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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. $$\begin{aligned} \Sigma(x-\bar{x})^{2} &=\Sigma x^{2}-2 \bar{x} \Sigma x+n \bar{x}^{2} \\ &=\Sigma x^{2}-2 n \bar{x}^{2}+n \bar{x}^{2} \\ &=\Sigma x^{2}-\frac{(\Sigma x)^{2}}{n} \end{aligned}$$

Short Answer

Expert verified
The expression simplifies to \( \Sigma x^{2} - \frac{(\Sigma x)^{2}}{n} \).

Step by step solution

01

Understand the Formula Components

The goal is to understand the formula: \( \Sigma(x-\bar{x})^{2} = \Sigma x^{2} - 2 \bar{x} \Sigma x + n \bar{x}^{2} \). This represents the computation of the sum of squared deviations from the mean of a sample, aiming to eventually express it in simpler form.
02

Calculate Sum of Squares

The expression \( \Sigma(x-\bar{x})^{2} \) indicates the sum of each data point\( x_i \) minus the sample mean \( \bar{x} \), squared. This is a core part of calculating variance.
03

Substitution in the Formula

Substitute \( \bar{x} \), which is the mean of the sample, calculated as \( \bar{x} = \frac{\Sigma x}{n} \) into the formula. Substituting this into the formula gives \( \Sigma x^{2} - 2 \bar{x} \Sigma x + n \bar{x}^{2} \).
04

Simplify using Mean Substitution

Start simplifying by noticing that \( n \bar{x}^{2} = \frac{(\Sigma x)^{2}}{n} \). Therefore, \(-2 \bar{x} \Sigma x \) becomes \(-2\frac{\Sigma x}{n} \cdot \Sigma x = -2 \frac{(\Sigma x)^{2}}{n} \). The simplified expression becomes \( \Sigma x^{2} - 2 \frac{(\Sigma x)^{2}}{n} + \frac{(\Sigma x)^{2}}{n} \).
05

Combine and Simplify further

Combine like terms \(-2 \frac{(\Sigma x)^{2}}{n} + \frac{(\Sigma x)^{2}}{n} = -\frac{(\Sigma x)^{2}}{n} \). Thus, we have \( \Sigma x^{2} - \frac{(\Sigma x)^{2}}{n} \), which simplifies to the final expression.

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Key Concepts

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

Understanding Variance
Variance is a statistical measure that provides insight into the spread of a dataset. It essentially tells us how much the individual data points differ from the mean of the sample. The larger the variance, the more spread out the data points are.
Consider it as a measure of how "spread out" the numbers are in a dataset:
  • To calculate variance, find the difference between each data point and the sample mean.
  • Square each of these differences to eliminate negative values.
  • Average the squared differences (depending on the context, you might divide by either the sample size or the sample size minus one).
For our exercise, the variance formula connects to the sum of these squared differences, which helps us to eventually simplify the problem and better understand the data's variability.
Sample Mean Simplified
The sample mean, denoted as \(\bar{x}\), is essentially the average of all the data points in a sample. It's calculated to find a central value around which the data is balanced.
Here's how you can easily calculate it:
  • Add up all the observed data values in the sample, represented as \(\Sigma x\).
  • Divide the sum by the total number of data points, \(n\).
The sample mean is important in our equation because it acts as a reference point. We measure each data point's deviation from this mean, which serves as the starting point for calculating variance and analyzing data distribution in statistics.
Role of Algebra in Formulas
Algebra allows us to manipulate and simplify mathematical expressions or equations efficiently. In our exercise, algebra plays a crucial role in simplifying complex expressions involving sum of squares.
Thanks to algebra, we can:
  • Rearrange and combine terms to make the expression clearer or simpler.
  • Substitute expressions like \(\bar{x}\) with its equivalent, \(\frac{\Sigma x}{n}\), to transform and solve the equation step by step.
  • Handle operations involving sums and differences smoothly, facilitating the simplification of variance-related formulas.
By applying algebraic rules and techniques, we break down the problem into manageable pieces and solve it effectively, enhancing our understanding of mathematical concepts and their practical applications.

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Most popular questions from this chapter

Interpretation A job-performance evaluation form has these categories: \(1=\) excellent; \(2=\) good; \(3=\) satisfactory; \(4=\) poor; \(5=\) unacceptable Based on 15 client reviews, one employee had median rating of \(4 ;\) mode rating of 1 The employee was pleased that most clients had rated her as excellent. The supervisor said improvement was needed because at least half the clients had rated the employee at the poor or unacceptable level. Comment on the different perspectives.

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.

Consider the numbers 2 3 4 5 5 (a) Compute the mode, median, and mean. (b) If the numbers represent codes for the colors of T-shirts ordered from a catalog, which average(s) would make sense? (c) If the numbers represent one-way mileages for trails to different lakes, which average(s) would make sense? (d) Suppose the numbers represent survey responses from 1 to \(5,\) with \(1=\) disagree strongly, \(2=\) disagree, \(3=\) agree, \(4=\) agree strongly, and \(5=\) agree very strongly. Which averages make sense?

For a given data set in which not all data values are equal, which value is smaller, \(s\) or \(\sigma ?\) Explain.

Wolf Packs 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: $$\begin{array}{ccccccccc}13 & 10 & 7 & 5 & 7 & 7 & 2 & 4 & 3 \\\2 & 3 & 15 & 4 & 4 & 2 & 8 & 7 & 8\end{array}$$ Compute the mean, median, and mode for the size of winter wolf packs.
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