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

1 pages89 viewsSpring 2019

School

Kent State UniversityDepartment

MathematicsCourse Code

MATH 10041Professor

Beverly M ReedChapter

3This

**preview**shows half of the first page. to view the full**1 pages of the document.**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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