Chapter 2 - Wessa.net · PDF filefrequency total number of data ... and 50% lie below it For any p, ... –Dot plot –Stem-and-leaf display –Histogram. Summary

Chapter 2

Methods for Describing Sets of Data

Objectives

Describe Data using Graphs

Describe Data using Charts

Describing Qualitative Data

•Qualitative data are nonnumeric in nature

•Best described by using Classes

•2 descriptive measures

class frequency – number of data points in a class

class relative = class frequency

frequency total number of data points in data set

class percentage – class relative frequency x 100

Describing Qualitative Data –

Displaying Descriptive Measures

Summary Table

Class

FrequencyClass percentage – class relative frequency x 100


Qualitative Data Displays

Bar Graph



Pie chart



Pareto Diagram

Graphical Methods for Describing

Quantitative Data

The Data

Company Percentage Company Percentage Company Percentage Company Percentage

1 13.5 14 9.5 27 8.2 39 6.5

2 8.4 15 8.1 28 6.9 40 7.5

3 10.5 16 13.5 29 7.2 41 7.1

4 9.0 17 9.9 30 8.2 42 13.2

5 9.2 18 6.9 31 9.6 43 7.7

6 9.7 19 7.5 32 7.2 44 5.9

7 6.6 20 11.1 33 8.8 45 5.2

8 10.6 21 8.2 34 11.3 46 5.6

9 10.1 22 8.0 35 8.5 47 11.7

10 7.1 23 7.7 36 9.4 48 6.0

11 8.0 24 7.4 37 10.5 49 7.8

12 7.9 25 6.5 38 6.9 50 6.5

13 6.8 26 9.5

Percentage of Revenues Spent on Research and Development


Quantitative Data

Dot Plot


Quantitative Data

Stem-and-Leaf Display


Quantitative Data

Histogram


Quantitative Data

More on Histograms

Number of Observations in Data Set Number of Classes

Less than 25 5-6

25-50 7-14

More than 50 15-20

Summation Notation

Used to simplify summation instructions

Each observation in a data set is identified

by a subscript

x1, x2, x3, x4, x5, …. xn

Notation used to sum the above numbers

together is

n

n

i

i xxxxxx

4321

1

Summation Notation

Data set of 1, 2, 3, 4

Are these the same? and

4

1

2

i

ix

24

1

i

ix

30169412

4

2

3

2

2

2

1

2

4

1

xxxxx

i

i

100104321222

24

1

4321

xxxxx

i

i

Numerical Measures of Central

Tendency

•Central Tendency – tendency of data to

center about certain numerical values

•3 commonly used measures of Central

Tendency

Mean

Median

Mode


Tendency

The Mean

•Arithmetic average of the elements of the data set

•Sample mean denoted by

•Population mean denoted by

•Calculated as

and

x

n

x

x

n

i

i

1

n

x

n

i

i

1


Tendency

The Median

•Middle number when observations are

arranged in order

•Median denoted by m

•Identified as the observation if n is

odd, and the mean of the and

observations if n is even

5.02

n

2

n1

2

n


Tendency

The Mode

•The most frequently occurring value in the

data set

•Data set can be multi-modal – have more

than one mode

•Data displayed in a histogram will have a

modal class – the class with the largest

frequency


Tendency

The Data set 1 3 5 6 8 8 9 11 12

Mean

Median is the or 5th observation, 8

Mode is 8

79

63

9

121198865311

n

x

x

n

i

i

5.02

n

Numerical Measures of Variability

•Variability – the spread of the data across

possible values

•3 commonly used measures of Central

Tendency

Range

Variance

Standard Deviation


The Range

•Largest measurement minus the smallest

measurement

•Loses sensitivity when data sets are large

These 2 distributions

have the same range.

How much does the

range tell you about

the data variability?


The Sample Variance (s2)

•The sum of the squared deviations from the

mean divided by (n-1). Expressed as units

squared

•Why square the deviations? The sum of the

deviations from the mean is zero

1

)(

1

2

2

n

xx

s

n

i

i


The Sample Standard Deviation (s)

•The positive square root of the sample

variance

•Expressed in the original units of

measurement

21

2

1

)(

sn

xx

s

n

i

i


Samples and Populations - Notation

Sample Population

Variance s2

Standard

Deviation s

2

Interpreting the Standard Deviation

How many observations fit within + n s of

the mean?

Chebyshev’s

Rule

Empirical

Rule

orNo useful info Approximately

68%

orAt least 75% Approximately

95%

or At least 8/9 Approximately

99.7%

2s2

3s3

1s1

Interpreting the Standard Deviation

You have purchased compact fluorescent light bulbs for your home.

Average life length is 500 hours, standard deviation is 24, and

frequency distribution for the life length is mound shaped. One of your

bulbs burns out at 450 hours. Would you send the bulb back for a

refund?

Interval Range % of observations

included

% of observations

excluded

476 - 524Approximately

68%

Approximately

32%


95%

Approximately

5%


99.7%

Approximately

0.3%

s1

s2

s3

Numerical Measures of Relative

Standing

Descriptive measures of relationship of a

measurement to the rest of the data

Common measures:

• percentile ranking or percentile score

• z-score


Standing

Percentile rankings make use of the pthpercentile

The median is an example of percentiles.

Median is the 50th percentile – 50 % of observations lie above it, and 50% lie below it

For any p, the pth percentile has p% of the measures lying below it, and (100-p)% above it


Standing

z-score – the distance between a

measurement x and the mean, expressed in

standard units

Use of standard units allows comparison

across data sets

xz

s

xxz


Standing

More on z-scores

Z-scores follow the empirical rule for

mounded distributions

Methods for Detecting Outliers

Outlier – an observation that is unusually large or small relative to the data values being described

Causes

• Invalid measurement

• Misclassified measurement

• A rare (chance) event

2 detection methods

• Box Plots

• z-scores


Box Plots

• based on quartiles, values that divide

the dataset into 4 groups

• Lower Quartile QL – 25th percentile

• Middle Quartile - median

• Upper Quartile QU – 75th percentile

• Interquartile Range (IQR) = QU - QL


Box Plots

Not on plot – inner and outer fences, which determine potential outliers

QU

(hinge)

QL

(hinge)

Median

Potential Outlier

Whiskers


Rules of thumb

•Box Plots

–measurements between inner and outer fences are suspect

–measurements beyond outer fences are highly suspect

•Z-scores

–Scores of 3 in mounded distributions (2 in highly skewed distributions) are considered outliers

Graphing Bivariate Relationships

Bivariate relationship – the relationship between

two quantitative variables

Graphically represented with the scattergram

The Time Series Plot

Time Series Data – data produced and monitored

over time

Graphically represented with the time series plot

Time on x axis Order on x axis

Distorting the Truth with Descriptive

Techniques

•Graphical techniques

–Scale manipulation

Same

data,

different

scales


Techniques


–More Scale manipulation


Techniques


–More Scale manipulation


Techniques

•Numerical techniques

–Mismatch of measure of central tendency and

distribution shape

Use of mean overstates average Use of mean understates average


Techniques


–Discussion of central tendency with no information on

variability

Which model would you

purchase if you knew only

the average MPG?

Would knowing the standard

deviation affect your choice?

Why?


Techniques


–Look past the pictures to the data they represent


–Is measure being used most appropriate for underlying

distribution?

–Are you provided with information on central tendency

and variability?

Summary

Graphical methods for Qualitative Data

–Pie chart

–Bar graph

–Pareto diagram

•Graphical methods for Quantitative Data

–Dot plot

–Stem-and-leaf display

–Histogram

Summary

Numerical measures of central tendency

–Mean

–Median

–Mode

•Numerical measures of variation

–Range

–Variance

–Standard Deviation

Summary

Distribution Rules

–Chebyshev’s Rule

–Empirical Rule

•Measures of relative standing

–Percentile scores

–z-scores

•Methods for detecting Outliers

–Box plots

–z-scores

Summary

Method for graphing the relationship

between two quantitative variables

–Scatterplot

Chapter 2 - Wessa.net · PDF filefrequency total number of data ... and 50% lie below it For any p, ... –Dot plot –Stem-and-leaf display –Histogram. Summary

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