Problem 14 - Calculus for Biology and Medicine

Question

Assume that a sample of size \(n\) has \(l\) distinct values \(x_{1}, x_{2}, \ldots, x_{l}\), where \(x_{k}\) occurs \(f_{k}\) times in the sample. Explain why the sample variance is given by the formula $$ S^{2}=\frac{1}{n-1}\left[\sum_{k=1}^{I} x_{k}^{2} f_{k}-\frac{1}{n}\left(\sum_{k=1}^{l} x_{k} f_{k}\right)^{2}\right] $$

Step-by-Step Solution

Answer

The sample variance formula accounts for repeated values by using frequencies, simplifying the calculation as shown.

Step 1: Understand the Expression for Sample Variance

The sample variance \( S^2 \) measures how spread out the values in the sample are from their mean. The formula for the variance is \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \), where \( \bar{x} \) is the sample mean. This formula can be expanded and rewritten in a different form to simplify calculations when the sample includes repeating values.

Step 2: Identify Key Elements in the Formula

Notice that we have repeating elements \( x_k \) occurring \( f_k \) times. The total number of observations in the sample is \( n = \sum_{k=1}^{l} f_k \). Also, the sample mean \( \bar{x} \) can be expressed as \( \bar{x} = \frac{1}{n} \sum_{k=1}^{l} x_k f_k \).

Step 3: Expand the Traditional Variance Formula

The variance formula \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \) can be expanded to \( S^2 = \frac{1}{n-1} \left( \sum_{i=1}^{n} x_i^2 - 2\bar{x}\sum_{i=1}^{n} x_i + n\bar{x}^2 \right) \).

Step 4: Simplify Using the Repeated Values

Incorporate the frequency of occurrences \( f_k \). Replace \( \sum_{i=1}^{n} x_i^2 \) with \( \sum_{k=1}^{l} x_k^2 f_k \) and \( \sum_{i=1}^{n} x_i \) with \( \sum_{k=1}^{l} x_k f_k \), rewriting the variance formula as:\[S^2 = \frac{1}{n-1} \left( \sum_{k=1}^{l} x_k^2 f_k - 2 \bar{x} \sum_{k=1}^{l} x_k f_k + n\bar{x}^2 \right)\]

Step 5: Substitute the Sample Mean

Substitute \( \bar{x} = \frac{1}{n} \sum_{k=1}^{l} x_k f_k \) into the expanded variance formula, simplifying to:\[S^2 = \frac{1}{n-1} \left( \sum_{k=1}^{l} x_k^2 f_k - 2 \frac{1}{n} \left(\sum_{k=1}^{l} x_k f_k\right)^2 + \frac{1}{n} \left(\sum_{k=1}^{l} x_k f_k \right)^2 \right)\]This further simplifies to:\[S^2 = \frac{1}{n-1} \left( \sum_{k=1}^{l} x_k^2 f_k - \frac{1}{n} \left( \sum_{k=1}^{l} x_k f_k \right)^2 \right)\]Thus, proving the given expression for sample variance.

Key Concepts

Sample Mean

The sample mean, often symbolized as \( \bar{x} \), is a central concept in statistics. It represents the average value of a dataset. To calculate the sample mean, you sum all the individual data points in your sample and divide by the total number of observations, \( n \). When you have data represented in a frequency distribution, the sample mean is calculated by multiplying each distinct value \( x_k \) with its corresponding frequency \( f_k \), summing these products, and then dividing by the total number of observations. This mathematical expression is \( \bar{x} = \frac{1}{n} \sum_{k=1}^{l} x_k f_k \). The sample mean provides a simple measure of a dataset's central tendency, offering insights into the average performance or characteristic described by the data.

Frequency Distribution

A frequency distribution is a way of summarizing data by showing how often each value (or range of values) appears in the dataset. It is particularly useful when dealing with large datasets or when values repeat. Each unique value \( x_k \) in the data is associated with a corresponding frequency \( f_k \), representing how many times that value occurs. Frequency distributions can be presented in tables, charts, or graphs. Understanding how often each data point occurs helps in identifying patterns, trends, and dispersions within the dataset. This simplification enables more efficient statistical calculations, such as determining the sample mean or variance.

Variance Formula

The variance formula is a critical tool in statistics used to quantify the spread or dispersion of a set of data points around their mean. For a sample, the variance, represented as \( S^2 \), is calculated by averaging the squared deviations of each observation from the sample mean. The traditional formula is \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \). However, when data includes repeated values, as in a frequency distribution, the formula can be rewritten to efficiently account for these repetitions. This alternative version is: \[ S^2 = \frac{1}{n-1} \left( \sum_{k=1}^{l} x_k^2 f_k - \frac{1}{n} \left( \sum_{k=1}^{l} x_k f_k \right)^2 \right) \]. This collects terms to simplify calculations, reducing computational complexity, and ensuring accurate results.

Statistical Calculation

Statistical calculation is the process of applying various computational techniques to data to extract useful information and insights. It encompasses measures of central tendency, like mean and median, as well as measures of variability, like variance and standard deviation. Calculations like these allow us to describe the characteristics of a dataset quantitatively and make inferences or predictions. When statistics are broken down into frequency distributions and sample means, the calculations become more intuitive and accessible. Notably, these statistical calculations are the backbone of data analysis, providing the metrics by which datasets can be analyzed, compared, and utilized in scientific research, business analysis, and everyday decision-making.