Transformation to normality when data is trimmed at a specific value So let's say I have an original dataset $x\sim N(58, 3.5)$, but then I go and take only the records with $x \geq 54$ and continue working with this trimmed data.
Now, as I understand, this trimmed data set is not normally distributed since I have cut a big chunk of data and I can't apply statistical methods relevant for normally distributed data?
I assume, in this case I'll need to transform the dataset to normal, but I'm not exactly sure how to do it in this particular case. Any suggestions?
Thanks!
 A: I wouldn't want to use t methods to make confidence intervals or test
hypotheses based on samples of $n=20$ truncated observations.

*

*The first histogram below shows the a histogram along with a density curve of the (skewed) truncated normal distribution.

*The second histogram is for means of 20 truncated values;
it seems roughly normal, but means do fail a Shapiro-Wilk normality test.

*However, the scatterplot shows that means and standard deviations
of truncated samples of $n=20$ are not independent (correlation $r \approx 0.37),$
so that "t-ratios" cannot have Student's t distributions. [For normal data, $\bar X$ and $S$ are independent random variables.]

.
set.seed(725)
q = pnorm(54, 58, 3.5); q
[1] 0.126549
x = qnorm(runif(20*10^5, q, 1), 58, 3.5)
MAT = matrix(x, nrow=10^5, ncol=20)
a = rowMeans(MAT)
shapiro.test(a[1:5000])$p.val
[1] 2.480093e-05
s = apply(MAT, 1, sd)
mean(s)
[1] 2.833662
cor(a, s)
[1] 0.3752128
t = (a - mean(a))*sqrt(20)/s; ks.test(t, pt, 19)$p.val
[1] 0            # 't.stat' not Student's t distributed
qt(.975, 19)
[1] 2.093024
mean(t < -2.093); mean(t > 2.093)
[1] 0.03545      # tails 'unbalanced'
[1] 0.01703

par(mfrow=c(1,3))
 hist(x, prob=T, col="skyblue2", main="Truncated Normal")
  curve(dnorm(x, 58, 3.5)/(1-q), add=T, col="red")
 hist(a, prob=T, br=30, col="skyblue2", main="Means of 20")
  curve(dnorm(x, mean(a), sd(a)), add=T, col="red")
 plot(a, s, pch=".")
par(mfrow=c(1,1))


