn)
(armose <- arm / armse)
(geo <- geomean(x))
(har <- harmean(x))
(qua <- quamean(x))
(win <- winmean(x))
(tri <- trimean(x))
(midr <- midrange(x))
midm <- array(NA,dim=8)
for (j in 1:8) midm[j] <- midmean(x,j)
midm
bitmap(file='test1.png')
lb <- win[,1] - 2*win[,2]
ub <- win[,1] + 2*win[,2]
if ((ylimmin == '') | (ylimmax == '')) plot(win[,1],type='b',main=main, xlab='j', pch=19, ylab='Winsorized Mean(j/n)', ylim=c(min(lb),max(ub))) else plot(win[,1],type='l',main=main, xlab='j', pch=19, ylab='Winsorized Mean(j/n)', ylim=c(ylimmin,ylimmax))
lines(ub,lty=3)
lines(lb,lty=3)
grid()
dev.off()
bitmap(file='test2.png')
lb <- tri[,1] - 2*tri[,2]
ub <- tri[,1] + 2*tri[,2]
if ((ylimmin == '') | (ylimmax == '')) plot(tri[,1],type='b',main=main, xlab='j', pch=19, ylab='Trimmed Mean(j/n)', ylim=c(min(lb),max(ub))) else plot(tri[,1],type='l',main=main, xlab='j', pch=19, ylab='Trimmed Mean(j/n)', ylim=c(ylimmin,ylimmax))
lines(ub,lty=3)
lines(lb,lty=3)
grid()
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Central Tendency - Ungrouped Data',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Measure',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.element(a,'S.E.',header=TRUE)
a<-table.element(a,'Value/S.E.',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,hyperlink('arithmetic_mean.htm', 'Arithmetic Mean', 'click to view the definition of the Arithmetic Mean'),header=TRUE)
a<-table.element(a,arm)
a<-table.element(a,hyperlink('arithmetic_mean_standard_error.htm', armse, 'click to view the definition of the Standard Error of the Arithmetic Mean'))
a<-table.element(a,armose)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,hyperlink('geometric_mean.htm', 'Geometric Mean', 'click to view the definition of the Geometric Mean'),header=TRUE)
a<-table.element(a,geo)
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,hyperlink('harmonic_mean.htm', 'Harmonic Mean', 'click to view the definition of the Harmonic Mean'),header=TRUE)
a<-table.element(a,har)
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,hyperlink('quadratic_mean.htm', 'Quadratic Mean', 'click to view the definition of the Quadratic Mean'),header=TRUE)
a<-table.element(a,qua)
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
for (j in 1:length(win[,1])) {
a<-table.row.start(a)
mylabel <- paste('Winsorized Mean (',j)
mylabel <- paste(mylabel,'/')
mylabel <- paste(mylabel,length(win[,1]))
mylabel <- paste(mylabel,')')
a<-table.element(a,hyperlink('winsorized_mean.htm', mylabel, 'click to view the definition of the Winsorized Mean'),header=TRUE)
a<-table.element(a,win[j,1])
a<-table.element(a,win[j,2])
a<-table.element(a,win[j,1]/win[j,2])
a<-table.row.end(a)
}
for (j in 1:length(tri[,1])) {
a<-table.row.start(a)
mylabel <- paste('Trimmed Mean (',j)
mylabel <- paste(mylabel,'/')
mylabel <- paste(mylabel,length(tri[,1]))
mylabel <- paste(mylabel,')')
a<-table.element(a,hyperlink('arithmetic_mean.htm', mylabel, 'click to view the definition of the Trimmed Mean'),header=TRUE)
a<-table.element(a,tri[j,1])
a<-table.element(a,tri[j,2])
a<-table.element(a,tri[j,1]/tri[j,2])
a<-table.row.end(a)
}
a<-table.row.start(a)
a<-table.element(a,hyperlink('median_1.htm', 'Median', 'click to view the definition of the Median'),header=TRUE)
a<-table.element(a,median(x))
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,hyperlink('midrange.htm', 'Midrange', 'click to view the definition of the Midrange'),header=TRUE)
a<-table.element(a,midr)
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_1.htm','Weighted Average at Xnp',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[1])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_2.htm','Weighted Average at X(n+1)p',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[2])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_3.htm','Empirical Distribution Function',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[3])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_4.htm','Empirical Distribution Function - Averaging',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[4])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_5.htm','Empirical Distribution Function - Interpolation',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[5])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_6.htm','Closest Observation',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[6])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_7.htm','True Basic - Statistics Graphics Toolkit',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[7])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
mymid <- hyperlink('midmean.htm', 'Midmean', 'click to view the definition of the Midmean')
mylabel <- paste(mymid,hyperlink('method_8.htm','MS Excel (old versions)',''),sep=' - ')
a<-table.element(a,mylabel,header=TRUE)
a<-table.element(a,midm[8])
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Number of observations',header=TRUE)
a<-table.element(a,length(x))
a<-table.element(a,'')
a<-table.element(a,'')
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable.tab')