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Chapter 2: Problem 122

Largest and Smallest Standard Deviation Using only the whole numbers 1 through 9 as possible data values, create a dataset with \(n=6\) and \(\bar{x}=5\) and with: (a) Standard deviation as small as possible. (b) Standard deviation as large as possible.

Short Answer

Expert verified
For a set of numbers between 1-9, each containing 6 numbers having a mean of 5; The set with the smallest standard deviation could be \{4,5,5,5,5,6\} yielding a standard deviation of approximately \(0.577\), and the set with the largest deviation could be \{1,1,9,9,5,5\} with a standard deviation of approximately \(3.266\).

Step by step solution

01

Developing a set with a small deviation

To have the smallest possible standard deviation, we need all the numbers to be as close to each other (and ideally equal to) the mean as possible. Since we need a mean of 5, and we have six numbers, a viable set could be \{4,5,5,5,5,6\}. This data set has a mean of 5 and all numbers are very close to the mean, resulting in a small standard deviation.
02

Calculating the standard deviation for the first set

Recall that standard deviation is calculated by finding the square root of the average of the squared differences from the mean. Here, the differences from the mean are: \(-1, 0, 0, 0, 0, 1\). The squares of these differences are: \(1, 0, 0, 0, 0, 1\). The mean of these squares is \(\frac{1+0+0+0+0+1}{6} = \frac{2}{6} = \frac{1}{3}\). Taking the square root of this, the standard deviation is approximately \(0.577\) (rounded to three decimal places)
03

Developing a set with a large deviation

To have the largest possible standard deviation, we need to choose numbers that are as far from the mean as possible. The extreme numbers in our range (1 and 9) are good candidates for this. However, we still need to balance out these extremes to maintain our average of 5. One possible solution for the set could be \{1,1,9,9,5,5\}. In this set, two numbers are far below the mean, two are far above and two are equal to the mean, resulting in a larger standard deviation.
04

Calculating the standard deviation for the second set

Using the formula for standard deviation again, our differences from the mean are: \(-4, -4, 4, 4, 0, 0\). The squares are: \(16, 16, 16, 16, 0, 0\). The mean of these squares is \(\frac{16+16+16+16+0+0}{6} = \frac{64}{6} = \frac{32}{3}\). Taking the square root of this, the standard deviation is approximately \(3.266\) (rounded to three decimal places)

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Key Concepts

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

Understanding Standard Deviation
Standard deviation is a measure of how spread out the values in a dataset are around the mean. It tells us how much the values deviate from the average.
If the standard deviation is small, it means the values are clustered closely around the mean. A large standard deviation indicates that the values are more spread out.
To calculate standard deviation:
  • Find the mean of the dataset.
  • Subtract the mean from each data point to get the differences.
  • Square each difference to eliminate negative values.
  • Calculate the average of these squared differences.
  • Take the square root of that average to get the standard deviation.
This process helps us understand how consistent or varied data is. In this exercise, different datasets with varying standard deviations help illustrate this concept.
Essence of Dataset Construction
Constructing a dataset involves choosing data points that meet specific criteria. Here, the requirements were to use whole numbers from 1 to 9, have a total of six numbers, and achieve a mean (average) of 5.
A thoughtfully constructed dataset aligns the numbers such that they fulfill the given conditions. For example, if we want a small standard deviation, we choose numbers close to the mean. For a large standard deviation, we select numbers that are more spaced out but still maintain the required mean.
Some key strategies include:
  • Using repeated numbers near the mean for small deviation.
  • Selecting extreme values balanced by numbers around the mean for large deviation.
Being mindful of these strategies allows one to creatively manipulate numbers to meet specific mathematical requirements.
The Process of Mean Calculation
The mean, often called the average, is a fundamental concept in statistics. It gives a central value for a dataset. To calculate the mean, you sum up all the data points and divide by the total number of points.
The steps to calculate the mean are:
  • Add all the numbers in the dataset.
  • Divide the total by the number of values.
In our exercise, the mean needed to be 5. Therefore, any dataset constructed was checked by adding all its values and dividing by 6 (since there are six numbers). Consistency in finding the mean ensures that datasets meet the required conditions, providing a reference point around which variations like standard deviation are calculated.

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