::Free Statistics and Forecasting Software::
v1.1.22-r6
 
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:: Central Tendency - Free Statistics Software (Calculator) ::
All rights reserved. The non-commercial (academic) use of this software is free of charge. The only thing that is asked in return is to cite this software when results are used in publications.

This free online software (calculator) computes the following statistics: arithmetic mean, geometric mean, harmonic mean, quadratic mean, winsorized mean, trimmed mean, median, midrange, and midmean.

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Cite this software as:
Wessa, P., (2008), Central Tendency (v1.0.3) in Free Statistics Software (v1.1.22-r6), Office for Research Development and Education, URL http://www.wessa.net/rwasp_centraltendency.wasp/
The R code is based on :
Borghers, E, and P. Wessa, Statistics - Econometrics - Forecasting, Office for Research Development and Education, http://www.xycoon.com/

Source code of R module
1geomean <- function(x) {
2return(exp(mean(log(x))))
3}
4harmean <- function(x) {
5return(1/mean(1/x))
6}
7quamean <- function(x) {
8return(sqrt(mean(x*x)))
9}
10winmean <- function(x) {
11x <-sort(x[!is.na(x)])
12n<-length(x)
13denom <- 3
14nodenom <- n/denom
15if (nodenom>40) denom <- n/40
16sqrtn = sqrt(n)
17roundnodenom = floor(nodenom)
18win <- array(NA,dim=c(roundnodenom,2))
19for (j in 1:roundnodenom) {
20win[j,1] <- (j*x[j+1]+sum(x[(j+1):(n-j)])+j*x[n-j])/n
21win[j,2] <- sd(c(rep(x[j+1],j),x[(j+1):(n-j)],rep(x[n-j],j)))/sqrtn
22}
23return(win)
24}
25trimean <- function(x) {
26x <-sort(x[!is.na(x)])
27n<-length(x)
28denom <- 3
29nodenom <- n/denom
30if (nodenom>40) denom <- n/40
31sqrtn = sqrt(n)
32roundnodenom = floor(nodenom)
33tri <- array(NA,dim=c(roundnodenom,2))
34for (j in 1:roundnodenom) {
35tri[j,1] <- mean(x,trim=j/n)
36tri[j,2] <- sd(x[(j+1):(n-j)]) / sqrt(n-j*2)
37}
38return(tri)
39}
40midrange <- function(x) {
41return((max(x)+min(x))/2)
42}
43q1 <- function(data,n,p,i,f) {
44np <- n*p;
45i <<- floor(np)
46f <<- np - i
47qvalue <- (1-f)*data[i] + f*data[i+1]
48}
49q2 <- function(data,n,p,i,f) {
50np <- (n+1)*p
51i <<- floor(np)
52f <<- np - i
53qvalue <- (1-f)*data[i] + f*data[i+1]
54}
55q3 <- function(data,n,p,i,f) {
56np <- n*p
57i <<- floor(np)
58f <<- np - i
59if (f==0) {
60qvalue <- data[i]
61} else {
62qvalue <- data[i+1]
63}
64}
65q4 <- function(data,n,p,i,f) {
66np <- n*p
67i <<- floor(np)
68f <<- np - i
69if (f==0) {
70qvalue <- (data[i]+data[i+1])/2
71} else {
72qvalue <- data[i+1]
73}
74}
75q5 <- function(data,n,p,i,f) {
76np <- (n-1)*p
77i <<- floor(np)
78f <<- np - i
79if (f==0) {
80qvalue <- data[i+1]
81} else {
82qvalue <- data[i+1] + f*(data[i+2]-data[i+1])
83}
84}
85q6 <- function(data,n,p,i,f) {
86np <- n*p+0.5
87i <<- floor(np)
88f <<- np - i
89qvalue <- data[i]
90}
91q7 <- function(data,n,p,i,f) {
92np <- (n+1)*p
93i <<- floor(np)
94f <<- np - i
95if (f==0) {
96qvalue <- data[i]
97} else {
98qvalue <- f*data[i] + (1-f)*data[i+1]
99}
100}
101q8 <- function(data,n,p,i,f) {
102np <- (n+1)*p
103i <<- floor(np)
104f <<- np - i
105if (f==0) {
106qvalue <- data[i]
107} else {
108if (f == 0.5) {
109qvalue <- (data[i]+data[i+1])/2
110} else {
111if (f < 0.5) {
112qvalue <- data[i]
113} else {
114qvalue <- data[i+1]
115}
116}
117}
118}
119midmean <- function(x,def) {
120x <-sort(x[!is.na(x)])
121n<-length(x)
122if (def==1) {
123qvalue1 <- q1(x,n,0.25,i,f)
124qvalue3 <- q1(x,n,0.75,i,f)
125}
126if (def==2) {
127qvalue1 <- q2(x,n,0.25,i,f)
128qvalue3 <- q2(x,n,0.75,i,f)
129}
130if (def==3) {
131qvalue1 <- q3(x,n,0.25,i,f)
132qvalue3 <- q3(x,n,0.75,i,f)
133}
134if (def==4) {
135qvalue1 <- q4(x,n,0.25,i,f)
136qvalue3 <- q4(x,n,0.75,i,f)
137}
138if (def==5) {
139qvalue1 <- q5(x,n,0.25,i,f)
140qvalue3 <- q5(x,n,0.75,i,f)
141}
142if (def==6) {
143qvalue1 <- q6(x,n,0.25,i,f)
144qvalue3 <- q6(x,n,0.75,i,f)
145}
146if (def==7) {
147qvalue1 <- q7(x,n,0.25,i,f)
148qvalue3 <- q7(x,n,0.75,i,f)
149}
150if (def==8) {
151qvalue1 <- q8(x,n,0.25,i,f)
152qvalue3 <- q8(x,n,0.75,i,f)
153}
154midm <- 0
155myn <- 0
156roundno4 <- round(n/4)
157round3no4 <- round(3*n/4)
158for (i in 1:n) {
159if ((x[i]>=qvalue1) & (x[i]<=qvalue3)){
160midm = midm + x[i]
161myn = myn + 1
162}
163}
164midm = midm / myn
165return(midm)
166}
167(arm <- mean(x))
168sqrtn <- sqrt(length(x))
169(armse <- sd(x) / sqrtn)
170(armose <- arm / armse)
171(geo <- geomean(x))
172(har <- harmean(x))
173(qua <- quamean(x))
174(win <- winmean(x))
175(tri <- trimean(x))
176(midr <- midrange(x))
177midm <- array(NA,dim=8)
178for (j in 1:8) midm[j] <- midmean(x,j)
179midm
180bitmap(file="test1.png")
181lb <- win[,1] - 2*win[,2]
182ub <- win[,1] + 2*win[,2]
183if ((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))
184lines(ub,lty=3)
185lines(lb,lty=3)
186grid()
187dev.off()
188bitmap(file="test2.png")
189lb <- tri[,1] - 2*tri[,2]
190ub <- tri[,1] + 2*tri[,2]
191if ((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))
192lines(ub,lty=3)
193lines(lb,lty=3)
194grid()
195dev.off()
196load(file="createtable")
197a<-table.start()
198a<-table.row.start(a)
199a<-table.element(a,"Central Tendency - Ungrouped Data",4,TRUE)
200a<-table.row.end(a)
201a<-table.row.start(a)
202a<-table.element(a,"Measure",header=TRUE)
203a<-table.element(a,"Value",header=TRUE)
204a<-table.element(a,"S.E.",header=TRUE)
205a<-table.element(a,"Value/S.E.",header=TRUE)
206a<-table.row.end(a)
207a<-table.row.start(a)
208a<-table.element(a,hyperlink("http://www.xycoon.com/arithmetic_mean.htm", "Arithmetic Mean", "click to view the definition of the Arithmetic Mean"),header=TRUE)
209a<-table.element(a,arm)
210a<-table.element(a,hyperlink("http://www.xycoon.com/arithmetic_mean_standard_error.htm", armse, "click to view the definition of the Standard Error of the Arithmetic Mean"))
211a<-table.element(a,armose)
212a<-table.row.end(a)
213a<-table.row.start(a)
214a<-table.element(a,hyperlink("http://www.xycoon.com/geometric_mean.htm", "Geometric Mean", "click to view the definition of the Geometric Mean"),header=TRUE)
215a<-table.element(a,geo)
216a<-table.element(a,"")
217a<-table.element(a,"")
218a<-table.row.end(a)
219a<-table.row.start(a)
220a<-table.element(a,hyperlink("http://www.xycoon.com/harmonic_mean.htm", "Harmonic Mean", "click to view the definition of the Harmonic Mean"),header=TRUE)
221a<-table.element(a,har)
222a<-table.element(a,"")
223a<-table.element(a,"")
224a<-table.row.end(a)
225a<-table.row.start(a)
226a<-table.element(a,hyperlink("http://www.xycoon.com/quadratic_mean.htm", "Quadratic Mean", "click to view the definition of the Quadratic Mean"),header=TRUE)
227a<-table.element(a,qua)
228a<-table.element(a,"")
229a<-table.element(a,"")
230a<-table.row.end(a)
231for (j in 1:length(win[,1])) {
232a<-table.row.start(a)
233mylabel <- paste("Winsorized Mean (",j)
234mylabel <- paste(mylabel,"/")
235mylabel <- paste(mylabel,length(win[,1]))
236mylabel <- paste(mylabel,")")
237a<-table.element(a,hyperlink("http://www.xycoon.com/winsorized_mean.htm", mylabel, "click to view the definition of the Winsorized Mean"),header=TRUE)
238a<-table.element(a,win[j,1])
239a<-table.element(a,win[j,2])
240a<-table.element(a,win[j,1]/win[j,2])
241a<-table.row.end(a)
242}
243for (j in 1:length(tri[,1])) {
244a<-table.row.start(a)
245mylabel <- paste("Trimmed Mean (",j)
246mylabel <- paste(mylabel,"/")
247mylabel <- paste(mylabel,length(tri[,1]))
248mylabel <- paste(mylabel,")")
249a<-table.element(a,hyperlink("http://www.xycoon.com/arithmetic_mean.htm", mylabel, "click to view the definition of the Trimmed Mean"),header=TRUE)
250a<-table.element(a,tri[j,1])
251a<-table.element(a,tri[j,2])
252a<-table.element(a,tri[j,1]/tri[j,2])
253a<-table.row.end(a)
254}
255a<-table.row.start(a)
256a<-table.element(a,hyperlink("http://www.xycoon.com/median_1.htm", "Median", "click to view the definition of the Median"),header=TRUE)
257a<-table.element(a,median(x))
258a<-table.element(a,"")
259a<-table.element(a,"")
260a<-table.row.end(a)
261a<-table.row.start(a)
262a<-table.element(a,hyperlink("http://www.xycoon.com/midrange.htm", "Midrange", "click to view the definition of the Midrange"),header=TRUE)
263a<-table.element(a,midr)
264a<-table.element(a,"")
265a<-table.element(a,"")
266a<-table.row.end(a)
267a<-table.row.start(a)
268mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
269mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_1.htm","Weighted Average at Xnp",""),sep=" - ")
270a<-table.element(a,mylabel,header=TRUE)
271a<-table.element(a,midm[1])
272a<-table.element(a,"")
273a<-table.element(a,"")
274a<-table.row.end(a)
275a<-table.row.start(a)
276mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
277mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_2.htm","Weighted Average at X(n+1)p",""),sep=" - ")
278a<-table.element(a,mylabel,header=TRUE)
279a<-table.element(a,midm[2])
280a<-table.element(a,"")
281a<-table.element(a,"")
282a<-table.row.end(a)
283a<-table.row.start(a)
284mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
285mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_3.htm","Empirical Distribution Function",""),sep=" - ")
286a<-table.element(a,mylabel,header=TRUE)
287a<-table.element(a,midm[3])
288a<-table.element(a,"")
289a<-table.element(a,"")
290a<-table.row.end(a)
291a<-table.row.start(a)
292mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
293mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_4.htm","Empirical Distribution Function - Averaging",""),sep=" - ")
294a<-table.element(a,mylabel,header=TRUE)
295a<-table.element(a,midm[4])
296a<-table.element(a,"")
297a<-table.element(a,"")
298a<-table.row.end(a)
299a<-table.row.start(a)
300mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
301mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_5.htm","Empirical Distribution Function - Interpolation",""),sep=" - ")
302a<-table.element(a,mylabel,header=TRUE)
303a<-table.element(a,midm[5])
304a<-table.element(a,"")
305a<-table.element(a,"")
306a<-table.row.end(a)
307a<-table.row.start(a)
308mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
309mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_6.htm","Closest Observation",""),sep=" - ")
310a<-table.element(a,mylabel,header=TRUE)
311a<-table.element(a,midm[6])
312a<-table.element(a,"")
313a<-table.element(a,"")
314a<-table.row.end(a)
315a<-table.row.start(a)
316mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
317mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_7.htm","True Basic - Statistics Graphics Toolkit",""),sep=" - ")
318a<-table.element(a,mylabel,header=TRUE)
319a<-table.element(a,midm[7])
320a<-table.element(a,"")
321a<-table.element(a,"")
322a<-table.row.end(a)
323a<-table.row.start(a)
324mymid <- hyperlink("http://www.xycoon.com/midmean.htm", "Midmean", "click to view the definition of the Midmean")
325mylabel <- paste(mymid,hyperlink("http://www.xycoon.com/method_8.htm","MS Excel (old versions)",""),sep=" - ")
326a<-table.element(a,mylabel,header=TRUE)
327a<-table.element(a,midm[8])
328a<-table.element(a,"")
329a<-table.element(a,"")
330a<-table.row.end(a)
331a<-table.row.start(a)
332a<-table.element(a,"Number of observations",header=TRUE)
333a<-table.element(a,length(x))
334a<-table.element(a,"")
335a<-table.element(a,"")
336a<-table.row.end(a)
337a<-table.end(a)
338table.save(a,file="mytable.tab")
Delete history
Server DateModuleCommand
Sat, 17 May 2008 14:41:58 -0600Start of session-

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To cite Wessa.net in publications use:
Wessa, P. (2008), Free Statistics Software, Office for Research Development and Education,
version 1.1.22-r6, URL http://www.wessa.net/

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Software Version : 1.1.22-r6
Algorithms & Software : Prof. dr. P. Wessa
Facilities : Resa R&D - Office for Research, Development, and Education
Server : www.wessa.net

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