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Chapter 1: Problem 36

Consider a sample \(x_{1}, x_{2}, \ldots, x_{n}\) and suppose that the values of \(\bar{x}, s^{2}\), and \(s\) have been calculated. a. Let \(y_{i}=x_{i}-\bar{x}\) for \(i=1, \ldots, n\). How do the values of for the \(x_{i}\) 's? Explain. b. Let \(z_{i}=\left(x_{i}-\bar{x}\right) / s\) for \(i=1, \ldots, n\). What are the values of the sample variance and sample standard deviation for the \(z_{i} \mathrm{~s}\) ?

Short Answer

Expert verified
a. Values of \(y_i\) have mean 0, same variance and standard deviation as \(x_i\); b. For \(z_i\), variance and standard deviation are 1.

Step by step solution

01

Understanding the mean-centering transformation for y

Given the transformation \(y_i = x_i - \bar{x}\), we are shifting each data point \(x_i\) from a sample by the sample mean \(\bar{x}\). This operation does not change the spread or variability of the data but simply shifts the entire dataset such that the new mean of the \(y_i\) values is 0. This is because the mean of the transformed values is \(\bar{y} = \frac{1}{n}\sum_{i=1}^n y_i = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x}) = 0\).
02

Evaluating the variance and standard deviation for y

Since the transformation \(y_i = x_i - \bar{x}\) involves only shifting the data, the variance \(s_y^2\) and standard deviation \(s_y\) of the \(y_i\) values do not change from those of the original \(x_i\) values. Thus, we have \(s_y^2 = s^2\) and \(s_y = s\).
03

Understanding the standardization transformation for z

In the transformation \(z_i = (x_i - \bar{x}) / s\), each data point is mean-centered and scaled by the standard deviation \(s\). This standardization results in each \(z_i\) value representing the number of standard deviations away from the mean \(\bar{x}\). These new values, \(z_i\), have a new mean of 0 and altered variance and standard deviation.
04

Calculating the variance and standard deviation for z

For standardized values \(z_i\), the variance \(s_z^2\) can be determined by recognizing that dividing by the standard deviation \(s\) scales the variance by \(s^2\). Thus, \(s_z^2 = \frac{s^2}{s^2} = 1\). Similarly, the standard deviation \(s_z\) of \(z_i\) values is \(\sqrt{s_z^2} = \sqrt{1} = 1\). Consequently, the \(z_i\) values have a variance and standard deviation of 1.

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Key Concepts

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

Mean-Centering
Mean-centering, as the name suggests, involves adjusting data points by subtracting the sample mean \(\bar{x}\) from each value in the dataset. This simple yet powerful transformation shifts the entire dataset so that the new mean becomes zero.
  • Imagine placing all data points on a number line. Mean-centering shifts the entire line left or right so that the center is now at zero.
  • This operation is crucial in many statistical methods because it simplifies analysis and emphasizes deviations from the average value.
By ensuring that the mean of the adjusted dataset \(\bar{y}\) is 0, mean-centering offers a clearer picture of how each value deviates from the average.
Standardization
Standardization goes a step further than mean-centering by scaling the mean-centered values by the sample standard deviation \(s\). Think of it as converting each value into a standard score or z-score, which tells you how far that value is from the mean in terms of standard deviations.
  • First, we subtract the mean from each data point, giving us the mean-centered value.
  • Next, we divide this result by the standard deviation \(s\), producing a standardized value.
The beauty of standardization is that it transforms the data so it has a mean of 0 and a standard deviation of 1. This process is widely used to prepare data for machine learning algorithms, ensuring that all features contribute equally to the analysis.
Sample Variance
Sample variance \(s^2\) quantifies the spread of a set of data points around their mean. It measures how much each data point deviates from the average value.
  • The formula for sample variance is \(s^2 = \frac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2\), where \(n\) is the number of observations.
  • This value represents the average of the squared deviations from the mean, providing insight into data variability.
After mean-centering, the variance of the new values \(y_i\) remains the same as the original dataset. This is because variance is unaffected by shifts in the data's location鈥攐nly its spread matters.
Sample Standard Deviation
The sample standard deviation \(s\) is a measure of the amount of variation or dispersion in a set of values. Unlike variance, which uses squared units, standard deviation is expressed in the same units as the data, making it more intuitive.
  • It is calculated by taking the square root of the sample variance \(s^2\).
  • A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation suggests more spread out data.
In the context of standardized values \(z_i\), the sample standard deviation is always 1. This makes it a perfect tool for comparing datasets with different units or ranges, as it normalizes the differences between them.

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Most popular questions from this chapter

Automated electron backscattered diffraction is now being used in the study of fracture phenomena. The following information on misorientation angle (degrees) was extracted from the article "Observations on the Faceted Initiation Site in the Dwell-Fatigue Tested Ti-6242 Alloy: Crystallographic Orientation and Size Effects (Metallurgical and Materials Trans., 2006: 1507-1518). \(\begin{array}{lcccc}\text { Class: } & 0-<5 & 5-<10 & 10-<15 & 15-<20 \\\ \text { Rel freq: } & .177 & .166 & .175 & .136 \\ \text { Class: } & 20-<30 & 30-<40 & 40-<60 & 60-<90 \\ \text { Rel freq: } & .194 & .078 & .044 & .030\end{array}\) a. Is it true that more than \(50 \%\) of the sampled angles are smaller than \(15^{\circ}\), as asserted in the paper? b. What proportion of the sampled angles are at least \(30^{\circ}\) ? c. Roughly what proportion of angles are between \(10^{\circ}\) and \(25^{\circ}\) ? d. Construct a histogram and comment on any interesting features.

The accompanying specific gravity values for various wood types used in construction appeared in the article "Bolted Connection Design Values Based on European Yield Model" (J. of Structural Engr., 1993: 2169-2186): \(\begin{array}{lllllllll}.31 & .35 & .36 & .36 & .37 & .38 & .40 & .40 & .40 \\\ .41 & .41 & .42 & .42 & .42 & .42 & .42 & .43 & .44 \\ .45 & .46 & .46 & .47 & .48 & .48 & .48 & .51 & .54 \\ .54 & .55 & .58 & .62 & .66 & .66 & .67 & .68 & .75\end{array}\) Construct a stem-and-leaf display using repeated stems (see the previous exercise), and comment on any interesting features of the display.

Do running times of American movies differ somehow from running times of French movies? The author investigated this question by randomly selecting 25 recent movies of each type, resulting in the following running times: Am: \(94 \quad 90 \quad 95 \quad 93 \quad 128 \quad 95 \quad 125 \quad 91 \quad 104\) \(\begin{array}{llllllllllll}110 & 92 & 113 & 116 & 90 & 97 & 103 & 95 & 120 & 109 & 91 & 138\end{array}\) Fr: \(123116 \quad 90 \quad 158 \quad 12211912590 \quad 96 \quad 94\) \(\begin{array}{llllllllllll}106 & 95 & 125 & 122 & 103 & 96 & 111 & 81 & 113 & 128 & 93 & 92\end{array}\) Construct a comparative stem-and-leaf display by listing stems in the middle of your paper and then placing the Am leaves out to the left and the Fr leaves out to the right. Then comment on interesting features of the display.

Give one possible sample of size 4 from each of the following populations: a. All daily newspapers published in the United States b. All companies listed on the New York Stock Exchange c. All students at your college or university d. All grade point averages of students at your college or university

The minimum injection pressure (psi) for injection molding specimens of high amylose corn was determined for eight different specimens (higher pressure corresponds to greater processing difficulty), resulting in the following observations (from "Thermoplastic Starch Blends with a Polyethylene-Co-Vinyl Alcohol: Processability and Physical Properties," Polymer Engr. and Science, 1994: 17-23): \(\begin{array}{llllllll}15.0 & 13.0 & 18.0 & 14.5 & 12.0 & 11.0 & 8.9 & 8.0\end{array}\) a. Determine the values of the sample mean, sample median, and \(12.5 \%\) trimmed mean, and compare these values. b. By how much could the smallest sample observation, currently \(8.0\), be increased without affecting the value of the sample median? c. Suppose we want the values of the sample mean and median when the observations are expressed in kilograms per square inch (ksi) rather than psi. Is it necessary to reexpress each observation in ksi, or can the values calculated in part (a) be used directly? [Hint: \(\mathrm{lkg}=2.2 \mathrm{lb} .]\)
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