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
Describing Qualitative Data –
Qualitative Data Displays
Bar Graph
Describing Qualitative Data –
Qualitative Data Displays
Pie chart
Describing Qualitative Data –
Qualitative Data Displays
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
Graphical Methods for Describing
Quantitative Data
Dot Plot
Graphical Methods for Describing
Quantitative Data
Stem-and-Leaf Display
Graphical Methods for Describing
Quantitative Data
Histogram
Graphical Methods for Describing
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
Numerical Measures of Central
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
Numerical Measures of Central
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
Numerical Measures of Central
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
Numerical Measures of Central
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
Numerical Measures of Variability
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?
Numerical Measures of 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
Numerical Measures of Variability
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
Numerical Measures of Variability
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%
452 - 548Approximately
95%
Approximately
5%
428 - 572Approximately
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
Numerical Measures of Relative
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
Numerical Measures of Relative
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
Numerical Measures of Relative
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
Methods for Detecting Outliers
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
Methods for Detecting Outliers
Box Plots
Not on plot – inner and outer fences, which determine potential outliers
QU
(hinge)
QL
(hinge)
Median
Potential Outlier
Whiskers
Methods for Detecting Outliers
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
Distorting the Truth with Descriptive
Techniques
•Graphical techniques
–More Scale manipulation
Distorting the Truth with Descriptive
Techniques
•Graphical techniques
–More Scale manipulation
Distorting the Truth with Descriptive
Techniques
•Numerical techniques
–Mismatch of measure of central tendency and
distribution shape
Use of mean overstates average Use of mean understates average
Distorting the Truth with Descriptive
Techniques
•Numerical 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?
Distorting the Truth with Descriptive
Techniques
•Graphical techniques
–Look past the pictures to the data they represent
•Numerical techniques
–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