S
O
C
S
Shape
Outliers
Center
Spread


We can use a graphs to look at the shape of the
quantitative variable distribution.
An example of a bell-shaped or normal distribution
which appear often in nature:
Symmetric
Mean, median,
mode roughly
equal
Scores from an easy
exam, skewed left.
Non-symmetric
Mean < Median < Mode
Scores from a hard
exam, skewed right.
Non-symmetric
Mean > Median > Mode
 Shape
described by number of peaks (mode)
 An
outlier is an extreme value of the data
(extremely high or extremely low). It is
an observation value that is significantly
different from the rest of the data. There
may be more than one outlier in a set of
data.
Possible Reasons for Outliers:
1. An error was made while taking the
measurement or entering it into the computer.
2. The individual belongs to a different group than
the bulk of individuals measured.
3. The outlier is a legitimate, though extreme data
value.
 We
can identify an outlier if it is
• Less Q1 – 1.5×IQR or
• Greater than Q3 + 1.5×IQR
 Make
a box and whisker plot of the data
and identify any outliers.
• 10, 12, 11, 15, 11, 14, 13, 17, 12, 22, 14, 11


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Australia: $2.20
Canada: $2.02
Germany: $4.58
Mexico: $2.09
United States: $1.59
Japan: $3.47
Taiwan: $2.16
a. Make a box and whisker
plot for the gasoline prices.
a. Which countries, if any, had
gasoline prices that can be
considered outliers?
 Measures
of center: mean, median, mode
Mean
Average of the
data set
Median
The middle
value of a data
set arranged
from smallest to
largest
Mode
The data value
that occurs the
most often, is a
common
measure of
center for
categorical data
 When
describing data, you must decide
which number is the most appropriate
description of the center.

Mean Median applet:
http://bcs.whfreeman.com/tps3e/content/cat_020/applets/mea
nmedian.html
 Use
the mean on symmetric data and the
median on skewed data or data with outliers
Range
Max value
subtract
minimum value
(spread of all
data)
Interquartile
Range
Mean Absolute
Deviation
Variance and
Standard
Deviation
Interquartile range
(IQR) : shows middle
50% of data
Average distance
between each
data value and
the mean
a measure of the
“average”
deviation of all
observations
from the mean.
IQR = Q3 – Q1
Not affected as much
by outliers
Use when measure of
center is median
Use when
measure of
center is mean
 Complete
a 5 number summary and box
and whisker plot for the following data.
• Number of hours spent on internet per week:
12, 4, 16, 18, 1, 6, 10, 8
To calculate Mean Absolute Value
Deviation:
Calculate the mean for the data set.
Find the distance between each data value
and the mean. That is, find the absolute value
of the difference between each data value
and the mean.
Find the average of those distances.
 Find
the mean absolute value of the
following data set:
• 52, 48, 60, 55, 59, 54, 58, 62
A measure of spread is the Standard
Deviation: a measure of the “average”
deviation of all observations from the
mean.
The symbol for
Standard
Deviation is σ
(the Greek letter
sigma).
To calculate Standard Deviation:
Calculate the mean for the data set.
Determine each observation’s deviation: subtract the mean
from each data point. (𝑥 − 𝑥).
Square each deviation.
“Average” the squared-deviations by totaling the squareddeviation and dividing the total squared deviation by (n-1).
This quantity is the Variance.
Square root the result to determine the Standard Deviation.
 Calculate
the standard deviation of the
following test scores:
• 15, 20, 21, 20, 36, 15, 25, 15
The
shape of the data’s distribution!
 If
data are symmetric, with no serious
outliers, use range and standard deviation.
 If
data are skewed, and/or have serious
outliers, use IQR.
 Quantitative
Data: through graphs
 Categorical Data: through two way
frequency tables
Multiple bar graphs
Multiple box and
whisker plots
 These
tables examine the relationships
between the two categorical variables.
 A two-way frequency table will deal with
two variables
Relative frequency is the ratio of the value of a
subtotal to the value of the total.
Create a two-way frequency table for
the following problem.