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Chapter 3: Problem 85

Construct two sets of numbers with at least five numbers in each set with the following characteristics: The means are the same, but the standard deviation of one of the sets is smaller than that of the other. Report the mean and both standard deviations.

Short Answer

Expert verified
The two sets with the same mean but different standard deviations could be \{9, 10, 10, 10, 11\} and \{5, 8, 10, 12, 15\}, with standard deviations of approximately 0.8 and 3.6, respectively.

Step by step solution

01

Calculate the mean

Choose a common mean for both sets. For simplicity, let's choose the mean to be 10.
02

Create the first set

For the first set with a smaller standard deviation, you need numbers close to the mean. An example might be \{9, 10, 10, 10, 11\}. All these numbers are close to 10, which means they won't diverge much from the mean, resulting in smaller standard deviation.
03

Create the second set

For the second set with a larger standard deviation, you need some numbers further from the mean. An example might be \{5, 8, 10, 12, 15\}. These numbers diverge more from the mean compared to the first set, hence shall result in a larger standard deviation.
04

Calculate standard deviations

Now, calculate the standard deviation for both sets. For the first set, the standard deviation would be approximately 0.8. For the second set, it would be around 3.6.

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Key Concepts

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

Standard Deviation
Standard deviation is a measure of how spread out numbers in a data set are. It tells us the average distance between each data point and the mean. If the numbers are closer to the mean, the standard deviation is smaller. Conversely, if the numbers vary greatly from the mean, the standard deviation is larger.

To calculate the standard deviation:
  • Find the mean of your data set.
  • Subtract the mean from each number to get the deviation for each data point.
  • Square each deviation to eliminate negative numbers.
  • Find the average of these squared deviations, known as the variance.
  • Take the square root of the variance to get the standard deviation.
In our example, the first set of numbers like \( \{9, 10, 10, 10, 11\} \) is closer to the mean. Therefore, it has a smaller standard deviation, approximately 0.8.

The second set \( \{5, 8, 10, 12, 15\} \) has numbers that are spread farther from the mean. Thus, its standard deviation is larger, around 3.6.
Mean
The mean, often called the average, is one of the most common measures of central tendency in data sets. It is calculated by adding up all the numbers in a set and then dividing by the number of numbers.
  • Sum up all numbers in the data set.
  • Divide the sum by the number of data points.
In our problem, we decided on a common mean of 10 for both data sets. When creating both sets, each set has numbers that average out to 10.

For example, in the first set \( \{9, 10, 10, 10, 11\} \):- Sum up the numbers: \(9 + 10 + 10 + 10 + 11 = 50\)- Divide by the total number of numbers, which is 5. Thus, the mean is \( \frac{50}{5} = 10 \). This simplicity ensures that both sets have the same mean, providing a clear focus on observing differences in standard deviation.
Data Sets
Data sets are collections of numbers that we analyze to find common patterns, trends, or statistical properties like mean and standard deviation. Each data set can be structured based on the needs of the analysis or the study's goals.

In our example, we created two data sets:
  • The first set \( \{9, 10, 10, 10, 11\} \) was structured to have less variation around the mean. This resulted in a smaller standard deviation. This design demonstrates data consistency and uniformity.
  • The second set \( \{5, 8, 10, 12, 15\} \) was created to display a greater range of numbers with some significantly deviating from the mean. This results in a larger standard deviation, showing greater data variability.
Understanding how to construct and analyze these data sets helps in grasping not only statistical concepts but also how they influence interpretation of data outcomes. Structuring data sets effectively is crucial for accurate statistical analysis and meaningful results.

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Most popular questions from this chapter

In 2017 a pollution index was calculated for a sample of cities in the eastern states using data on air and water pollution. Assume the distribution of pollution indices is unimodal and symmetric. The mean of the distribution was \(35.9\) points with a standard deviation of \(11.6\) points. (Source: numbeo. com) see Guidance page \(142 .\) a. What percentage of eastern cities would you expect to have a pollution index between \(12.7\) and \(59.1\) points? b. What percentage of castern cities would you expect to have a pollution index between \(24.3\) and \(47.5\) points? c. The pollution index for New York, in 2017 was \(58.7\) points. Based on this distribution, was this unusually high? Explain.

Four siblings are \(2,6,9\), and 10 years old. a. Calculate the mean of their current ages. Round to the nearest tenth. b. Without doing any calculation, predict the mean of their ages 10 years from now. Check your prediction by calculating their mean age in 10 years (when they are \(12,16,19\), and 20 years old). c. Calculate the standard deviation of their current ages. Round to the nearest tenth. d. Without doing any calculation, predict the standard deviation of their ages 10 years from now. Check your prediction by calculating the standard deviation of their ages in 10 years. c. Adding 10 years to each of the siblings ages had different effects on the mean and the standard deviation. Why did one of these values change while the other remained unchanged? How does adding the same value to cach number in a data set affect the mean and standard deviation?

Standard Deviation Is it possible for a standard deviation to be negative? Explain.

Data were collected on cereals stocked by a supermarket. A portion of the data table is shown below. Data includes shelf location (top, middle, bottom), name, manufacturer, type (hot or cold), and amount of calories, sodium and fiber per serving. Write one or two investigative questions that could be answered using these data.

A dieter recorded the number of calories he consumed at lunch for one week. As you can see, a mistake was made on one entry. The calories are listed in increasing order: $$ 331,374,387,392,405,4200 $$ When the error is corrected by removing the extra 0 , will the median calories change? Will the mean? Explain without doing any calculations.
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