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Chapter 3: Problem 4

Mean and Median. Egure \(3.7\) displays three density curves, each with three points marked on it. At which of these points on each curve do the mean and the median fall?

Short Answer

Expert verified
The mean is at the balance point; the median splits the area in half.

Step by step solution

01

Understand the Density Curve

A density curve is a graph that shows the general pattern of a distribution. It represents the distribution of a continuous probability variable across a given range. The total area under the curve is equal to 1.
02

Identify Density Curve Characteristics

In a density curve, the mean is the balance point. If the curve were made of solid material, the mean would be the point where it balances. The median is the point that divides the area under the curve into two equal halves.
03

Locate the Median

For each density curve, find the point that splits the curve into two areas of equal size. This point serves as the median of the distribution.
04

Locate the Mean

For each density curve, identify the point that would balance the curve if it were physically placed on a fulcrum. This point serves as the mean of the distribution.
05

Compare Points on Each Curve

Compare the identified mean and median points on each density curve. Remember that if the curve is symmetric, the mean and median will coincide. If the curve is skewed, the mean will be pulled towards the tail.

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Key Concepts

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

Mean and Median
In the context of density curves, the mean and median are important measures of central tendency.
  • The mean of a distribution represented by a density curve is the "balance point." It's the point where the curve would perfectly balance if made of a solid material.
  • The median is the "midpoint" of the distribution. It's the point that divides the area under the curve into two equal parts.
When dealing with density curves that are symmetric, meaning they are mirror images on either side of a central point, both the mean and median will align at this central point.
On the other hand, in skewed distributions, these two points will diverge. Identifying where the mean and median are located on a density curve helps in understanding the shape and balance of the distribution.
Continuous Probability Distribution
A continuous probability distribution is a probability distribution that can take an infinite number of possible outcomes. Unlike discrete distributions, which deal with countable outcomes, continuous distributions cover outcomes that take on a range of values.
  • The range is often an interval on the real number line.
  • Key examples include the normal distribution, uniform distribution, and exponential distribution.
As these distributions are represented by density curves, the total area under the curve is always equal to 1. This represents the entirety of possible outcomes in the distribution being modeled.
Understanding continuous probability distributions is crucial for modeling real-world phenomena that can't be counted in discrete steps and require a smooth probability curve.
Skewness
Skewness in a distribution refers to the asymmetry from the normal distribution in a set of data. A distribution can be skewed to the left (negative skewness) or to the right (positive skewness).
  • In a positively skewed distribution, the mean is located to the right of the peak, and it drags towards the tail end.
  • In a negatively skewed distribution, the mean falls to the left of the peak.
The level of skewness impacts the relationship between the mean and median. Typically, in positively skewed distributions, the mean is greater than the median, while in negatively skewed distributions, the mean is less than the median.
Recognizing skewness helps to interpret and predict behaviors in data used across various statistical analyses.
Symmetric Distribution
A symmetric distribution is one where the left and right sides of the distribution are mirror images of each other. This often occurs in theoretical distributions such as the normal distribution.
  • When a distribution is perfectly symmetric, the mean and median coincide.
  • This implies that the skewness is zero, signifying no skewness.
Symmetry in distributions implies a regularity and predictability, which is valuable when analyzing data for trends.
In practical scenarios, perfectly symmetric distributions are rare. Most real-world data may show slight deviations or skewness. However, the concept of symmetry helps in understanding and using theoretical evaluative measures for data analysis.

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Most popular questions from this chapter

Understanding Density Curves. Remember that it is areas under a density curve, not the height of the curve, that give proportions in a distribution. To illustrate this, sketch a density curve that has a tall, thin peak at 0 on the horizontal axis but has most of its area close to 1 on the horizontal axis without a high peak at 1 .

The distribution of hours of sleep per weeknight among college students is found to be Normally distributed, with a mean of \(6.5\) hours and a standard deviation of 1 hour. The percentage of college students that sleep at least eight hours per weeknight is about a. \(95 \%\) b. \(6.7 \%\) c. \(2.5 \%\)

The Medical College Admissions Test. Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). The total score of the four sections on the test ranges from 472 to 528 . In spring of 2019 , the mean score was \(500.9\), with a standard deviation of \(10.6\). a. What are the median and the first and third quartiles of the MCAT scores? What is the interquartile range? b. Give the interval that contains the central \(80 \%\) of the MCAT scores.

Which of these variables is most likely to have a Normal distribution? a. Income per person for 150 different countries b. Sale prices of 200 homes in Santa Barbara, California c. Lengths of 100 newborns in Connecticut

Upper Arm Lengths. The upper arm length of males over 20 years old in the United States is approximately Normal with mean \(39.1\) centimeters \((\mathrm{cm})\) and standard deviation \(5.0 \mathrm{~cm}\). Use the 68-95-99.7 rule to answer the following questions. (Start by making a sketch like Figure 3.10.) a. What range of lengths covers the middle \(99.7 \%\) of this distribution? b. What percentage of men over 20 have upper arm lengths greater than \(44.1 \mathrm{~cm}\) ?
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