MATH3339_lect04_F

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Mathematics

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Feb 20, 2024

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MATH 3339 Statistics for the Sciences sec 2.3; 2.6 Wendy Wang, Ph.D. wwang60@central.uh.edu Lecture 4 - 3339 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 1 / 44
Outline 1 Understanding Standard Deviation 2 Calculating The Standard Deviation 3 Measures of Variability 4 Jointly Distributed Data 5 Scatterplots for Jointly Distributed Variables 6 Covariance and Correlation Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 2 / 44
Measuring Spread: The Standard Deviation Measures spread by looking at how far the observations are from their mean. Most common numerical description for the spread of a distribution. A larger standard deviation implies that the values have a wider spread from the mean. Denoted s when used with a sample. This is the one we calculate from a list of values. Denoted σ when used with a population. This is the "idealized" standard deviation. The standard deviation has the same units of measurements as the original observations. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 3 / 44
Definition of the Standard Deviation The standard deviation is the average distance each observation is from the mean. Using this list of values from a sample: 3, 3, 9, 15, 15 The mean is __. By definition, the average distance each of these values are from the mean is __. So the standard deviation is __. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 4 / 44
Values of the Standard Deviation The standard deviation is a value that is greater than or equal to zero. It is equal to zero only when all of the observations have the same value. By the definition of standard deviation determine s for the following list of values. ▶ 2, 2, 2, 2 : standard deviation = __ ▶ 125, 125, 125, 125, 125: standard deviation = __ Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 5 / 44
Adding or Subtracting a Value to the Observations Adding or subtracting the same value to all the original observations does not change the standard deviation of the list. Using this list of values: 3, 3, 3, 15, 15, 15 mean = 9, standard deviation = __. If we add 4 to all the values: 7, 7, 7, 19, 19, 19 mean = 13 , standard deviation = __ Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 6 / 44
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