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The sample mean, denoted as \(\bar{x}\), is the average of all observations in a sample. It is calculated by summing all the sample values and then dividing by the sample size \(n\). Formally, \(\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\). The key feature of the sample mean is its status as an unbiased estimator of the population mean \(\mu\). This implies that if we repeatedly take samples and compute their means, the average of these sample means will converge to the population mean. This property makes the sample mean a reliable and preferred statistic for estimating the central tendency of the entire population.

Sample variance, often represented as \(s^2\), quantifies the spread or dispersion of a set of data points. The formula to calculate sample variance is \(s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2\), where \((x_i - \bar{x})^2\) represents the squared differences between each data point \(x_i\) and the sample mean \(\bar{x}\). Unlike the population variance which divides by \(n\), the sample variance uses \(n-1\), a factor known as Bessel's correction, to ensure it is an unbiased estimator of the population variance \(\text{σ}^2\). This correction accounts for the fact that a sample's variability tends to be smaller than that of the entire population, making \(s^2\) particularly accurate for estimating \(\text{σ}^2\).

The sample proportion, denoted as \(\frac{x}{n}\), is used primarily with categorical data to estimate the proportion of a particular outcome within a sample. Here, \(x\) represents the number of times a specific event occurs, and \(n\) is the total number of trials or observations. For example, if you survey 100 people and 30 say they prefer chocolate ice cream, the sample proportion of chocolate ice cream preferences is \(\frac{30}{100} = 0.3\). This sample proportion is an unbiased estimator of the population proportion \(p\), meaning that over many samples, the average of the sample proportions will equal the true proportion in the population.

The sample median is the middle value of a data set when the values are arranged in ascending order. For a sample with an odd number of observations, the median is the center value. For an even number of observations, it is the average of the two middle values. While the sample median is a robust measure of central tendency, especially in skewed distributions, it is generally not an unbiased estimator of the population median. This discrepancy arises because the expected value of the sample median does not always equate to the population median, particularly in small or non-normally distributed samples.

The range of a sample is the difference between the maximum and minimum values within the data set. It is a simple measure of data spread. However, the sample range is not an unbiased estimator of the population range. This is due to its heavy dependency on sample size and the variability within the sample. As sample size increases, the range generally increases, making it less reliable for estimating the true range of the population. Thus, while the range can provide some insights into the data dispersion, it is often supplemented with other statistics for a more accurate picture.

The sample standard deviation, represented as \(s\), measures the spread of data points around the sample mean. It is the square root of the sample variance ( \(s = \sqrt{s^2}\) ). While it gives valuable insights into data variability similar to the sample variance, the sample standard deviation is not an unbiased estimator of the population standard deviation \(\text{σ}\). This occurs because taking the square root introduces a downward bias. Though popular for its interpretability, additional corrections or alternative measures are often needed to obtain an unbiased estimate of the population standard deviation.