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Chapter 3

# MATH 10041 Chapter Notes - Chapter 3: Random Variable, Standard Deviation, UnimodalityPremium

1 pages89 viewsSpring 2019

Department
Mathematics
Course Code
MATH 10041
Professor
Beverly M Reed
Chapter
3

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3.1: Summaries for Symmetric Distributions
Symmetric distributions are those in which the left-hand side of the graph of the
distribution is roughly a mirror image of the right-hand side
Mean: the arithmetic average
The mean of a sample
o A numerical summary
o Measures the center of the distribution of a sample of data
o The mean identifies the “balancing point” of the distribution, which is the
arithmetic average of the values
o The mean represents the typical value in a set of data when the distribution is
roughly symmetric
Average:
Deviation: the distance each observation is from the mean
Standard Deviation: a number that measures how far away the typical observation is from the
mean
The standard deviation of a sample:
o A numerical summary
o Measures the spread of a distribution of a sample of data
o It measures the typical distance of observations from the mean
o To measure the amount
Variance: variance is the expectation of the squared deviation of a random variable from its
mean
3.2: The Empirical Rule
The Empirical Rule: if the distribution is unimodal and symmetric then:
Approx. 68% of the observations will be within one standard deviation of the mean
Approx. 95% of the observations will be within two standard deviations of the mean
Nearly all observations will be within two standard deviations of the mean
Standard Units: measure a value relative to the sample rather than to some absolute measure
Z-Score: a measurement converted into a standard
A standard observation
Converts a measurement into standard units
By measuring how many standard deviations is from the sample mean
To compare values from different groups, such as two exam scores from different
exams, or to compare values measured in different units, such as inches and pounds
Formula:
o Z= (x-mean)/deviation

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