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Rescale the data 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 \(\mathrm{c} \bar{x}\) 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 \(\underline{c} .\) 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

Understand the Problem

We are given two scenarios: (a) boosting scores by adding a constant and (b) converting income using a conversion rate (multiplying by a constant). Both scenarios require understanding how adding or multiplying by a constant affects the mean and standard deviation. We will also explore the effect of linear transformations on the distribution shape.

02

Rescale Scores by Adding a Constant (Part a)

In part (a), the mean of the exam scores is initially 57 and the standard deviation is 20. The scores are increased by adding 20 points to each score. According to the rule, adding a constant \(c\) to each observation changes the mean to \(\bar{x} + c\) (where \(c = 20\)) and leaves the standard deviation unchanged. The new mean becomes \(57 + 20 = 77\) and the standard deviation remains 20.

03

Rescale Income by Multiplying by a Constant (Part b)

In part (b), the annual income in dollars has a mean of \\(39,000 and a standard deviation of \\)15,000. We convert these values by multiplying by the conversion rate of 0.5 British pounds per dollar (not 2, since each pound equals \$2.00, making \(c = 0.5\)). Multiplying a sample by \(c\) changes the mean to \(c\bar{x}\) and the standard deviation to \(cs\). Thus, the mean in pounds becomes \(0.5 \times 39,000 = 19,500\) pounds, and the standard deviation becomes \(0.5 \times 15,000 = 7,500\) pounds.

04

Consider Effect on Distribution Shape (Part c)

A linear transformation (adding or multiplying by a constant) does not change the shape of the distribution. This means that properties such as skewness or kurtosis remain unchanged. The transformation affects only the scale (mean and spread) of the distribution, but not its overall shape.

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Key Concepts

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

Rescaling Data

Understanding how rescaling affects data is crucial in statistics. When we rescale data, we adjust the values by either adding or multiplying by a constant. This process is known as a linear transformation. Rescaling can be useful in various situations, such as when converting temperatures or currencies, as shown in our exercise examples.
- **Addition**: Adding a constant to each value modifies its position along the number line. Specifically, it increases the mean (average) of your dataset by that constant. However, the spread of the data, quantified by the standard deviation, stays the same. This is because every data point is shifted uniformly.
- **Multiplication**: On the contrary, when we multiply each observation by a positive constant, both the mean and the standard deviation are affected. The mean is scaled by the constant, becoming **c̄**. The standard deviation, reflecting the variation from the mean, is also multiplied by the same constant. This scaling results in broader (or narrower) data dispersion, depending on whether **c** is greater than or less than one.
Understanding this mechanism helps ensure accurate data interpretation, particularly in diverse contexts.

Mean and Standard Deviation

The mean and standard deviation are fundamental statistical tools that help describe data distributions. The mean, commonly known as the average, provides the central tendency of the data set, a representative value.
- **Mean Alterations**: In our exercise, when scores are adjusted by adding 20 points to each one, the new mean can be quickly computed by simply adding 20 to the original mean. In the case of converting currencies, the mean is altered by applying the conversion rate, which in this case doubles the unit of measurement but results in halving the value, as demonstrated by the transition from dollars to pounds.
- **Effect on Standard Deviation**: Standard deviation is a measure of the amount of variation or dispersion in a set of values. When dealing with addition, it remains unchanged. However, multiplication of the data by a constant alters the spread proportionately. This was evident with the income conversion, where both the mean and the standard deviation were halved by multiplying by 0.5.
Correctly applying these concepts ensures a consistent understanding of changes to data, as statistics leverage shifts to interpret underlying patterns effectively.

Distribution Shape

An intriguing aspect of linear transformations like rescaling is their neutral effect on distribution shapes. Despite shifting or stretching the scale, the shape—whether normal, skewed, or kurtotic—remains unchanged.
- **Why Shape Persists**: The preservation of shape happens because linear transformations adjust all data points consistently. They do not influence the relative proportions or structure within the data. For instance, if a set of exam scores were initially normally distributed, adding 20 points to each would still leave them normally distributed, just with a shifted center.
- **Implications of Unchanged Shape**: The constancy of shape is significant when performing data analyses. It implies the fundamental features of the dataset—including its symmetry, peakness, or skewness—are retained post-transformation. This property is vital in comparative analyses and in maintaining the integrity of subsequent statistical inferences. By recognizing how transformations interact with data distribution, one can better predict, model, and understand statistical trends without concerns of distortion from scale changes.