# expected value of sample from a skewed distribution

Let us say a market's prices are distributed with a skew.

Given a sample of 10000 products, I can estimate the median, arithmetic mean and weighted average or any other statistic.

Let us say I draw 100 products from this distribution, what would be their total expected/average value? I cannot use arithmetic mean * 100 or can I? Should I use median * 100? Or should one use 100 * weighted average?

PS:

Given Dave's answer and code below, I tried to simulate my situation. Basically I assumed a lognormal distribution to get a skew. I obtain some averages from a larger sample. I want to see which average to use to best estimate the total of N randomly sampled items. It turn out that, according to this (correct?) simulation the arithmetic mean is the better average in my situation. I guess this is some empirical evidence against my gut feeling that the median is the better/more representative average in my situation ...

p1 <- log(10)
p2 <- log(2)

# obtain estimates/averages from large sample from "the truth"
x <- rlnorm(1000000, p1, p2)
plot(density(x))
median <- median(x)
mean <- mean(x)

set.seed(2021)
N <- 100 # sample size
R <- 10000 # how many times to draw samples of size N
mean_estimate_se <- median_estimate_se <- rep(NA, R)
for (i in 1:R){

x <- rlnorm(N, p1, p2)

sum <- sum(x)

mean_estimate_se[i] <- ((mean * N) - sum)^2
median_estimate_se[i] <- ((median * N) - sum)^2

}

paste0("Error using arithmetic mean: ", round(mean(mean_estimate_se), 2))
paste0("Error using median: ", round(mean(median_estimate_se), 2))

round(mean(median_estimate_se) / mean(mean_estimate_se), 2) Whether the distribution is skewed or not, the usual sample mean is always an unbiased estimator for the population mean.$$^{\dagger}$$

$$\mathbb E\bigg[\bar X\bigg] = \mathbb E\Bigg[\frac{1}{n}\sum_{i = 1}^nX_i\Bigg]\\ = \frac{1}{n}\sum_{i = 1}^n \bigg( \mathbb E[X_i]\bigg) = \frac{1}{n} \sum_{i = 1}^n \mu = \frac{1}{n}n\mu = \mu$$

The usual sample mean is a perfectly defendable way to estimate the population mean.

Should gets into lots of fascinating ideas related to constructing estimators with desirable properties but might be a bit beyond where you want to operate. Depending on the particulars of the distribution, you might find the median to be a better estimator of the mean, perhaps based on the mean squared errors of those estimators.

$$^{\dagger}$$There's this annoying issues where the population mean might not exist. In that case, this calculation does not make sense, but you are trying to estimate a value that does not exist, which also does not make sense.

EDIT

In your comment, you mention that median might be a preferable estimator for a skewed distribution. Let's put that to the test in a simulation study.

set.seed(2021)
N <- 100 # sample size
R <- 1000 # how many times to draw samples of size N
means <- medians <- rep(NA, R)
for (i in 1:R){
parameter <- 13 # the parameter of the distribution
x <- rchisq(N, parameter) # this has a mean of parameter

means[i] <- mean(x)
medians[i] <- median(x)

}

mean((means - parameter)^2)
mean((medians - parameter)^2)


I get an MSE of the sample mean of $$0.25775941187491$$ and an MSE of the sample median of $$0.845599781948668$$, more than three times as big. For this $$\chi^2_3$$ distribution, at least in terms of the common MSE metric, the mean is a vastly superior estimator, despite the skewness. As noted in the comments, other distributions, such as $$t_3$$, will give the opposite conclusion (despite the symmetry of $$t_3$$).

(I had trouble coming up with a skewed distribution that gave superior performance of the empirical median, and I welcome suggestions in the comments. For superiority of the empirical mean for a distribution that lacks skewness, look no further than the normal distribution. (Noted in the comments, absolute value of a low-degree-of-freedom t-distribution should do the trick.))

• If you replace rchisq(, 13) by, say, rt(, 3) (which has an expectation of zero, a finite variance, and a symmetric distribution to boot), you will draw the opposite conclusion when comparing the sample mean to the median. Thus, your blanket statement "the mean is a vastly superior estimator" needs qualification. Indeed, researchers who have run extensive simulation studies have concluded that in many cases robust alternatives to the mean lose relatively little efficiency for near-Normal distributions and often have superior efficiency for less-than-Normal distributions.
– whuber
Sep 21 at 21:08
• If you want the expected sum of those $100$ values, sure, take $100\bar x$. You can tweak my simulation to estimate the sum of the $100$ observations instead of the mean, and it will reveal that, under the given conditions, sample mean has much lower mean squared error than sample median. // I haven’t though my through exactly how to do this, but there might be a way to use a bootstrap procedure to estimate which is better for your exact data. // Could it be that you don’t want to estimate a mean (expected value) but a median?
– Dave
Sep 22 at 10:28
• You have to be careful about what exactly you want. If what you want is to estimate the mean (or some multiple of the mean, such as $100\mu$), then there is an argument to use the sample mean. If you want to estimate the median (or some multiple of the median), then a different estimator might be appropriate. So what exactly do you want to estimate?
– Dave
Sep 22 at 15:45
• Re "superior performance of the empirical median:" take absolute values and use t distributions with small df.
– whuber
Sep 22 at 16:00
• You're more-or-less following what I am doing, just with log-normal instead of chi-squared, and you're coming to a similar conclusion. // Do note that you might prefer to estimate the median or sum of medians, rather than the mean or sum of means, though that is a decision for you (and a separate question if you are not sure which you want). // Quick note about your code: reconsider naming variables as existing functions like sum. I'm surprised that your code even compiled and didn't throw an error when you got to i=2 and had redefined sum to be a number instead of a function.
– Dave
Sep 23 at 12:41