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Chapter 3: Problem 53

Where Are the Quartiles? How many standard deviations above and below the mean do the quartiles of any Normal distribution lie? (Use the standard Normal distribution to answer this question.)

Short Answer

Expert verified
The quartiles lie approximately 0.6745 standard deviations from the mean in a normal distribution.

Step by step solution

01

Understanding the Problem

We need to determine how many standard deviations above and below the mean the quartiles lie in a normal distribution. This involves finding the z-scores that correspond to the first and third quartiles in a standard normal distribution.
02

Know Your Quartiles

In any distribution, the first quartile (Q1) is the 25th percentile and the third quartile (Q3) is the 75th percentile. We'll find the z-scores for these percentiles in the standard normal distribution.
03

Use Standard Normal Table or Calculator

Using a standard normal distribution table or calculator, find the z-score such that the area to the left is 0.25, which gives us Q1. Similarly, find the z-score for which the area to the left is 0.75, which gives us Q3.
04

Calculate the Z-scores

From the standard normal distribution, the z-score corresponding to the 25th percentile (Q1) is approximately -0.6745. Similarly, the z-score for the 75th percentile (Q3) is approximately 0.6745.
05

Interpretation in Terms of Standard Deviations

These z-scores indicate that Q1 is around -0.6745 standard deviations below the mean, and Q3 is around 0.6745 standard deviations above the mean.

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Key Concepts

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

Quartiles
Quartiles are key tools used in statistics to understand how data is distributed. Essentially, they divide a dataset into four equal parts. In any data set:
  • The first quartile (Q1) marks the 25th percentile, meaning 25% of the data falls below this value.
  • The second quartile (Q2) is the median or the 50th percentile, splitting the dataset into two equal halves.
  • The third quartile (Q3) represents the 75th percentile, which indicates that 75% of the data is below this point.
In a standard normal distribution, which is a symmetrical bell-shaped distribution, these quartiles help us understand the spread of data around the mean. The quartiles in such a distribution are typically measured in terms of z-scores, which we will explore in the following sections.
Standard Deviation
Standard deviation is a statistical measurement that describes how spread out the numbers in a data set are. Think of it as a way to quantify variation or dispersion of a set of values. In a normal distribution:
  • The standard deviation quantifies the amount of variation or deviation from the mean (average).
  • A smaller standard deviation indicates that the data points tend to be closer to the mean.
  • A larger standard deviation indicates that the data points are spread out over a broader range of values.
In any normal distribution, about 68% of values fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations, also known as the 68-95-99.7 rule. This rule is crucial as it helps us understand probabilities and make predictions based on the data.
Z-Scores
Z-scores, also known as standard scores or normal deviates, are a measure of how many standard deviations an element is from the mean of the distribution. They allow us to put different data points on a common scale, making it easier to compare them.
  • A z-score of 0 indicates that the data point is exactly at the mean.
  • Positive z-scores indicate data points above the mean, while negative z-scores indicate points below the mean.
To find z-scores for quartiles in a standard normal distribution, we look at specific percentiles. For example:
  • The first quartile (Q1) usually has a z-score of approximately -0.6745, indicating it's roughly 0.6745 standard deviations below the mean.
  • The third quartile (Q3) has a z-score of approximately 0.6745, meaning it's 0.6745 standard deviations above the mean.
These z-scores are essential for understanding how the quartiles relate to the rest of the data in a standard normal distribution.

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Most popular questions from this chapter

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. \(2.5 \%\)

Runners. In a study of exercise, a large group of male runners walk on a treadmill for 6 minutes. Their heart rates in beats per minute at the end vary from runner to runner according to the \(N(104,12.5)\) distribution. The heart rates for male nonrunners after the same exercise have the \(N(130,17)\) distribution. a. What percentage of the runners have heart rates above 140 ? b. What percentage of the nonrunners have heart rates above 140 ?

Weights Aren't Normal. The heights of people of the same sex and similar ages follow a Normal distribution reasonably closely. Weights, on the other hand, are not Normally distributed. The weights of men aged 20-29 in the United States have mean 186.8 pounds and median \(177.8\) pounds. The first and third quartiles are \(152.9\) pounds and \(208.5\) pounds, respectively. In addition, the bottom \(10 \%\) have weights less than or equal to \(137.6\) pounds while the top \(10 \%\) have weights greater than or equal to 247.2. What can you say about the shape of the weight distribution? Why?

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.

The proportion of observations from a standard Normal distribution that take values greater than \(1.78\) is about a. \(0.9554 .\) b. \(0.0446 .\) c. \(0.0375 .\)
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