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Contemporary Mathematics

8.4 Range and Standard Deviation

Contemporary Mathematics8.4 Range and Standard Deviation

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
A standardized test form with bubbles for options A, B, C, and D is shown. The answer choices for the first three questions are filled in.
Figure 8.30 Measures of spread help us get a better understanding of test scores. (credit: "Standardized test exams form with answers bubbled" by Marco Verch Professional Photographer/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Calculate the range of a dataset
  2. Calculate the standard deviation of a dataset

Measures of centrality like the mean can give us only part of the picture that a dataset paints. For example, let’s say you’ve just gotten the results of a standardized test back, and your score was 138. The mean score on the test is 120. So, your score is above average! But how good is it really? If all the scores were between 100 and 140, then you know your score must be among the best. But if the scores ranged from 0 to 200, then maybe 140 is good, but not great (though still above average). Knowing information about how the data are spread out can help us put a particular data value in better context. In this section, we’ll look at two numbers that help us describe the spread in the data: the range and the standard deviation. These numbers are called measures of dispersion.

The Range

Our first measure of dispersion is the range, or the difference between the maximum and minimum values in the set. It’s the measure we used in the standardized test example above.

Let’s look at a couple of examples.

Example 8.24

Finding the Range

You survey some of your friends to find out how many hours they work each week. Their responses are: 5, 20, 8, 10, 35, 12. What is the range?

Your Turn 8.24

1.
On your morning commute, you decide to record how long you have to wait each time you get caught at a red light. Here are the times in seconds: 12, 58, 35, 79, 21. What is the range?

For large datasets, finding the maximum and minimum values can be daunting. There are two ways to do it in a spreadsheet. First, you can ask the spreadsheet program to sort the data from smallest to largest, then find the first and last numbers on the sorted list. The second method uses built-in functions to find the minimum and maximum.

In either method, once you’ve found the maximum and minimum, all you have to do is subtract to find the range.

Example 8.25

Finding the Range with Google Sheets

The data in “AvgSAT” contains the average SAT score for students attending every institution of higher learning in the US for which data is available. What is the range of these average SAT scores?

Your Turn 8.25

1.
The file “InState” contains in-state tuition costs (in dollars) for every institution of higher learning in the US for which data is available. What is the range of these costs?

The range is very easy to compute, but it depends only on two of the data values in the entire set. If there happens to be just one unusually high or low data value, then the range might give a distorted measure of dispersion. Our next measure takes every single data value into account, making it more reliable.

The Standard Deviation

The standard deviation is a measure of dispersion that can be interpreted as approximately the average distance of every data value from the mean. (This distance from the mean is the “deviation” in “standard deviation.”)

FORMULA

The standard deviation is computed as follows:

s=(x-x¯)2n-1s=(x-x¯)2n-1

Here, xx represents each data value, x¯x¯ is the mean of the data values, nn is the number of data values, and the capital sigma (ΣΣ) indicates that we take a sum.

To compute the standard deviation using the formula, we follow the steps below:

  1. Compute the mean of all the data values.
  2. Subtract the mean from each data value.
  3. Square those differences.
  4. Add up the results in step 3.
  5. Divide the result in step 4 by n-1 n-1
  6. Take the square root of the result in step 5.

Let’s see that process in action.

Example 8.26

Computing the Standard Deviation

You surveyed some of your friends to find out how many hours they work each week. Their responses were: 5, 20, 8, 10, 35, 12. What is the standard deviation?

Your Turn 8.26

1.
On your morning commute, you decide to record how long you have to wait each time you get caught at a red light. Here are the times in seconds: 12, 58, 35, 79, 21.
What is the standard deviation?

The computation for the standard deviation is complicated, even for just a small dataset. We’d never want to compute it without technology for a large dataset! Luckily, technology makes this calculation easy.

Example 8.27

Finding the Standard Deviation with Google Sheets

The data in “AvgSAT” contains the average SAT score for students attending every institution of higher learning in the US for which data is available. What is the standard deviation of these average SAT scores?

Your Turn 8.27

1.
The file “InState” contains in-state tuition costs (in dollars) for every institution of higher learning in the US for which data is available. What is the standard deviation of these costs?

Check Your Understanding

24.
Given the data 1, 4, 5, 5, and 10, find the range.
25.
Given the data 1, 4, 5, 5, and 10, find the standard deviation using the process outlined in the definition.

Employees at a college help desk track the number of people who request assistance each week, as listed below:

142 153 158 156 141 143
139 158 156 146 137 153
136 127 157 148 132 139
155 167 143 168 133 157
138 156 164 130 148 136
26.
Compute the range.
27.
Compute the standard deviation.

The following are data on the admission rates of the different branch campuses in the University of California system, along with the out-of-state tuition and fee cost.

Campus Admission Rate Cost ($)
Berkeley 0.1484 43,176
Davis 0.4107 43,394
Irvine 0.2876 42,692
Los Angeles 0.1404 42,218
Merced 0.6617 42,530
Riverside 0.5057 42,819
San Diego 0.3006 43,159
Santa Barbara 0.322 43,383
Santa Cruz 0.4737 42,952
(source: https://data.ed.gov)
28.
Compute the range of the admission rate.
29.
Compute the standard deviation of the admission rate.
Using the data from Table 8.21, find the:
30.
Range of the cost.
31.
Standard deviation of the cost.
Given the data 81, 84, 89, 85, 86, 84:
1 .
Compute the range.
2 .
Compute the standard deviation using the formula.
Given the data 127, 167, 156, 158, 156, 157:
3 .
Compute the range.
4 .
Compute the standard deviation using the formula.

For the following exercises, use the data found in “TNSchools”, which has data on many institutions of higher education in the state of Tennessee. Here are what the columns represent:

Column Name Description
AdmRate Proportion of applicants that are admitted
UGEnr Number of undergraduate students
PTUG Proportion of undergraduates who attend part-time
InState Tuition and fees for in-state students
OutState Tuition and fees for out-of-state students
FacSal Mean monthly faculty salary
Pell Proportion of students receiving Pell Grants
MedDebt Median student loan debt at degree completion
StartAge Mean age at the time of entry
Female Proportion of students who identify as female
(source: https://data.ed.gov)
5 .
Find the range of admissions rates.
6 .
Find the standard deviation of admissions rates.
7 .
Find the range of undergraduate enrollments.
8 .
Find the standard deviation of undergraduate enrollments.
9 .
Find the range of the proportions of undergraduates attending part-time.
10 .
Find the standard deviation of the proportions of undergraduates attending part-time.
11 .
Find the range of in-state tuition costs.
12 .
Find the standard deviation of in-state tuition costs.
13 .
Find the range of median debt.
14 .
Find the standard deviation of median debt.

For the following exercises, use the table below, which gives the final results for the 2021 National Women’s Soccer League season. The columns are standings points (PTS; teams earn three points for a win and one point for a tie), wins (W), losses (L), ties (T), goals scored by that team (GF), and goals scored against that team (GA).

Team PTS W L T GF GA
Portland Thorns FC 44 13 6 5 33 17
OL Reign 42 13 8 3 37 24
Washington Spirit 39 11 7 6 29 26
Chicago Red Stars 38 11 8 5 28 28
NJ/NY Gotham FC 35 8 5 11 29 21
North Carolina Courage 33 9 9 6 28 23
Houston Dash 32 9 10 5 31 31
Orlando Pride 28 7 10 7 27 32
Racing Louisville FC 22 5 12 7 21 40
Kansas City Current 16 3 14 7 15 36
15 .
Compute the standard deviation and range of points.
16 .
Compute the standard deviation and range of wins.
17 .
Compute the standard deviation and range of losses.
18 .
Compute the standard deviation and range of ties.
19 .
Compute the standard deviation and range of goals scored (GF).
20 .
Compute the standard deviation and range of goals against (GA).
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