note.w02
This is an old revision of the document!
Sampling Distribution and z-test
rm(list=ls())
rnorm2 <- function(n,mean,sd){
mean+sd*scale(rnorm(n))
}
se <- function(sample) {
sd(sample)/sqrt(length(sample))
}
ss <- function(x) {
sum((x-mean(x))^2)
}
################################
N.p <- 1000000
m.p <- 100
sd.p <- 10
p1 <- rnorm2(N.p, m.p, sd.p)
mean(p1)
sd(p1)
p2 <- rnorm2(N.p, m.p+5, sd.p)
mean(p2)
sd(p2)
m.p1 <- mean(p1)
sd.p1 <- sd(p1)
var(p1)
hist(p1, breaks=50, col = rgb(1, 1, 1, 0.5),
main = "histogram of p1 and p2",)
abline(v=mean(p1), col="black", lwd=3)
hist(p2, add=T, breaks=50, col=rgb(1,1,.5,.5))
abline(v=mean(p2), col="red", lwd=3)
hist(p1, breaks=50, col=rgb(0,.5,.5,.5))
abline(v=mean(p1),lwd=2)
abline(v=mean(p1)-sd(p1), lwd=2)
abline(v=mean(p1)+sd(p1), lwd=2)
abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')
# area bet black = 68%
# between red = 95%
# between green = 99%
pnorm(m.p1+sd.p1, m.p1, sd.p1)
pnorm(m.p1-sd.p1, m.p1, sd.p1)
pnorm(m.p1+sd.p1, m.p1, sd.p1) -
pnorm(m.p1-sd.p1, m.p1, sd.p1)
pnorm(m.p1+2*sd.p1, m.p1, sd.p1) -
pnorm(m.p1-2*sd.p1, m.p1, sd.p1)
pnorm(m.p1+3*sd.p1, m.p1, sd.p1) -
pnorm(m.p1-3*sd.p1, m.p1, sd.p1)
m.p1
sd.p1
(m.p1-m.p1)/sd.p1
((m.p1-sd.p1) - m.p1) / sd.p1
(120-100)/10
pnorm(1)-pnorm(-1)
pnorm(2)-pnorm(-2)
pnorm(3)-pnorm(3)
1-pnorm(-2)*2
pnorm(2)-pnorm(-2)
pnorm(120, 100, 10)
pnorm(2)-pnorm(-2)
zscore <- (120-100)/10
pnorm(zscore)-pnorm(-zscore)
zscore
# reminder.
pnorm(-1)
pnorm(1, 0, 1, lower.tail = F)
pnorm(110, 100, 10, lower.tail = F)
zscore <- (110-100)/10
pnorm(zscore, lower.tail = F)
pnorm(118, 100, 10, lower.tail = F)
pnorm(18/10, lower.tail = F)
z.p1 <- (p1-mean(p1))/sd(p1)
mean(z.p1)
round(mean(z.p1),10)
sd(z.p1)
pnorm(1.8)-pnorm(-1.8)
hist(z.p1, breaks=50, col=rgb(1,0,0,0))
abline(v=c(m.p1, -1.8, 1.8), col='red')
1-(pnorm(1.8)-pnorm(-1.8))
pnorm(1)-pnorm(-1)
1-(pnorm(-1)*2)
pnorm(2)-pnorm(-2)
1-(pnorm(-2)*2)
1-(pnorm(-3)*2)
#
hist(p1, breaks=50, col=rgb(.9,.9,.9,.9))
abline(v=mean(p1),lwd=2)
abline(v=mean(p1)-sd(p1), lwd=2)
abline(v=mean(p1)+sd(p1), lwd=2)
abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')
# 68%
a <- qnorm(.32/2)
b <- qnorm(1-.32/2)
c(a, b)
c(-1, 1)
# note that
.32/2
pnorm(-1)
qnorm(.32/2)
qnorm(pnorm(-1))
# 95%
c <- qnorm(.05/2)
d <- qnorm(1-.05/2)
c(c, d)
c(-2,2)
# 99%
e <- qnorm(.01/2)
f <- qnorm(1-.01/2)
c(e,f)
c(-3,3)
pnorm(b)-pnorm(a)
c(a, b)
pnorm(d)-pnorm(c)
c(c, d)
pnorm(f)-pnorm(e)
c(e, f)
qnorm(.5)
qnorm(1)
################################
s.size <- 50
means.temp <- c()
s1 <- sample(p1, s.size, replace = T)
mean(s1)
means.temp <- append(means.temp, mean(s1))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp
iter <- 1000000
# means <- c()
means <- rep(NA, iter)
for (i in 1:iter) {
# means <- append(means, mean(sample(p1, s.size, replace = T)))
means[i] <- mean(sample(p1, s.size, replace = T))
}
length(means)
mean(means)
sd(means)
se.s <- sd(means)
hist(means, breaks=50,
xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)),
col=rgb(1, 1, 1, .5))
abline(v=mean(means), col="black", lwd=3)
# now we want to get sd of this distribution
lo1 <- mean(means)-se.s
hi1 <- mean(means)+se.s
lo2 <- mean(means)-2*se.s
hi2 <- mean(means)+2*se.s
lo3 <- mean(means)-3*se.s
hi3 <- mean(means)+3*se.s
abline(v=mean(means), col="black", lwd=2)
# abline(v=mean(p2), colo='darkgreen', lwd=2)
abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3),
col=c("red","red", "blue", "blue", "orange", "orange"),
lwd=2)
# meanwhile . . . .
se.s
se.z <- sqrt(var(p1)/s.size)
se.z <- c(se.z)
se.z
se.z.adjusted <- sqrt(var(s1)/s.size)
se.z.adjusted
# sd of sample means (sd(means))
# = se.s
# when iter value goes to
# infinite value:
# mean(means) = mean(p1)
# and
# sd(means) = sd(p1) / sqrt(s.size)
# that is, se.s = se.z
# This is called CLT (Central Limit Theorem)
# see http://commres.net/wiki/cetral_limit_theorem
mean(means)
mean(p1)
sd(means)
var(p1)
# remember we started talking sample size 10
sqrt(var(p1)/s.size)
se.z
sd(means)
se.s
se.z
# because CLT
loz1 <- mean(p1)-se.z
hiz1 <- mean(p1)+se.z
loz2 <- mean(p1)-2*se.z
hiz2 <- mean(p1)+2*se.z
loz3 <- mean(p1)-3*se.z
hiz3 <- mean(p1)+3*se.z
c(lo1, loz1)
c(lo2, loz2)
c(lo3, loz3)
c(hi1, hiz1)
c(hi2, hiz2)
c(hi3, hiz3)
hist(means, breaks=50,
xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)),
col = rgb(1, 1, 1, .5))
abline(v=mean(means), col="black", lwd=3)
# abline(v=mean(p2), colo='darkgreen', lwd=3)
abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3),
col=c("darkgreen","darkgreen", "blue", "blue", "orange", "orange"),
lwd=2)
round(c(lo1, hi1))
round(c(lo2, hi2))
round(c(lo3, hi3))
round(c(loz1, hiz1))
round(c(loz2, hiz2))
round(c(loz3, hiz3))
m.sample.i.got <- mean(means)+ 1.5*sd(means)
m.sample.i.got
hist(means, breaks=30,
xlim = c(mean(means)-7*sd(means), mean(means)+10*sd(means)),
col = rgb(1, 1, 1, .5))
abline(v=mean(means), col="black", lwd=3)
abline(v=m.sample.i.got, col='darkgreen', lwd=3)
# what is the probablity of getting
# greater than
# m.sample.i.got?
m.sample.i.got
pnorm(m.sample.i.got, mean(means), sd(means), lower.tail = F)
pnorm(m.sample.i.got, mean(p1), se.z, lower.tail = F)
# then, what is the probabilty of getting
# greater than m.sample.i.got and
# less than corresponding value, which is
# mean(means) - m.sample.i.got - mean(means)
# (green line)
tmp <- mean(means) - (m.sample.i.got - mean(means))
abline(v=tmp, col='red', lwd=3)
2 * pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)
m.sample.i.got
### one more time
# this time, with a story
mean(p2)
sd(p2)
s.from.p2 <- sample(p2, s.size)
m.s.from.p2 <- mean(s.from.p2)
m.s.from.p2
se.s
se.z
sd(means)
tmp <- mean(means) - (m.s.from.p2 - mean(means))
tmp
hist(means, breaks=30,
xlim = c(tmp-4*sd(means), m.s.from.p2+4*sd(means)),
col = rgb(1, 1, 1, .5))
abline(v=mean(means), col="black", lwd=3)
abline(v=m.s.from.p2, col='blue', lwd=3)
# what is the probablity of getting
# greater than
# m.sample.i.got?
m.s.from.p2
pnorm(m.s.from.p2, mean(p1), se.z, lower.tail = F)
# then, what is the probabilty of getting
# greater than m.sample.i.got and
# less than corresponding value, which is
# mean(means) - m.sample.i.got - mean(means)
# (green line)
abline(v=tmp, col='red', lwd=3)
2 * pnorm(m.s.from.p2, mean(p1), se.z, lower.tail = F)
se.z
sd(s.from.p2)/sqrt(s.size)
se.z.adjusted <- sqrt(var(s.from.p2)/s.size)
se.z.adjusted
2 * pnorm(m.s.from.p2, mean(p1), se.z.adjusted,
lower.tail = F)
z.cal <- (m.s.from.p2 - mean(p1))/se.z.adjusted
z.cal
pnorm(z.cal, lower.tail = F)*2
pt(z.cal, 49, lower.tail = F)*2
t.test(s.from.p2, mu=mean(p1), var.equal = T)
output
> rm(list=ls())
> rnorm2 <- function(n,mean,sd){
+ mean+sd*scale(rnorm(n))
+ }
>
> se <- function(sample) {
+ sd(sample)/sqrt(length(sample))
+ }
>
> ss <- function(x) {
+ sum((x-mean(x))^2)
+ }
>
> ################################
> N.p <- 1000000
> m.p <- 100
> sd.p <- 10
>
> p1 <- rnorm2(N.p, m.p, sd.p)
> mean(p1)
[1] 100
> sd(p1)
[1] 10
>
> p2 <- rnorm2(N.p, m.p+5, sd.p)
> mean(p2)
[1] 105
> sd(p2)
[1] 10
>
> m.p1 <- mean(p1)
> sd.p1 <- sd(p1)
> var(p1)
[,1]
[1,] 100
>
>
> hist(p1, breaks=50, col = rgb(1, 1, 1, 0.5),
+ main = "histogram of p1 and p2",)
> abline(v=mean(p1), col="black", lwd=3)
> hist(p2, add=T, breaks=50, col=rgb(1,1,.5,.5))
> abline(v=mean(p2), col="red", lwd=3)
>
>
> hist(p1, breaks=50, col=rgb(0,.5,.5,.5))
> abline(v=mean(p1),lwd=2)
> abline(v=mean(p1)-sd(p1), lwd=2)
> abline(v=mean(p1)+sd(p1), lwd=2)
> abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
> abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')
>
> # area bet black = 68%
> # between red = 95%
> # between green = 99%
>
> pnorm(m.p1+sd.p1, m.p1, sd.p1)
[1] 0.8413447
> pnorm(m.p1-sd.p1, m.p1, sd.p1)
[1] 0.1586553
> pnorm(m.p1+sd.p1, m.p1, sd.p1) -
+ pnorm(m.p1-sd.p1, m.p1, sd.p1)
[1] 0.6826895
>
> pnorm(m.p1+2*sd.p1, m.p1, sd.p1) -
+ pnorm(m.p1-2*sd.p1, m.p1, sd.p1)
[1] 0.9544997
>
> pnorm(m.p1+3*sd.p1, m.p1, sd.p1) -
+ pnorm(m.p1-3*sd.p1, m.p1, sd.p1)
[1] 0.9973002
>
> m.p1
[1] 100
> sd.p1
[1] 10
>
> (m.p1-m.p1)/sd.p1
[1] 0
> ((m.p1-sd.p1) - m.p1) / sd.p1
[1] -1
> (120-100)/10
[1] 2
> pnorm(1)-pnorm(-1)
[1] 0.6826895
> pnorm(2)-pnorm(-2)
[1] 0.9544997
> pnorm(3)-pnorm(3)
[1] 0
>
> 1-pnorm(-2)*2
[1] 0.9544997
> pnorm(2)-pnorm(-2)
[1] 0.9544997
>
> pnorm(120, 100, 10)
[1] 0.9772499
> pnorm(2)-pnorm(-2)
[1] 0.9544997
>
> zscore <- (120-100)/10
> pnorm(zscore)-pnorm(-zscore)
[1] 0.9544997
> zscore
[1] 2
>
> # reminder.
> pnorm(-1)
[1] 0.1586553
> pnorm(1, 0, 1, lower.tail = F)
[1] 0.1586553
> pnorm(110, 100, 10, lower.tail = F)
[1] 0.1586553
> zscore <- (110-100)/10
> pnorm(zscore, lower.tail = F)
[1] 0.1586553
>
> pnorm(118, 100, 10, lower.tail = F)
[1] 0.03593032
> pnorm(18/10, lower.tail = F)
[1] 0.03593032
>
> z.p1 <- (p1-mean(p1))/sd(p1)
> mean(z.p1)
[1] 8.189457e-18
> round(mean(z.p1),10)
[1] 0
> sd(z.p1)
[1] 1
> pnorm(1.8)-pnorm(-1.8)
[1] 0.9281394
>
> hist(z.p1, breaks=50, col=rgb(1,0,0,0))
> abline(v=c(m.p1, -1.8, 1.8), col='red')
> 1-(pnorm(1.8)-pnorm(-1.8))
[1] 0.07186064
>
>
> pnorm(1)-pnorm(-1)
[1] 0.6826895
> 1-(pnorm(-1)*2)
[1] 0.6826895
>
> pnorm(2)-pnorm(-2)
[1] 0.9544997
> 1-(pnorm(-2)*2)
[1] 0.9544997
>
> 1-(pnorm(-3)*2)
[1] 0.9973002
>
> #
> hist(p1, breaks=50, col=rgb(.9,.9,.9,.9))
> abline(v=mean(p1),lwd=2)
> abline(v=mean(p1)-sd(p1), lwd=2)
> abline(v=mean(p1)+sd(p1), lwd=2)
> abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
> abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')
>
> # 68%
> a <- qnorm(.32/2)
> b <- qnorm(1-.32/2)
> c(a, b)
[1] -0.9944579 0.9944579
> c(-1, 1)
[1] -1 1
> # note that
> .32/2
[1] 0.16
> pnorm(-1)
[1] 0.1586553
> qnorm(.32/2)
[1] -0.9944579
> qnorm(pnorm(-1))
[1] -1
>
> # 95%
> c <- qnorm(.05/2)
> d <- qnorm(1-.05/2)
> c(c, d)
[1] -1.959964 1.959964
> c(-2,2)
[1] -2 2
>
> # 99%
> e <- qnorm(.01/2)
> f <- qnorm(1-.01/2)
> c(e,f)
[1] -2.575829 2.575829
> c(-3,3)
[1] -3 3
>
>
> pnorm(b)-pnorm(a)
[1] 0.68
> c(a, b)
[1] -0.9944579 0.9944579
> pnorm(d)-pnorm(c)
[1] 0.95
> c(c, d)
[1] -1.959964 1.959964
> pnorm(f)-pnorm(e)
[1] 0.99
> c(e, f)
[1] -2.575829 2.575829
>
> qnorm(.5)
[1] 0
> qnorm(1)
[1] Inf
>
>
> ################################
> s.size <- 50
>
> means.temp <- c()
> s1 <- sample(p1, s.size, replace = T)
> mean(s1)
[1] 98.76098
> means.temp <- append(means.temp, mean(s1))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp
[1] 98.76098 99.90935 99.29643 99.66014 101.93822
>
> iter <- 1000000
> # means <- c()
> means <- rep(NA, iter)
> for (i in 1:iter) {
+ # means <- append(means, mean(sample(p1, s.size, replace = T)))
+ means[i] <- mean(sample(p1, s.size, replace = T))
+ }
> length(means)
[1] 1000000
> mean(means)
[1] 99.99824
> sd(means)
[1] 1.414035
> se.s <- sd(means)
>
> hist(means, breaks=50,
+ xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)),
+ col=rgb(1, 1, 1, .5))
> abline(v=mean(means), col="black", lwd=3)
> # now we want to get sd of this distribution
> lo1 <- mean(means)-se.s
> hi1 <- mean(means)+se.s
> lo2 <- mean(means)-2*se.s
> hi2 <- mean(means)+2*se.s
> lo3 <- mean(means)-3*se.s
> hi3 <- mean(means)+3*se.s
> abline(v=mean(means), col="black", lwd=2)
> # abline(v=mean(p2), colo='darkgreen', lwd=2)
> abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3),
+ col=c("red","red", "blue", "blue", "orange", "orange"),
+ lwd=2)
>
> # meanwhile . . . .
> se.s
[1] 1.414035
>
> se.z <- sqrt(var(p1)/s.size)
> se.z <- c(se.z)
> se.z
[1] 1.414214
>
> se.z.adjusted <- sqrt(var(s1)/s.size)
> se.z.adjusted
[1] 1.370464
>
> # sd of sample means (sd(means))
> # = se.s
>
> # when iter value goes to
> # infinite value:
> # mean(means) = mean(p1)
> # and
> # sd(means) = sd(p1) / sqrt(s.size)
> # that is, se.s = se.z
> # This is called CLT (Central Limit Theorem)
> # see http://commres.net/wiki/cetral_limit_theorem
>
> mean(means)
[1] 99.99824
> mean(p1)
[1] 100
> sd(means)
[1] 1.414035
> var(p1)
[,1]
[1,] 100
> # remember we started talking sample size 10
> sqrt(var(p1)/s.size)
[,1]
[1,] 1.414214
> se.z
[1] 1.414214
>
> sd(means)
[1] 1.414035
> se.s
[1] 1.414035
> se.z
[1] 1.414214
>
> # because CLT
> loz1 <- mean(p1)-se.z
> hiz1 <- mean(p1)+se.z
> loz2 <- mean(p1)-2*se.z
> hiz2 <- mean(p1)+2*se.z
> loz3 <- mean(p1)-3*se.z
> hiz3 <- mean(p1)+3*se.z
>
> c(lo1, loz1)
[1] 98.58421 98.58579
> c(lo2, loz2)
[1] 97.17017 97.17157
> c(lo3, loz3)
[1] 95.75614 95.75736
>
> c(hi1, hiz1)
[1] 101.4123 101.4142
> c(hi2, hiz2)
[1] 102.8263 102.8284
> c(hi3, hiz3)
[1] 104.2403 104.2426
>
>
> hist(means, breaks=50,
+ xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)),
+ col = rgb(1, 1, 1, .5))
> abline(v=mean(means), col="black", lwd=3)
> # abline(v=mean(p2), colo='darkgreen', lwd=3)
> abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3),
+ col=c("darkgreen","darkgreen", "blue", "blue", "orange", "orange"),
+ lwd=2)
>
> round(c(lo1, hi1))
[1] 99 101
> round(c(lo2, hi2))
[1] 97 103
> round(c(lo3, hi3))
[1] 96 104
>
> round(c(loz1, hiz1))
[1] 99 101
> round(c(loz2, hiz2))
[1] 97 103
> round(c(loz3, hiz3))
[1] 96 104
>
> m.sample.i.got <- mean(means)+ 1.5*sd(means)
> m.sample.i.got
[1] 102.1193
>
> hist(means, breaks=30,
+ xlim = c(mean(means)-7*sd(means), mean(means)+10*sd(means)),
+ col = rgb(1, 1, 1, .5))
> abline(v=mean(means), col="black", lwd=3)
> abline(v=m.sample.i.got, col='darkgreen', lwd=3)
>
> # what is the probablity of getting
> # greater than
> # m.sample.i.got?
> m.sample.i.got
[1] 102.1193
> pnorm(m.sample.i.got, mean(means), sd(means), lower.tail = F)
[1] 0.0668072
> pnorm(m.sample.i.got, mean(p1), se.z, lower.tail = F)
[1] 0.0669929
>
> # then, what is the probabilty of getting
> # greater than m.sample.i.got and
> # less than corresponding value, which is
> # mean(means) - m.sample.i.got - mean(means)
> # (green line)
> tmp <- mean(means) - (m.sample.i.got - mean(means))
> abline(v=tmp, col='red', lwd=3)
> 2 * pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)
[1] 0.1339368
> m.sample.i.got
[1] 102.1193
>
> ### one more time
> # this time, with a story
> mean(p2)
[1] 105
> sd(p2)
[1] 10
> s.from.p2 <- sample(p2, s.size)
> m.s.from.p2 <- mean(s.from.p2)
> m.s.from.p2
[1] 103.1283
>
> se.s
[1] 1.414035
> se.z
[1] 1.414214
> sd(means)
[1] 1.414035
>
> tmp <- mean(means) - (m.s.from.p2 - mean(means))
> tmp
[1] 96.86822
>
> hist(means, breaks=30,
+ xlim = c(tmp-4*sd(means), m.s.from.p2+4*sd(means)),
+ col = rgb(1, 1, 1, .5))
> abline(v=mean(means), col="black", lwd=3)
> abline(v=m.s.from.p2, col='blue', lwd=3)
>
> # what is the probablity of getting
> # greater than
> # m.sample.i.got?
> m.s.from.p2
[1] 103.1283
> pnorm(m.s.from.p2, mean(p1), se.z, lower.tail = F)
[1] 0.01348266
>
> # then, what is the probabilty of getting
> # greater than m.sample.i.got and
> # less than corresponding value, which is
> # mean(means) - m.sample.i.got - mean(means)
> # (green line)
> abline(v=tmp, col='red', lwd=3)
> 2 * pnorm(m.s.from.p2, mean(p1), se.z, lower.tail = F)
[1] 0.02696533
>
> se.z
[1] 1.414214
> sd(s.from.p2)/sqrt(s.size)
[1] 1.414296
> se.z.adjusted <- sqrt(var(s.from.p2)/s.size)
> se.z.adjusted
[1] 1.414296
> 2 * pnorm(m.s.from.p2, mean(p1), se.z.adjusted,
+ lower.tail = F)
[1] 0.02697421
>
> z.cal <- (m.s.from.p2 - mean(p1))/se.z.adjusted
> z.cal
[1] 2.211891
> pnorm(z.cal, lower.tail = F)*2
[1] 0.02697421
>
>
> pt(z.cal, 49, lower.tail = F)*2
[1] 0.03166797
> t.test(s.from.p2, mu=mean(p1), var.equal = T)
One Sample t-test
data: s.from.p2
t = 2.2119, df = 49, p-value = 0.03167
alternative hypothesis: true mean is not equal to 100
95 percent confidence interval:
100.2861 105.9704
sample estimates:
mean of x
103.1283
>
note.w02.1757554571.txt.gz · Last modified: 2025/09/11 10:36 by hkimscil