# Does central limit theorem help in making inference only about the population mean and not other parameters?

As per the weak law of large numbers, if your sample size is large, your mean of the sample is likely to be closer to the population mean than in a smaller sample. Additionally the CLT tells us the approximate sampling distribution of the sample mean tends to a normal curve for larger and larger Ns.

But what about other population parameters, such as the variance or perhaps other measures? Do we know that sample estimates tend to the true value and that their distribution is approximately normal in larger samples?

• FYI the WLLN (or SLLN) tell us that the sample mean goes to the population in large N, the CLT tells us that sample mean in repeated sampling is approximately normally distributed, so we can make inference about the true mean even if our estimates aren't exactly "close". Feb 19, 2021 at 16:45
• can we make any inferences about the true variance using sample variances? making inference about just the population mean does not describe the population fully.We need to know about other parameters such as variance, standard deviation etc. of the population. That was my question, which got edited Feb 19, 2021 at 19:06
• Of course you can make inference about the variance. The sample variance is a biased estimator without degrees of freedom correction, but it is consistent and asymptotically has a normal sampling distribution. And you can make inference about the mean and variance simultaneously... if it's explicitly stated in the study objectives/hypotheses. FWIW, we rarely care about estimating the population variance because the sample variance can be used as a "plug-in" estimate. Your question was edited to correct incorrect statements, and you didn't exclusively ask about variance in the original. Feb 19, 2021 at 19:24
• "Does central limit theorem help in making inference only about the population mean and not other parameters? " The "other parameters" in my question refers to SD, variance etc. anyways thanks for your response. so can we say that the distribution of all sample variances will have its mean close to that of the polulation variance? Feb 20, 2021 at 5:32
• What exactly is the etc. in "SD, variance, etc."? Is there a question? Anyway, the sample variance is a consistent estimator of the population variance, meaning in larger samples you're more likely to have a "close" estimate. But in a finite sample, or in any sample really, the estimate can be "not close" but "way off" as a consequence of randomness. That's why you summarize the uncertainty with confidence intervals. Feb 22, 2021 at 15:56

What you state is rather the law of large numbers. The CLT is rather concerned with the resulting asymptotic distribution. But I think I get what you want to know. The law of large numbers ensures the consistency for the average/mean calculation and you question what ensures the consistency of the variance etc. estimation. In general, when calculating these statistics, one swaps out theoretical moments by empirical moments. For instance, if in your formula appears E(X) we estimate empirically (1/T)*sum(X). This is also done when calculating the variance, SD etc. Hence, the consistency of the first moment (sample mean) also ensures the consistency of higher moments if they are finite.

• ie $1/T \sum(X^3)$ is just the sample mean of a different random variable $X^3$ Feb 19, 2021 at 16:11
• yes thanks for the clearification. @seanv507 Feb 19, 2021 at 16:15
• No, my question is while the CLT helps in making inference about population mean based on sample mean, what about other parameters like the population variance? how do we estimate that using sample distributions? is there an equivalent CLT for variance? Feb 19, 2021 at 19:04
• okay for clarification: CLT states that as the observation number goes to infinity the average of a random variable is asymptotically normal distributed even if the random variable itself is originally not normally distributed. Built on that distribution the inference is conducted. As any moment/central moment (includes also your precious variance) is calculated using expectations/theoretical moments which are replaced by empirical moments (sample mean), the same CLT applies. Feb 19, 2021 at 19:19