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Chapter 1: Problem 9

a. If a constant \(c\) is added to each \(x_{i}\) in a sample, yielding \(y_{i}=x_{j}+c\), how do the sample mean and median of the \(y_{s} s\) relate to the mean and median of the \(x s\) ? Verify your conjectures. b. If each \(x_{i}\) is multiplied by a constant \(c\), yielding \(y_{i}=c x_{j}\), answer the question of part (a). Again, verify your conjectures.

Short Answer

Expert verified
Adding or multiplying by a constant changes the mean and median accordingly: add/multiply transformations apply directly to both.

Step by step solution

01

Understanding the Problem

The problem asks us to analyze how adding or multiplying each data point in a sample by a constant affects the mean and median of the sample. We need to address this in two parts.
02

Adding a Constant to Each Sample Value

Suppose we have a set of sample points \(x_1, x_2, \ldots, x_n\) with a mean \(\bar{x}\) and median \(M_x\). When a constant \(c\) is added to each \(x_i\), we transform the data to \(y_i = x_i + c\). We need to determine the new mean and median of the \(y\) values.
03

Deriving the New Mean After Addition

The mean of the \(y\) values is given by: \[ \bar{y} = \frac{1}{n}(y_1 + y_2 + \cdots + y_n) = \frac{1}{n}((x_1+c) + (x_2+c) + \cdots + (x_n+c)) = \bar{x} + c \] Thus, adding \(c\) to each data point increases the mean by \(c\).
04

Deriving the New Median After Addition

The median is the middle value in the ordered data set. Since adding \(c\) to each data point does not change their order, the new median is \(M_y = M_x + c\). Thus, the median also increases by \(c\).
05

Multiplying Each Sample Value by a Constant

Consider the same sample \(x_1, x_2, \ldots, x_n\) with mean \(\bar{x}\) and median \(M_x\). Now we multiply each \(x_i\) by a constant \(c\), transforming the data to \(y_i = c \cdot x_i\). We need to calculate the new mean and median of the \(y\) values.
06

Deriving the New Mean After Multiplication

The mean of the \(y\) values is: \[ \bar{y} = \frac{1}{n}(c \cdot x_1 + c \cdot x_2 + \cdots + c \cdot x_n) = c \cdot \frac{1}{n}(x_1 + x_2 + \cdots + x_n) = c \cdot \bar{x} \] Thus, multiplying by \(c\) scales the mean by \(c\).
07

Deriving the New Median After Multiplication

As in addition, multiplying by a constant \(c\) does not change the order of the data points. So the new median is \(M_y = c \cdot M_x\). Thus, the median is also scaled by \(c\).
08

Verifying Conjectures

In both scenarios, working through the calculations confirms that the sample mean and median undergo predictable transformations. When added a constant, both increase by that constant. When multiplied by a constant, both are scaled by that constant.

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Key Concepts

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

Sample Mean
The sample mean is a fundamental concept in statistics, representing the average value of a data set. To find the sample mean, you add up all the values in your sample (let's call this sum \(S_x\)) and then divide by the number of data points \(n\). Mathematically, it is represented as \(\bar{x} = \frac{S_x}{n}\).
Here's why it's useful:
  • It provides a central value for your data, helping you understand the typical value in your sample.
  • It's sensitive to every data point, meaning changes to any value will affect the mean.
When you transform your data by adding a constant \(c\), the new sample mean becomes \(\bar{y} = \bar{x} + c\).
If you multiply each data point by a constant \(c\), the sample mean scales accordingly, changing to \(\bar{y} = c \cdot \bar{x}\).
This sensitivity to transformations is one reason why the mean serves as an important measure for data analysis.
Sample Median
In statistics, the sample median is the middle value of a data set when it is organized in ascending order. It splits the data set into two equal halves.
Here's why the median is important:
  • It represents the central tendency of the data without being affected by outliers or skewed data.
If you add a constant \(c\) to each data point, the sample median will shift by the same constant. Therefore, the new median is \(M_y = M_x + c\).
Similarly, when you multiply each data point by a constant \(c\), the median changes in kind to become \(M_y = c \cdot M_x\).
Since it is derived from the order of the data points, these transformations maintain this order, meaning the median is always an intuitive representation after such transformations.
Data Transformation
Data transformation involves modifying data to achieve desired analytical results, often making the data easier to understand or analyze. With respect to statistical measures like the mean and median, transformations can change how these values are interpreted but keep their fundamental properties intact.
Transformations can take many forms, including:
  • Adding a constant value \(c\) to each data point. This will shift all measurements equally, affecting the sample mean and median while maintaining their relative positions.
  • Multiplying each data point by a constant \(c\). This scales the entire data set, proportionately changing both the mean and the median.
The insights offered by understanding data transformation are valuable in various fields, such as economics, engineering, and social sciences, allowing us to model real-world phenomena accurately. By knowing how transformations impact statistical measures, we gain a clearer picture of our data's behavior and underlying patterns.

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Most popular questions from this chapter

The accompanying observations on stabilized viscosity (cP) for specimens of a certain grade of asphalt with \(18 \%\) rubber added are from the article "Viscosity Characteristics of Rubber-Modified Asphalts" (J. of Materials in Civil Engr., 1996: 153-156): \(27812900 \quad 3013 \quad 2856 \quad 2888\) a. What are the values of the sample mean and sample median? b. Calculate the sample variance using the computational formula. [Hint; First subtract a convenient number from each observation.]

a. For what value of \(c\) is the quantity \(\sum\left(x_{f}-c\right)^{2}\) minimized? [Hint: Take the derivative with respect to \(c\), set equal to 0 , and solve.] b. Using the result of part (a), which of the two quantities \(\Sigma\left(x_{f}-\bar{x}\right)^{2}\) and \(\Sigma\left(x_{f}-\mu\right)^{2}\) will be smaller than the other (assuming that \(\bar{x} \neq \mu)^{2}\) ?

The article "Oxygen Consumption During Fire Suppression: Error of Heart Rate Estimation" (Ergonomics, 1991: 1469-1474) reported the following data on oxygen consumption ( \(\mathrm{mL} / \mathrm{kg} / \mathrm{min})\) for a sample of ten firefighters performing a fire-suppression simulation: \(\begin{array}{llllllllll}29.5 & 49.3 & 30.6 & 28.2 & 28.0 & 26.3 & 33.9 & 29.4 & 23.5 & 31.6\end{array}\) Compute the following: a. The sample range b. The sample variance \(s^{2}\) from the definition (i.e., by first computing deviations, then squaring them, etc.) c. The sample standard deviation d. \(s^{2}\) using the shortcut method

A sample of \(n=10\) automobiles was selected, and each was subjected to a \(5-m p h\) crash test. Denoting a car with no visible damage by \(\mathrm{S}\) (for success) and a car with such damage by \(\mathrm{F}\), results were as follows: S \(S \quad F \quad S \quad S \quad S \quad F \quad F \quad S \quad S\) a. What is the value of the sample proportion of successes \(x / n ?\) b. Replace each S with a 1 and each \(\mathrm{F}\) with a 0 . Then calculate \(\bar{x}\) for this numerically coded sample. How does \(\bar{x}\) compare to \(x / n\) ? c. Suppose it is decided to include 15 more cars in the experiment. How many of these would have to be \(S\) 's to give \(x / n=.80\) for the entire sample of 25 cars?

Let \(\bar{x}_{n}\) and \(s_{n}^{2}\) denote the sample mean and variance for the sample \(x_{1}, \ldots, x_{n}\) and let \(\bar{x}_{n+1}\) and \(s_{n+1}^{2}\) denote these quantities when an additional observation \(x_{n+1}\) is added to the sample. a. Show how \(\bar{x}_{n+1}\) can be computed from \(\bar{x}_{n}\) and \(x_{n+1^{*}}\). b. Show that $$ n s_{n+1}^{2}=(n-1) s_{n}^{2}+\frac{n}{n+1}\left(x_{n+1}-\bar{x}_{n}\right)^{2} $$ so that \(s_{n+1}^{2}\) can be computed from \(x_{n+1}, \bar{x}_{n}\), and \(s_{n}^{2}\) c. Suppose that a sample of 15 strands of drapery yarn has resulted in a sample mean thread elongation of \(12.58 \mathrm{~mm}\) and a sample standard deviation of \(512 \mathrm{~mm}\). A \(16^{\text {th }}\) strand results in an elongation value of \(11.8\). What are the values of the sample mean and sample standard deviation for all 16 elongation observations?
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