How do i estimate how many items in a sample are outside 1 standard deviation of the mean? My question is how to estimate how many samples are above 1 standard deviation of the mean. I do not know individual values, hence why it's an estimate.
The sum of x is 243 sum of x^2 is 2317, n = 30, range is 13(4 to 17).
I have calculated mean (8.1) and standard deviation (3.41). From this information, how do I estimate how many of the sample are between 11.51 and 17?
The distribution is clearly not normal since the lowest value is only 4.1 below the mean, whilst the highest value is 8.9 above it, so I'm guessing that less than 16% of values are above 11.51. This would imply that less than 0.16×30 = 4.8 of the samples are in that range. Is this correct? Any guidance appreciated.

Extended Comment with simulated samples from various distributions:
Continuing from @whuber's Comment and yours, here are examples of simulated samples of size $n = 1000$ from
normal, uniform, exponential, gamma, beta, Poisson, and
binomial populations.
Some are skewed, some have support
other than the whole real line, and the last two are discrete.
The part of the Empirical Rule for the proportion of observations
exceeding one sample standard deviation above
the sample mean (right-sided) holds "fairly well" for all of them: Proportions are
between 12% and 22%.
set.seed(2021)

x  = rnorm(10^3, 100, 15)
mean(x > mean(x)+sd(x))
[1] 0.161

x  = runif(10^3, 30, 50)
mean(x > mean(x)+sd(x))
[1] 0.211

x  = rexp(10^3, 5)
mean(x > mean(x)+sd(x))
[1] 0.135

x  = rgamma(10^3, 3, .05)
mean(x > mean(x)+sd(x))
[1] 0.15

x = rbeta(10^3, 3, 4)
mean(x > mean(x)+sd(x))
[1] 0.162

x  = rpois(10^3, 10)
mean(x > mean(x)+sd(x))
[1] 0.155

x = rbinom(10^3, 30, .3)
mean(x > mean(x)+sd(x))
[1] 0.169

