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Chapter 3: Problem 27

If you like mathematical puzzles or love algebra, try this! Otherwise, just trust that the computational formula for the sum of squares is correct. We have a sample of \(x\) values. The sample size is \(n\). Fill in the details for the following steps. $$ \begin{aligned} \Sigma(x-\bar{x})^{2} &=\Sigma x^{2}-2 \bar{x} \sum x+n \bar{x}^{2} \\ &=\Sigma x^{2}-2 n \bar{x}^{2}+n \bar{x}^{2} \\ &=\Sigma x^{2}-\frac{(\Sigma x)^{2}}{n} \end{aligned} $$

Short Answer

Expert verified
The formula for sum of squares simplifies to \( \Sigma x^{2} - \frac{(\Sigma x)^{2}}{n} \) by expanding and substituting the mean formula.

Step by step solution

01

Understand the Formula

The formula we are looking at aims to compute the sum of squares of deviations of sample values from their mean, denoted as \( \Sigma(x-\bar{x})^{2} \). This is a common calculation in statistics for finding variance and involves squaring the difference between each sample value and the mean, then summing these squares.
02

Expand the Sum of Squares Formula

The formula can be expanded as \( \Sigma(x-\bar{x})^{2} = \Sigma x^{2} - 2\bar{x} \sum x + n \bar{x}^{2} \). Here, \( \Sigma x^{2} \) is the sum of squares of the actual \(x\) values, \( 2\bar{x} \sum x \) is twice the mean multiplied by the sum of \(x\) values, and \( n\bar{x}^{2} \) is the mean squared, multiplied by the number of observations.
03

Substitute the Mean in the Formula

Recall that the mean \( \bar{x} \) is defined as \( \frac{\Sigma x}{n} \). Replace \( \bar{x} \) in the previously expanded formula, which becomes \( \Sigma x^{2} - 2n \left(\frac{\Sigma x}{n}\right)^{2} + n \left(\frac{\Sigma x}{n}\right)^{2} \). Simplifying this results in \( \Sigma x^{2} - \frac{(\Sigma x)^{2}}{n} \).
04

Simplification

Notice that in the term \( -2\bar{x}\sum x + n\bar{x}^{2} \), when substituting \( \bar{x} = \frac{\Sigma x}{n} \), the coefficients combine, resulting in the final expression \( \Sigma x^{2} - \frac{(\Sigma x)^{2}}{n} \). This shows how variance is computed using sums of squares of raw data and squared sums of data, adjusted by the sample size.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Squares
In statistics, the **sum of squares** is a crucial concept. It measures how data vary or spread out from a central value, usually the mean. By squaring the differences between each value and the mean, then adding them up, we account for all deviations:
  • Calculate differences: Subtract the mean from each data point.
  • Square each difference: This ensures that all values are positive, emphasizing larger deviations more.
  • Sum these squares: This gives us the sum of squares.
This measurement helps in computing variance, telling us how consistent or varied our data is around the mean. Variance, the average of these squared deviations, provides insight into the data's reliability. This term is essential in many areas of statistics, especially in advanced analyses like regression.
Sample Mean
The **sample mean** is a simple yet powerful concept. It represents the average of a data set and is a central component in understanding data behavior. To find it, follow these easy steps:
  • Add up all the data points.
  • Divide this sum by the number of observations, denoted as \( n \).
Mathematically, this is represented as \( \bar{x} = \frac{\Sigma x}{n} \), where \( \Sigma x \) is the sum of all data points. The sample mean offers a single value representation of the dataset, making it easier to grasp large data quickly. It's an invaluable starting point for deeper statistical evaluations such as variance and standard deviation and heavily influences the structure of various statistical formulas.
Algebraic Manipulation
When handling statistical formulas, **algebraic manipulation** plays a vital role in simplifying complex equations. It involves rearranging and simplifying mathematical expressions to make them more interpretable:
  • **Substituting values:** Replace variables with known quantities to see how parts of the expression relate.
  • **Combining like terms:** This is crucial for simplification, especially when dealing with polynomials or multi-part equations.
  • **Rearranging terms:** Allows you to make clearer conclusions about the data and the formula's behavior.
For instance, when transforming the expression to compute variance, we substitute \( \bar{x} = \frac{\Sigma x}{n} \) back into our equations. By doing this, we derive an easier-to-understand form like \( \Sigma x^{2} - \frac{(\Sigma x)^{2}}{n} \), revealing core statistical insights. Mastery of these techniques assists in exploring the underlying meaning of statistical calculations.

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Most popular questions from this chapter

Consider sample data with \(\bar{x}=15\) and \(s=3\). (a) Compute the coefficient of variation. (b) Compute a \(75 \%\) Chebyshev interval around the sample mean.

Consider the mode, median, and mean. Which average represents the middle value of a data distribution? Which average represents the most frequent value of a distribution? Which average takes all the specific values into account?

Clayton and Timothy took different sections of Introduction to Economics. Each section had a different final exam. Timothy scored 83 out of 100 and had a percentile rank in his class of 72 . Clayton scored 85 out of 100 but his percentile rank in his class was 70 . Who performed better with respect to the rest of the students in the class, Clayton or Timothy? Explain your answer.

How large is a wolf pack? The following information is from a random sample of winter wolf packs in regions of Alaska, Minnesota, Michigan, Wisconsin, Canada, and Finland (Source: The Wolf, by L. D. Mech, University of Minnesota Press). Winter pack size: \(\begin{array}{rrrrrrrrr}13 & 10 & 7 & 5 & 7 & 7 & 2 & 4 & 3 \\ 2 & 3 & 15 & 4 & 4 & 2 & 8 & 7 & 8\end{array}\) Compute the mean, median, and mode for the size of winter wolf packs.

When data consist of rates of change, such as speeds, the harmonic mean is an appropriate measure of central tendency. For \(n\) data values, Harmonic mean \(=\frac{n}{\sum \frac{1}{x}}, \quad\) assuming no data value is 0 Suppose you drive 60 miles per hour for 100 miles, then 75 miles per hour for 100 miles. Use the harmonic mean to find your average speed.
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