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The mean and standard deviation of a sample may change if data are rescaled (for instance, temperature changed from Fahrenheit to Celsius). For a sample with mean \(\bar{x}\), adding a constant \(c\) to each observation changes the mean to \(\bar{x}+c,\) and the standard deviation \(s\) is unchanged. Multiplying each observation by \(c>0\) changes the mean to \(c \bar{x}\) and the standard deviation to \(c s\) a. Scores on a difficult exam have a mean of 57 and a standard deviation of \(20 .\) The teacher boosts all the scores by 20 points before awarding grades. Report the mean and standard deviation of the boosted scores. Explain which rule you used and identify \(c .\) b. Suppose that annual income for some group has a mean of $$\$ 39,000$$ and a standard deviation of $$\$ 15,000$$. Values are converted to British pounds for presentation to a British audience. If one British pound equals $$\$ 2.00,$$ report the mean and standard deviation in British currency. Explain which rule above you used and identify \(c\). c. Adding a constant and/or multiplying by a constant is called a linear transformation of the data. Do linear transformations change the shape of the distribution? Explain your reasoning.

Step by step solution

01

Calculate Mean After Adding a Constant

For part (a), the rule for adding a constant states that if you add a constant \(c\) to all observations, the mean becomes \(\bar{x} + c\). Since the original mean is 57 and we add 20 to each score, the new mean is \(57 + 20 = 77\).

02

Determine Standard Deviation After Adding a Constant

When a constant is added to each observation, the standard deviation \(s\) remains unchanged. Therefore, the standard deviation of the boosted scores is still 20.

03

Explanation for Adding a Constant

In part (a), the constant \(c = 20\), and by adding \(c\) to each observation, only the mean changes to \(\bar{x} + c = 77\), while the standard deviation remains \(s = 20\).

04

Calculate Mean After Multiplying by a Constant

For part (b), converting dollars to pounds involves multiplication by \(c\). If each observation is multiplied by \(c\), the mean becomes \(c \bar{x}\). With \(\bar{x} = 39,000\) dollars and \(c = 0.5\), the mean in pounds is \(0.5 \times 39,000 = 19,500\) pounds.

05

Determine Standard Deviation After Multiplying by a Constant

When each observation is multiplied by \(c\), the standard deviation changes to \(c s\). For \(s = 15,000\) dollars and \(c = 0.5\), the new standard deviation is \(0.5 \times 15,000 = 7,500\) pounds.

06

Explanation for Multiplying by a Constant

In part (b), the constant used is \(c = 0.5\), due to the exchange rate from dollars to pounds. Multiplying each observation by \(c\) changes the mean to \(19,500\) pounds, and the standard deviation to \(7,500\) pounds.

07

Effect of Linear Transformations on Distribution Shape

For part (c), linear transformations affect mean and standard deviation but do not change the overall shape of the distribution. These transformations shift or scale the data while maintaining the relative positions and spacing of data points.

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Key Concepts

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

Mean and Standard Deviation

In statistics, the mean and standard deviation are two fundamental concepts used to describe the characteristics of a data set. The mean, often referred to as the average, is the sum of all observations divided by the total number of observations. Standard deviation, on the other hand, indicates how spread out the observations are from the mean. It measures the amount of variation or dispersion in a set of values.

When data undergoes a rescaling, such as adding or multiplying by a constant, the characteristics of the mean and standard deviation change accordingly. Specifically, when a constant, say \(c\), is added to every observation, the mean becomes \(\bar{x} + c\). However, the standard deviation remains unchanged. This is because adding a constant shifts the entire data set without affecting the distances between values.

Conversely, when each observation is multiplied by a constant \(c\), both the mean and the standard deviation are multiplied by \(c\). This leads to the mean being\( c\bar{x} \) and the standard deviation being \( cs \). Rescaling in this manner stretches or shrinks the distribution, which affects both location and spread.

Effect on Distribution Shape

A common question that arises with linear transformations is how they affect the shape of the distribution of a data set. Linear transformations include operations like adding or multiplying each data point by a constant.

Interestingly, adding a constant value to a data set shifts the whole distribution horizontally on a number line but does not alter the actual shape of the distribution. Similarly, while multiplying changes the scale of the data set, the proportional distances between data points remain the same.

• So, adding a constant \(c\) essentially shifts the distribution without changing its shape.
• Multiplying by a constant \(c\) stretches or compresses the distribution, yet the general form or shape remains unchanged.

Linear transformations leave the relative position and spacing of points stable, hence the shape such as bell-shaped, skewed, or uniform remains constant.

Data Rescaling in Statistics

Data rescaling in statistics refers to the process of adjusting data using a constant factor. This is crucial when converting units or changing scales, such as turning temperatures from Fahrenheit to Celsius or converting currencies.

Rescaling is not just about numerical changes; it has meaningful implications for interpretation and comparison. Adding a constant to data may correlate it more directly with other datasets under the same conditions or circumstances.

• This process adjusts the mean by the additive constant while leaving the dispersion of data unchanged.
• Conversely, multiplying resizes the dispersion along with mean changes.

Rescaling ensures that data remains comparable despite changes in measurement units, which is highly important in statistical reporting and data analysis across various domains. Keeping track of the constants used and understanding their effects helps in clear, consistent data communication.