4 ms·

Is it true that the standard deviation is the "expected" distance from the mean of a randomly selected value? Like, if the mean is 1 and the SD is 1, and I was

Is it true that the standard deviation is the "expected" distance from the mean of a randomly selected value? Like, if the mean is 1 and the SD is 1, and I was to take a sample and take the average of the absolute values of the samples, would that tend towards 1 as I take more samples? Edit: I think I've just described "mean absolute error"... seems to be related but the "root mean square deviation" is the one more closely related to SD. I think... Not sure where the "n" comes in.

LogicalRisk 3y ago
"Where does n come in?" I *think* this might help to answer your question for where n comes from. It helps me at least think about it. The definition of the variance of the standard error V[\bar{x}] = V[X]/n. You can back this out from the definition of variance, the property that V[aX] = a^2V[X], and that variances are additive with independent draws. Take the square root of that and you have the standard error. Why this "feels" right to me via an example. Suppose we want to know the average height of the US population. Intuitively, we think that (assuming a representative sample) we'll do "better" in the sense of a tighter distribution around our best guess (mean of sample) of the population value if we sample 1000 people as opposed to sampling 10. This is related to the distribution and would function as our "best guess" about the dispersion of a variable in the same units as the original. Both of them are sampling to try and guess the average. Since \bar{X} is itself a random variable, it has a distribution, and that distribution should probably include something about the sampling process we used to characterize it. Mean absolute error would be E[|X-mu|] since the true mean of the distribution is a constant.
atkindel 3y ago
Your thinking here is pretty close to right, but it sounds like you may be a bit confused about the difference between "standard deviation" and "standard error". They're closely related, but it's very important to keep them distinct. In your example, you have a random variable X that comes from some distribution parameterized by its mean m and its variance s^2. We can write this random variable as x ~ ?(m=1, s^2=1). Continuing your example, suppose we sample a sequence of n draws of this random variable X = [x1, x2, ... xn] and then take their sample mean f(X) = 1/n * sum(X). The value of the sample mean will vary depending on the sample we drew, so we want to know the properties of the sampling distribution of f(X), particularly as we increase n. The standard error is the standard deviation of the sampling distribution of f(X).

LogicalRisk 3y ago

"Where does n come in?" I *think* this might help to answer your question for where n comes from. It helps me at least think about it. The definition of the variance of the standard error V[\bar{x}] = V[X]/n. You can back this out from the definition of variance, the property that V[aX] = a^2V[X], and that variances are additive with independent draws. Take the square root of that and you have the standard error. Why this "feels" right to me via an example. Suppose we want to know the average height of the US population. Intuitively, we think that (assuming a representative sample) we'll do "better" in the sense of a tighter distribution around our best guess (mean of sample) of the population value if we sample 1000 people as opposed to sampling 10. This is related to the distribution and would function as our "best guess" about the dispersion of a variable in the same units as the original. Both of them are sampling to try and guess the average. Since \bar{X} is itself a random variable, it has a distribution, and that distribution should probably include something about the sampling process we used to characterize it. Mean absolute error would be E[|X-mu|] since the true mean of the distribution is a constant.

atkindel 3y ago

Your thinking here is pretty close to right, but it sounds like you may be a bit confused about the difference between "standard deviation" and "standard error". They're closely related, but it's very important to keep them distinct. In your example, you have a random variable X that comes from some distribution parameterized by its mean m and its variance s^2. We can write this random variable as x ~ ?(m=1, s^2=1). Continuing your example, suppose we sample a sequence of n draws of this random variable X = [x1, x2, ... xn] and then take their sample mean f(X) = 1/n * sum(X). The value of the sample mean will vary depending on the sample we drew, so we want to know the properties of the sampling distribution of f(X), particularly as we increase n. The standard error is the standard deviation of the sampling distribution of f(X).