The mean is the first statistic we learn, the cornerstone of many analyses. But the question is how well do we understand its estimation? For statisticians, estimating the mean is more than just summing and dividing. It involves navigating assumptions, choosing appropriate methods, and understanding the implications of our choices. Let us delve deeper into the art and science of estimating the mean.
Table of Contents
The Simple Sample Mean: A Foundation
The Formula of sample mean $\overline{x}= \frac{\sum\limits_{i=1}^n x_i}{n}$​​. The sample mean is the unbiased estimator of population mean ($\mu$) under ideal conditions (simple random sampling, independent and identically distributed data). Violating the assumption can lead to biased estimates. For large samples, the distribution of the sample mean approximates a normal distribution, regardless of the population distribution due to the Central Limit Theorem (CLT).
Weighted Means
Beyond Simple Random Sampling, For weighted means, observations have varying importance (e.g., survey data with different sampling weights). The formula of weighted mean is $ \overline{x}_w = \frac{\sum\limits_{i=1}^n w_ix_i}{\sum\limits_{i=1}^n w_i}$. Weighted means are used in Survey sampling, and dealing with non-response. In Stratified Sampling, estimate the mean when the population is divided into strata for getting reduced variance, and improved precision. In cluster sampling have unique challenges of estimating the mean with cluster sampling, where observations are grouped.
Robust Estimation
Robust Estimation is required when the sample mean is vulnerable to extreme values. The alternative of the sample mean is the median which emphasizes its robustness to outliers. The trimmed mean is also used to balance out the robustness and efficiency.
Confidence Intervals for Estimating the Mean
Confidence Intervals make use of standard error to estimate the mean to reflect the precision of the estimate. For small samples, t-distribution while for large samples, z-distribution is used for the construction of confidence intervals. Bootstrapping (a non-parametric method) can also be used for constructing confidence intervals, especially useful when assumptions are violated.
Point Estimate: To estimate the population mean $\mu$ for a random variable $x$ using a sample of values, the best possible point estimate is the sample mean $\overline{x}$.
Interval Estimate: An interval estimate for mean $\mu$ is constructed by starting with sample mean $\overline{x}$ and adding a margin of error (S.E.) above and below the mean $\overline{x}$. The interval is of the form $(\overline{x} – SE, \overline{x} + SE)$.
Example: Suppose that the mean height of Pakistani men is between 67.5 and 70.5 inches with a level of confidence of $c = 0.90$. To estimate the men’s height, the sample mean $\overline{x}$ is 69 inches with a margin of error = 1.5 inches. That is, $(\overline{x} – SE, \overline{x}+SE) = (69 – 1.5, 69+1.5) = (67.5, 70.5)$.
Note that the margin of error used for constructing an interval estimate depends on the level of confidence interval. A larger level of confidence will result in a larger margin of error and hence a wider interval.
Calculating Margin of Error for a Large Sample Data
If a random variable $x$ is normally distributed (with a known population standard deviation $\sigma$) or if the sample size $n$ is at least 30 (we will apply Central Limit Theorem, which will guarantee that),
- $\overline{x}$ is approximately normally distributed
- $\mu_{\overline{x}}$ = \mu$
- $\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}$
The mean value of $\overline{x}$ is equal to the population mean $\mu$ being estimated. Given the desired level of confidence $c$, it is try to find the amount of error $E$ necessary to ensure that the probability of $\overline{x}$ being within $E$ of the mean is $c$.
There are always two critical $Z$-scores ($\pm z_c$ which give the appropriate probability for the standard normal distribution), and the corresponding probability for the distribution of $\overline{x}$ is $z_c \times \sigma_{\overline{x}}$ or
$$E=z_c \frac{\sigma}{\sqrt{n}}$$
Usually, $\sigma$ is not known, but if $n\ge 30$ then the sample standard deviation $s$ is generally a reasonable estimate.
Dealing with Missing Data
When dealing with missing data, one can impute mean. Imputing the mean is simple but it can underestimate variance. One can also perform multiple imputations to account for the uncertainty.
Bayesian Estimation
In Bayesian estimation, the prior and posterior distributions are used for estimating the mean by incorporating prior information, updated beliefs about the mean, and handling uncertainty.
Summary
Estimating the mean is a fundamental statistical task, but it requires careful consideration of assumptions, data characteristics, and the goals of the analysis. By understanding the nuances of different estimation methods, statisticians can provide more accurate and reliable insights.
Exploratory Data Analysis in R Language
