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Chapter 3Statistics for Describing,

Exploring, and Comparing Data

3-1 Overview

3-2 Measures of Center

3-3 Measures of Variation

3-4 Measures of Relative Standing

3-5 Exploratory Data Analysis (EDA)

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Section 3-1 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

Overview

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Section 3-2Measures of Center

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Key Concept

When describing, exploring, and comparing data sets, these characteristics are usually extremely important: center, variation, distribution, outliers, and changes over time.

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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

Definition

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Notation

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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Notation

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

x =n

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

x

Nµ =

x

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Definitions

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

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

is not affected by an extreme value

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Finding the Median

If the number of values is odd, the median is the number located in the exact middle of the list.

If the number of values is even, the median is found by computing the mean of the two middle numbers.

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5.40 1.10 0.42 0.73 0.48 1.10 0.66

0.42 0.48 0.66 0.73 1.10 1.10 5.40 (in order - odd number of values)

exact middle MEDIAN is 0.73

5.40 1.10 0.42 0.73 0.48 1.10

0.42 0.48 0.73 1.10 1.10 5.40

0.73 + 1.10

2

(in order - even number of values – no exact middleshared by two numbers)

MEDIAN is 0.915

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Definitions Mode

the value that occurs most frequently

Mode is not always unique

A data set may be:Bimodal

MultimodalNo Mode

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

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a. 5.40 1.10 0.42 0.73 0.48 1.10

b. 27 27 27 55 55 55 88 88 99

c. 1 2 3 6 7 8 9 10

Mode - Examples

Mode is 1.10

Bimodal - 27 & 55

No Mode

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Midrange

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

Definition

Midrange =maximum value + minimum value

2

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

Round-off Rule for Measures of Center

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You do!

Here are the volumes (in ounces) or randomly selected cans of Coke:

12.3 12.1 12.2 12.3 12.2

Find the mean, median, mode, and midrange.

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Assume that in each class, all sample values are equal to the class midpoint.

Mean from a Frequency Distribution

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use class midpoint of classes for variable x

Mean from a Frequency Distribution

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Find the Mean from a Frequency Distribution

Age of Actress Frequency (f) Class Midpoint (x)

f*x

21-30 28 25.5 714

31-40 30 35.5 1065

41-50 12 45.5 546

51-60 2 55.5 111

61-70 2 65.5 131

71-80 2 75.5 151

Totals 76 2718

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Weighted Mean

x =w

(w • x)

In some cases, values vary in their degree of importance, so they are weighted accordingly.

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Example

• We have three test sores: 85, 90, 75

• The first test counts for 20%, the second for 30%, and the third for 50% of the final grade.

• Find the weighted mean.

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Best Measure of Center

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Symmetricdistribution of data is symmetric if

the left half of its histogram is roughly a mirror image of its right half

Skeweddistribution of data is skewed if it is

not symmetric and if it extends more to one side than the other

Definitions

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Skewness

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Heart Rate ActivityIs there a difference between male and

female heart rates?

Male: 60 67 59 64 80 55 72 84 59 67 69

Female: 83 56 57 63 60 69 70 86 70 57 67 75 72 75 57 76 69 79 84 75 56

Compare: 1) The sample sizes

2) The Means and the Medians (centers)

3) The IQR, the 5 # Summaries and the standard deviations (spreads)

4) Look at the shape: symmetric, skewed, or irregular?