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Chapter 3: Problem 21

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
\( \Sigma(x-\bar{x})^2 = \Sigma x^2 - \frac{(\Sigma x)^2}{n} \).

Step by step solution

01

Understanding the Problem

The problem revolves around decomposing and understanding how to calculate the sum of squares for a given sample of data. The sum of squares is used to measure the variance, a key statistical metric.
02

Define the Formula Components

We are given the expression \( \Sigma(x-\bar{x})^2 \) and need to break it down in terms of known quantities. Here, \( \Sigma x^2 \) is the sum of squares of the sample values, and \( \bar{x} \) represents the sample mean, calculated as \( \bar{x} = \frac{\Sigma x}{n} \).
03

Expand the Squares

Let's expand the expression \( \Sigma(x-\bar{x})^2 \). Using the identity \((a-b)^2 = a^2 - 2ab + b^2\), we can expand \((x-\bar{x})^2\) to get \(x^2 - 2x\bar{x} + \bar{x}^2\).
04

Substitute Back into the Expression

Substitute the expanded form into the sum: \( \Sigma(x^2 - 2x\bar{x} + \bar{x}^2) = \Sigma x^2 - 2\bar{x} \Sigma x + n \bar{x}^2 \), assuming \(\Sigma \bar{x}^2 = n \bar{x}^2\) because it represents adding up the mean squared for each sample.
05

Simplify Using Sample Mean

Replace \(\bar{x}\) with \(\frac{\Sigma x}{n}\) in the expression \( -2\bar{x} \Sigma x + n \bar{x}^2\). This gives \(-2\frac{\Sigma x}{n}\Sigma x + n\left(\frac{\Sigma x}{n}\right)^2\).
06

Simplify and Arrange Terms

Further simplify the expression: \(-2\frac{(\Sigma x)^2}{n} + \frac{n(\Sigma x)^2}{n^2}\). This simplifies to \(-2\frac{(\Sigma x)^2}{n} + \frac{(\Sigma x)^2}{n}\).
07

Final Simplification

Combine the terms: \( \Sigma x^2 - \frac{(\Sigma x)^2}{n} \). This expression represents the sum of squares of deviations from the mean, completing the derivation.

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Key Concepts

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

Sample Variance
The concept of sample variance is central to understanding variability within a given set of data. It gives us a numerical estimate of how much the data values differ from the mean. Essentially, the sample variance is calculated to discern the spread or dispersion in your data.
The formula for sample variance is given by:
\[ s^2 = \frac{\Sigma (x_i - \bar{x})^2}{n-1} \]where:
  • \(\Sigma (x_i - \bar{x})^2\) is the sum of squares of deviations.
  • \(\bar{x}\) is the sample mean.
  • \(n\) is the number of data points in the sample.
This calculation is crucial because it lays the groundwork for many other statistical analyses, helping us understand and interpret data more effectively.
Algebraic Manipulation
Algebraic manipulation plays a key role in simplifying and breaking down expressions, allowing us to see underlying patterns or relationships within equations. When tackling formulas in statistics, like the sum of squares, using algebraic manipulation helps in transforming the expressions into more workable forms.
In the case of the sum of squares, we start with an expression involving deviations, \((x - \bar{x})^2\), and break it down using the property
\((a-b)^2 = a^2 - 2ab + b^2\).
This simplifies to:
\[ x^2 - 2x\bar{x} + \bar{x}^2 \]Thus, algebraic manipulation allows us to take complex statistical formulas and simplify them to forms which are easier to analyze.
Statistical Analysis
Statistical analysis involves collecting and scrutinizing every data sample in a set of items from which samples can be drawn. This process can involve several steps and use various methods, such as descriptive statistics or inferential statistics, to draw conclusions.
Our focus here is on using the sum of squares in statistical analysis, as a crucial component both in variability assessment like variance, and in deeper analyses such as regression analysis.
  • Sum of squares helps to assess the impact of individual variables.
  • Understanding the variability reveals insights on data consistency.
By dissecting data mathematically, statistical analysis builds a framework for decision-making and prediction.
Decomposition in Algebra
Decomposition in algebra refers to breaking down a complex equation or expression into smaller, more manageable parts. This is essential in understanding components and performing step-by-step calculations, especially in statistical contexts like calculating variance.
In our example, we began with:
\[ \Sigma (x - \bar{x})^2 = \Sigma x^2 - 2\bar{x}\Sigma x + n\bar{x}^2 \]By decomposing, it becomes possible to see how each part contributes, allowing for substituting and simplifying using known values like the mean, \(\bar{x}\). This breakdown not only assists in completing calculations but also deepens comprehension of how each component interacts within the equation.

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Most popular questions from this chapter

For mallard ducks and Canada geese, what percentage of nests are successful (at least one offspring survives)? Studies in Montana, Illinois, Wyoming, Utah, and California gave the following percentages of successful nests (Reference: The Wildlife Society Press, Washington, D.C.). \(x\) : Percentage success for mallard duck nests 56 85 52 13 39 y: Percentage success for Canada goose nests 24 53 60 \(\begin{array}{cc}69 & 18\end{array}\) (a) Use a calculator to verify that \(\Sigma x=245 ; \Sigma x^{2}=14,755 ; \Sigma y=224 ;\) and \(\Sigma y^{2}=12,070\) (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\), the percent of successful mallard nests. (c) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(y\), the percent of successful Canada goose nests. (d) Use the results of parts (b) and (c) to compute the coefficient of variation for successful mallard nests and Canada goose nests. Write a brief explanation of the meaning of these numbers. What do these results say about the nesting success rates for mallards compared to Canada geese? Would you say one group of data is more or less consistent than the other? Explain.

Critical Thinking: Data Transformation In this problem, we explore the effect on the standard deviation of adding the same constant to each data value in a data set. Consider the data set \(5,9,10,11,15\). (a) Use the defining formula, the computation formula, or a calculator to compute \(s\). (b) Add 5 to each data value to get the new data set \(10,14,15,16,20 .\) Compute \(s\). (c) Compare the results of parts (a) and (b). In general, how do you think the standard deviation of a data set changes if the same constant is added to each data value?

Some data sets include values so high or so low that they seem to stand apart from the rest of the data. These data are called outliers. Outliers may represent data collection errors, data entry errors, or simply valid but unusual data values. It is important to identify outliers in the data set and examine the outliers carefully to determine if they are in error. One way to detect outliers is to use a box-and-whisker plot. Data values that fall beyond the limits Lower limit: \(Q_{1}-1.5 \times(I Q R)\) Upper limit: \(Q_{3}+1.5 \times(I Q R)\) where \(I Q R\) is the interquartile range, are suspected outliers. In the computer software package Minitab, values beyond these limits are plotted with asterisks \(\left(^{*}\right)\) Students from a statistics class were asked to record their heights in inches. The heights (as recorded) were \(\begin{array}{llllllllllll}65 & 72 & 68 & 64 & 60 & 55 & 73 & 71 & 52 & 63 & 61 & 74 \\ 69 & 67 & 74 & 50 & 4 & 75 & 67 & 62 & 66 & 80 & 64 & 65\end{array}\) (a) Make a box-and-whisker plot of the data. (b) Find the value of the interquartile range \((I Q R)\). (c) Multiply the \(I Q R\) by \(1.5\) and find the lower and upper limits. (d) Are there any data values below the lower limit? above the upper limit? List any suspected outliers. What might be some explanations for the outliers?

When a distribution is mound-shaped symmetrical, what is the general relationship among the values of the mean, median, and mode?

What is the relationship between the variance and the standard deviation for a sample data set?
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