Problem 15 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

The following rule is useful for the computation of the sample variance (and standard deviation). Show that $$ \frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}_{n}\right)^{2}=\left(\frac{1}{n} \sum_{i=1}^{n} x_{i}^{2}\right)-\left(\bar{x}_{n}\right)^{2} $$ where \(\bar{x}_{n}=\left(\sum_{i=1}^{n} x_{i}\right) / n\).

Step-by-Step Solution

Answer

The given rule for the sample variance and the expression are equivalent and verified through expansion and simplification.

Step 1: Understand the variables and terms

We are given two expressions to equate. The first is the definition of sample variance: \( \frac{1}{n} \sum_{i=1}^{n}(x_{i}-\bar{x}_{n})^{2}\). The second expression is \(\left(\frac{1}{n}\sum_{i=1}^{n} x_{i}^{2}\right) - \left(\bar{x}_{n}\right)^{2}\). \(\bar{x}_{n}\) is the mean, defined as \(\bar{x}_{n}=\frac{1}{n}\sum_{i=1}^{n} x_{i}\). Each \(x_i\) represents a data point within the set.

Step 2: Expand the variance formula

Start with the variance formula: \( \frac{1}{n} \sum_{i=1}^{n}(x_{i}-\bar{x}_{n})^{2} \). Expand this expression as \( \frac{1}{n} \sum_{i=1}^{n}(x_{i}^2 - 2x_i\bar{x}_{n} + \bar{x}_{n}^2)\). This uses the identity \((a-b)^2 = a^2 - 2ab + b^2\).

Step 3: Distribute the summation

Apply the sum distributive property: \( \frac{1}{n} \left( \sum_{i=1}^{n} x_{i}^2 - 2\bar{x}_{n}\sum_{i=1}^{n} x_{i} + \sum_{i=1}^{n} \bar{x}_{n}^2 \right) \). This separates the expression into three distinct summations inside one summation.

Step 4: Simplify each term inside the summation

For the second term, notice that \( \sum_{i=1}^{n} x_i = n\bar{x}_{n} \), which simplifies \(-2\bar{x}_n\sum_{i=1}^{n} x_i\) to \(-2n\bar{x}_n^2\). The third term simplifies due to constant \(\bar{x}_n^2\), as \(n\bar{x}_n^2\) because \(\sum_{i=1}^{n} 1 = n\).

Step 5: Combine and simplify to find equality

Combine back the inner sums: \(\frac{1}{n} \left( \sum_{i=1}^{n} x_{i}^2 - 2n\bar{x}_n^2 + n\bar{x}_n^2 \right) = \frac{1}{n} \sum_{i=1}^{n} x_{i}^2 - \frac{n\bar{x}_n^2}{n}\). This simplifies to \( \frac{1}{n} \sum_{i=1}^{n} x_{i}^2 - \bar{x}_n^2 \). Thus, the equality is proved.

Key Concepts

Standard Deviation

The standard deviation is a key concept in statistics used to measure the amount of variation or dispersion in a set of data points. It provides insights into how spread out the data points are around the mean (average). If the data points are close to the mean, the standard deviation is smaller. If they are spread out over a wider range, the standard deviation is larger.

The formula for standard deviation is based on the square root of the variance. The variance is the average of the squared differences from the mean. So, the standard deviation is essentially the square root of the average squared distances from the mean. Mathematically, for a sample of data, the standard deviation is represented by the formula:

• ds = \(\sqrt{\frac{1}{n - 1}\sum_{i=1}^{n}(x_i - \bar{x})^2}\\)

where \(\bar{x}\) is the sample mean, \(n\) is the number of data points, and \(x_i\) represents each individual data point in the sample.Understanding standard deviation is important because it helps you interpret data variability and understand how much the scores deviate from the mean.

Summation Formula

The summation formula is an essential concept used frequently in mathematics and statistics, especially when dealing with groups of numbers. It denotes the operation of adding together a sequence of numbers.The basic summation formula is expressed using the Greek letter sigma (\(\Sigma\)), which represents the sum of all terms in a series. For example:

• \(\sum_{i=1}^{n} x_i\)

is the sum of all \(x_i\) from \(i=1\) to \(i=n\), where \(n\) is the total number of terms.When using summation formulas in statistics, they allow us to easily calculate total values or averages more efficiently. For instance, when computing the mean, you sum up all the data points and then divide by the number of points. Summation formulas also enable the breakdown of complex mathematical expressions into manageable parts, which is crucial for solving variance and standard deviation problems.

Mean

The mean, often referred to as the average, is the central value of a set of numbers. It's calculated by adding up all the data points and then dividing by the number of data points.Mathematically, the mean of a set of numbers \(x_1, x_2, ..., x_n\) is given by:

• \(\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i\)

Here \(\bar{x}\) represents the mean, \(n\) is the number of data points, and \(x_i\) are the individual data points.The mean is a fundamental concept in statistics and is used extensively in various calculations, including variance and standard deviation. It provides a single value that represents a general idea of the data's overall characteristics.It's essential to understand that while the mean provides a snapshot of the data's center, it doesn't give any information about the variability or spread of the data. That's why other statistics, like the median or the standard deviation, are also important when fully analyzing a dataset.