stats ch 3 definitions

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Created by Tom Wegleitner, Centreville, VirginiaEdited by Olga Pilipets, San Diego, California

Overview

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Descriptive Statistics summarize or describe the important

characteristics of a known set of data

Inferential Statistics use sample data to make inferences

(or generalizations) about a population

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Overview

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Key Concept

In order to be able to describe, analyze, and compare data sets we need further insight into the nature of data. In this chapter we are going to look at such extremely important characteristics of data: center and variation. We will also learn how to compare values from different data sets.

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Definition

Measure of Centerthe value at the center or middle of a data set

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Arithmetic Mean(Mean)

the measure of center obtained by adding the values and dividing the total by the number of values

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Definition

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denotes the sum of a set of values. x is the variable usually used to represent the

individual data values. n represents the number of values in a

sample. N represents the number of values in a

population.

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µ is pronounced ‘mu’ and denotes the mean of all values in a population

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Notation

x =n

xis pronounced ‘x-bar’ and denotes the mean of a set of sample values

x

Nµ =

x

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Medianthe middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

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Definitions

often denoted by x (pronounced ‘x-tilde’)~

is not affected by an extreme value

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Modethe value that occurs most frequently

Mode is not always unique A data set may be:

BimodalMultimodalNo Mode

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Definitions

Mode is the only measure of central tendency that can be used with nominal data

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Midrange

the value midway between the maximum and minimum values in the original data set

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Definition

Midrange =maximum value + minimum value

2

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Because this section introduces the concept of variation, which is something so important in statistics, this is one of the most important sections in the entire book.

Place a high priority on how to interpret values of standard deviation.

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The range of a set of data is the difference between the maximum value and the minimum value.

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Range = (maximum value) – (minimum value)

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The standard deviation of a set of sample values is a measure of variation of values about the mean.

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(x - x)2

n - 1s =

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2 (x - µ)

N =

This formula is similar to the previous formula, but instead, the population mean and population size are used.

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Population variance: Square of the population standard deviation

The variance of a set of values is a measure of variation equal to the square of the standard

deviation.

Sample variance: Square of the sample standard deviation s

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standard deviation squared

s

2

2

}Notation

Sample variance

Population variance

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Carry one more decimal place than is present in the original set of data.

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Round only the final answer, not values in the middle of a calculation.

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Estimation of Standard DeviationRange Rule of Thumb

For estimating a value of the standard deviation s,Use

Where range = (maximum value) – (minimum value)

Range4

s

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Estimation of Standard DeviationRange Rule of Thumb

For interpreting a known value of the standard deviation s, find rough estimates of the minimum and maximum “usual” sample values by using:

Minimum “usual” value (mean) – 2 X (standard deviation) =Maximum “usual” value (mean) + 2 X (standard deviation) =

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Definition

Empirical (68-95-99.7) RuleFor data sets having a distribution that is approximately bell shaped, the following properties apply:

About 68% of all values fall within 1 standard deviation of the mean.

About 95% of all values fall within 2 standard deviations of the mean.

About 99.7% of all values fall within 3 standard deviations of the mean.

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The Empirical Rule

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The Empirical Rule

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The Empirical Rule

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z Score (or standardized value)the number of standard deviations

that a given value x is above or below the mean

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Definition

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Sample

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Population

x - µz =

Round z to 2 decimal places

Measures of Position z score

z = x - xs

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Interpreting Z Scores

Whenever a value is less than the mean, its corresponding z score is negative

Ordinary values: z score between –2 and 2 Unusual Values: z score < -2 or z score > 2