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Chapter 8: Problem 29

give an example of: A distribution with a mean of \(1 / 2\) and median \(1 / 2\)

Short Answer

Expert verified
A simple example is the set \( \{0, 1, 0.5\} \), where both mean and median are \( \frac{1}{2} \).

Step by step solution

01

Understand Mean and Median

To find a distribution with a given mean and median, recognize that both should be equal at \( \frac{1}{2} \). This implies a symmetric distribution or a carefully chosen set of values where the average and middle value (median) are the same.
02

Choose a Simple Set

Consider a simple set of numbers that are symmetrical around \( \frac{1}{2} \). An example set could be \( \{0, 1, 0.5\} \).
03

Calculate the Mean

For the set \( \{0, 1, 0.5\} \), the mean is calculated as follows:\[\text{Mean} = \frac{0 + 1 + 0.5}{3} = \frac{1.5}{3} = \frac{1}{2}.\]
04

Calculate the Median

To find the median of \( \{0, 1, 0.5\} \), first arrange the numbers in ascending order: \( \{0, 0.5, 1\} \). The median is the middle number, which is \( 0.5 \) or \( \frac{1}{2} \).

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Key Concepts

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

Understanding the Mean
The mean, often called the average, is a measure that helps summarize a set of numbers into a single value. This value represents the central point of a dataset. To find the mean, you sum up all the numbers and then divide by the count of the numbers.
For example, if you have the numbers 0, 1, and 0.5, the mean is calculated by adding them together to get 1.5, and then dividing by the total number of values, which is 3. This gives us \((0 + 1 + 0.5) / 3 = 1/2 \.\)

Think of the mean as the number that balances the weights of the other numbers, like how a seesaw balances when both sides are equal. This property makes it very useful, though it assumes equal importance of all points, which might not suit all data distributions.
Explaining the Median
The median offers another way to look at the center of a dataset. While the mean requires calculation, the median is simply the middle value, making it straightforward to determine for small sets.
To find the median, you arrange the numbers in order from smallest to largest and pick the middle one. If there is an even number of values, the median is the mean of the two central numbers.
Consider the numbers 0, 0.5, 1. Arranging them in order gives the same sequence, and since there are three values, the middle one is 0.5. Thus, the median here also equals \( \frac{1}{2} \).

The median is less sensitive to extremely high or low values, which makes it a better measure for data with outliers. It's ideal where the dataset might include anomalies.
Symmetric Distribution Characteristics
A symmetric distribution is balanced and mirrors itself around a central value. In this type of distribution, the mean and median are often the same, which clearly shows no skewness.
For example, in the dataset \(\{0, 1, 0.5\}\), both the mean and median are \(\frac{1}{2}\). This symmetry indicates that each side of the center has the same shape, leading to balance.

Some common traits of symmetric distributions include:
  • The mean equals the median.
  • The shape is mirrored on either side of the center.
  • It's often depicted as a bell curve or normal distribution.
Symmetric distributions are easy to understand and can simplify statistical analysis as outliers do not skew these distributions in one direction.

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Most popular questions from this chapter

Sketch the solid obtained by rotating each region around the indicated axis. Using the sketch, show how to approximate the volume of the solid by a Riemann sum, and hence find the volume. Bounded by \(y=x^{3}, x=1, y=-1 .\) Axis: \(y=-1\).

Concern the region bounded by \(y=x^{2}\) \(y=1,\) and the \(y\) -axis, for \(x \geq 0 .\) Find the volume of the solid. The solid obtained by rotating the region about the \(x\) axis.

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The probability of a transistor failing between \(t=a\) months and \(t=b\) months is given by \(c \int_{a}^{b} e^{-c t} d t,\) for some constant \(c\) (a) If the probability of failure within the first six months is \(10 \%,\) what is \(c ?\) (b) Given the value of \(c\) in part (a), what is the probability the transistor fails within the second six months?
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