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 $$ S^{2}=\frac{1}{n-1}\left[\sum_{k=1}^{l} x_{k}^{2} f_{k}-\frac{1}{n}\left(\sum_{k=1}^{l} x_{k} f_{k}\right)^{2}\right] $$

Step-by-Step Solution

Answer

The sample variance formula is derived by expanding the variance definition using mean and frequency counts.

Step 1: Recall the Formula for Sample Variance

The formula for the sample variance \( S^2 \) is generally given in terms of deviations from the sample mean. Specifically, it is \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \), where \( x_i \) are the sample values and \( \bar{x} \) is the sample mean.

Step 2: Express the Sample Mean

The sample mean \( \bar{x} \) is calculated as \( \bar{x} = \frac{1}{n} \sum_{k=1}^{l} x_{k} f_{k} \), where each distinct value \( x_k \) appears \( f_k \) times in the sample.

Step 3: Expand the Variance Formula

Substitute \( \bar{x} \) into the variance formula: \[ S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{n-1} \left[ \sum_{i=1}^{n} x_i^2 - 2x_i \bar{x} + \bar{x}^2 \right] \].

Step 4: Rewrite the Variance using Counts\(f_k\)

Recognize that \( \sum_{i=1}^{n} x_i^2 = \sum_{k=1}^{l} x_{k}^2 f_{k} \) and \( \sum_{i=1}^{n} x_i \bar{x} = \bar{x} \sum_{k=1}^{l} x_{k} f_{k} = n \bar{x}^2 \). Substituting these into the variance formula gives: \[ S^2 = \frac{1}{n-1} \left[ \sum_{k=1}^{l} x_{k}^2 f_{k} - n \bar{x}^2 \right] \].

Step 5: Simplify using the Mean Expression

Substitute \( \bar{x} = \frac{1}{n} \sum_{k=1}^{l} x_{k} f_{k} \) into the expression for \( n \bar{x}^2 \): \( n \bar{x}^2 = \left(\sum_{k=1}^{l} x_{k} f_{k}\right)^2 / n \). Replace this in the variance expression to obtain: \[ S^2 = \frac{1}{n-1} \left[ \sum_{k=1}^{l} x_{k}^2 f_{k} - \frac{1}{n} \left(\sum_{k=1}^{l} x_{k} f_{k}\right)^2 \right] \].

Step 6: Finalize the Derivation

Notice that the finalized formula matches the given expression for the sample variance: \[ S^2 = \frac{1}{n-1} \left[ \sum_{k=1}^{l} x_{k}^2 f_{k} - \frac{1}{n} \left(\sum_{k=1}^{l} x_{k} f_{k} \right)^2 \right] \], which confirms our derivation.

Key Concepts

Mean Deviation

Mean deviation is an essential concept in understanding data variability. It measures the average distance between each data point and the mean of the dataset.
The mean deviation provides insights into the spread and dispersion of values. To calculate the mean deviation, perform the following steps:

• First, find the mean (average) of the dataset.
• Then, compute the absolute difference between each data point and the mean.
• Lastly, find the average of all these absolute differences.

These steps help determine how much the sample values deviate from the mean, on average. Mean deviation is considered less sensitive to extreme values compared to variance and standard deviation, which is useful when analyzing samples with outliers.

Statistical Formulas

Statistical formulas are vital tools that allow us to summarize and understand the characteristics of data.
Different statistical measures enable us to communicate complex data in a simplified manner, aiding in interpretation and decision-making. For variance and mean deviation, statistical formulas use arithmetic and algebraic manipulation to quantify data spread:

• Variance is calculated using squared deviations from the mean, highlighting data dispersion.
• Mean deviation measures the absolute differences from the mean, providing a simpler view of data spread.

Understanding the derivation and application of these formulas is crucial for interpreting their results correctly.

Variance Calculation

Variance calculation helps quantify the variability within a data sample. It demonstrates how spread out or dispersed the values are.
The sample variance formula in statistics is often articulated as:\[ S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]where:

• \( n \) represents the sample size.
• \( x_i \) are the individual data points.
• \( \bar{x} \) is the sample mean.

For a dataset with distinct values and frequencies, the formula adapts to:
\[ S^2 = \frac{1}{n-1} \left[ \sum_{k=1}^{l} x_{k}^2 f_{k} - \frac{1}{n} \left(\sum_{k=1}^{l} x_{k} f_{k}\right)^2 \right] \]This adjustment considers the frequency of each value, making it apt for grouped data.
Understanding variance provides insights into data consistency and predictability.

Sample Mean

The sample mean is a fundamental concept in statistics that represents the average of all observations in a dataset.
It is denoted as \( \bar{x} \) and serves as a measure of central tendency.To calculate the sample mean:

• Add up all the values in the sample.
• Divide the sum by the number of observations (\( n \)).

The formula is expressed as:\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]For samples with distinct values and frequencies, it becomes:\[ \bar{x} = \frac{1}{n} \sum_{k=1}^{l} x_{k} f_{k} \]The sample mean is central to many statistical analyses and calculations, including variance and deviation measurements. It provides a summary statistic that is often the first step in data analysis.