/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Can the sample standard deviatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 5

Can the sample standard deviation be equal to zero? If so, give an example.

Short Answer

Expert verified
Yes, the sample standard deviation can be zero, for example, in the dataset \( \{5, 5, 5, 5, 5\} \).

Step by step solution

01

Understanding Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion in a set of values. A standard deviation of zero indicates that all values in the dataset are identical, meaning there is no spread in the dataset.
02

Conditions for Standard Deviation of Zero

For the standard deviation to be zero, every value in the dataset must be the same. This is because the deviations of all individual data points from the mean must be zero.
03

Constructing an Example

Consider the dataset \( \{5, 5, 5, 5, 5\} \). In this dataset, every value is the same (5). The mean is also 5, and every data point minus the mean is zero, resulting in zero variance and, thus, zero standard deviation.
04

Calculating the Sample Standard Deviation

First, calculate the mean: \( \bar{x} = \frac{5+5+5+5+5}{5} = 5 \). Next, calculate each deviation from the mean, which all result in zero. Calculate the squared deviations and their average: \( \frac{0+0+0+0+0}{4} = 0 \). Therefore, the sample standard deviation is \( \sqrt{0} = 0 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Variance
Variance is a fundamental concept that helps us understand how data points in a dataset are spread out from the mean. It is calculated by taking the average of the squared differences between each data point and the mean. This gives us a sense of how much the data varies. If the variance is high, it means that data points are spread out over a wider range.

To calculate variance:
  • First, find the mean (\(\bar{x}\)) of your dataset.
  • Then, calculate each data point's deviation from the mean.
  • Square each of these deviations.
  • Finally, compute the average of these squared deviations.
If all values in a dataset are the same, each deviation from the mean will be zero. Consequently, the variance will also be zero. This emphasizes that there is no spread or variation in the data.
Standard Deviation Calculation
The standard deviation is closely related to variance but is a more intuitive measure of data spread since it is in the same units as the data. The standard deviation is simply the square root of the variance. This makes it easier to interpret, as it directly relates to the original scales of the data. Calculating the standard deviation is quite straightforward:
  • Determine the variance of your dataset.
  • Take the square root of the variance to obtain the standard deviation.
For example, consider a dataset with values \(\{5, 5, 5, 5, 5\}\). With a variance of 0 (since all deviations are 0), the standard deviation is also 0 after taking the square root. This indicates no variability between data points.
Dataset with Zero Variance
A dataset with zero variance is an interesting scenario where all data points are identical. This means that every value in the data set is exactly the same as the mean. Since the variance measures the average squared deviation from the mean, if all data points equal the mean, these deviations are all zero.

For instance, if you have a dataset like \(\{10, 10, 10, 10\}\), the mean is 10, and every data point's deviation from the mean is zero.
  • The squared deviations are also zero, leading to zero variance.
  • Therefore, the standard deviation is also zero.
This lack of variance indicates complete consistency in the data, without any variation or spread.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that the sample size \(n\) is such that the quantity \(n T / 100\) is not an integer. Develop a procedure for obtaining a trimmed mean in this case.

The following data are direct solar intensity measurements (watts/m \(^{2}\) ) on different days at a location in southern Spain: 562,869,708,775,775,704,809,856,655,806,878 909,918,558,768,870,918,940,946,661,820,898,935 \(952,957,693,835,905,939,955,960,498,653,730,\) and 753\. Calculate the sample mean and sample standard deviation. Prepare a dot diagram of these data. Indicate where the sample mean falls on this diagram. Give a practical interpretation of the sample mean.

Will the sample mean always be the most frequently occurring data value in the sample?

An article in the Journal of Physiology ["Response of Rat Muscle to Acute Resistance Exercise Defined by Transcriptional and Translational Profiling" (2002, Vol. 545, pp. \(27-41\) ) ] studied gene expression as a function of resistance exercise. Expression data (measures of gene activity) from one gene are shown in the following table. One group of rats was exercised for six hours while the other received no exercise. Compute the sample mean and standard deviation of the exercise and no-exercise groups separately. Construct a dot diagram for the exercise and no-exercise groups separately. Comment on any differences for the groups.

When will the median of a sample be equal to the sample mean?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.