I am interested in predicting a normal distribution, but not sure if this is possible.
I do not have information on the mean or standard deviation. However, I know the range of values, let's say from 0 to 10, and I know the sample size, let's say 1000, and I also know that value 8.1 is the 50th highest value.
Is it possible in any way from this information to infer more about what the distribution looks like?
Use maximum likelihood.
Generally, suppose you have sorted array of order statistics $x_{[i]}$ for $i$ in a subset $\mathcal{I}$ of $\{1,2,\ldots, n\}$. Augment this vector to include $x_{[0]}=-\infty$ and $x_{[n+1]}=\infty$. You are supposing the underlying distribution is $F_{\theta}$ with corresponding density $f_{\theta}$ and you wish to estimate $\theta$. The likelihood of these order statistics, up to a factor that will not vary with $\theta$, is a product of the $f_\theta{x_{[i]}}$ (excluding $x_{[0]}$ and $x_{[n+1]}$, where necessarily $f=0$) times all the powers
$$\left(F_\theta(x_{[i_{j+1}]}) - F_\theta(x_{[i_j]})\right)^{i_{j+1}-i_j-1}$$
for $j$ from $1$ onward. In the example of the question this would be
$$f_\theta(0)f_\theta(8.1)f_\theta(10)\ \left(F_\theta(8.1) - F_\theta(0)\right)^{949}\left(F_\theta(10) - F_\theta(8.1)\right)^{48}.$$
The rest is completely routine, provided you have at least as many order statistics as there are components of $\theta$: there typically will be a unique value of $\theta$ that minimizes this expression. In the question, $\theta$ has two components and there are three order statistics, so all is fine. The machinery of ML produces, in a standard way, estimated standard errors of the parameters, too.
To illustrate, the R code below estimates $\hat\mu=5.39794$ and $\hat\sigma=1.62553$ from the data in the question (namely, $(x_{[1]}, x_{[951]}, x_{[1000]}) = (0, 8.1, 10)$, with $n=1000$). Then, as a quick visual check that the estimates are reasonable, it generates $1000$ datasets of size $n=1000$ from this Normal distribution, records the $1,951,$ and $1000$ order statistics (as given in the question), plots their histograms, and superimposes the observed order statistics on those histograms. The fit is beautiful for orders $1$ and $951$ and reasonable for order $1000$ (the value of $10$ is around the twelfth percentile of this distribution--not too extreme).
#
# Negative log likelihood.
#
Lambda <- function(theta) {
mu <- theta[1]
log.sigma <- theta[2]
sigma <- exp(log.sigma)
f <- dnorm(values, mu, sigma, log=TRUE)
F <- log(diff(pnorm(c(-Inf, values, Inf), mu, sigma)))
-(sum(f) + sum(F * (diff(c(0, orders, n+1))-1)))
}
#
# The data.
#
n <- 1000
orders <- c(1, 951, 1000)
values <- c(0, 8.1, 10)
#
# Compute the estimate.
#
theta.start <- c(mu=mean(range(values)), log.sigma=log(diff(range(values))/6))
fit <- nlm(Lambda, theta.start) # It converged.
theta.hat <- fit$estimate
mu.hat <- theta.hat[1]
sigma.hat <- exp(theta.hat[2])
#
# Check the quality of the estimate visually.
#
n.sim <- 1e3
n <- 1000
sim <- apply(matrix(rnorm(n.sim*n, mu.hat, sigma.hat), n), 2, sort)[orders, ]
par(mfrow=c(1, length(orders)))
invisible(
sapply(1:length(orders), function(i) {
hist(sim[i,], freq=FALSE, xlab="Value", main=paste("Order", orders[i]))
abline(v=values[i], lwd=2, col="Red")
})
)
You have a sample of random numbers $x_1,x_2,\dots,x_{1000}$. What's know is $\min[x_i],\max[x_i]$ and 50th highest number A, which is 5th percentile in a sample 1000 long.
Given the information I'd go with $$\hat\mu=\frac{\min+\max}{2}=5$$, and $$\hat\sigma=\frac{A-\hat\mu}{1.65}=\frac{8.1-5}{1.65}\approx 1.9$$
The t-statistics 1.65 is derived from 5% one tail z-value. I bet that you can come up with a rigorous MLE procedure which would produce the same result
UPDATE
Here's another way of estimating the dispersion using @jth idea. The max and min upper bounds are: $$\max=\mu+\sigma(\sqrt{2\log n} +1)$$ Hence, $$\hat\sigma=\frac{\max-\min}{2(\sqrt{2\log n} +1)}=\frac{10}{2(\sqrt{2\log 1000}+1)}\approx 1.4$$
