Learning Objectives
In this chapter you will learn about
measures of central tendency
 levels of measurement
 measures of shape

Uses of Statistics
Statistics


provide information by organizing and
summarizing data
describe the nature of a sample
Description of a data involves

measures that best characterize a
frequency distribution
Measures of Central
Tendency
Descriptive statistics

measures that best characterize a
frequency distribution
the scores that are most “typical”
these measures describe scores that
group around a central value

Frequency Distribution
The next slide shows the number of
prisoners executed
 Items are listed in order from the
highest to the lowest value
 The symbol x stands for the value of
the variable


x = the number of inmates executed
f is the number of cases that assume a
certain value. f is the number of
states that have executed a number
of inmates.
Here, we see that one state has
executed 104 inmates. This is by far
the highest number.
fX is the sum of cases. A total of 303
offenders were executed in the U.S.
Table 3.1: Number of Inmates Executed Between 1977 and 1995
X
f
fX
104
1
104
36
1
36
29
1
29
22
1
22
20
1
20
17
1
17
11
1
11
8
1
8
7
1
7
6
1
6
5
3
15
4
3
12
3
1
3
2
3
6
1
5
5
0
25
0
Central Tendency
In a distribution, where do most of the
cases “cluster?”
Three measures of central tendency



mode
median
mean
The Mode
The mode



is the score that occurs most frequently
in a distribution
In our table, zero (0) is the mode
Twenty-five states (25 under the f
column) did not execute a single
convicted offender between 1977 and
1995
The Mode
Note - the mode IS NOT 104!



The mode is the more frequently
occurring category, in this case zero
The mode is ALSO NOT 25!
A frequency distribution may have more
than one mode
The Mode
If another value (number of states)
had a frequency of 25 in the table, it
would also have been the mode


frequency distribution with two modes
is termed bimodal
more than two modes, it is called
multimodal
Properties of the Mode
The mode




does not necessarily occur in or near
the center of a distribution
can occur anywhere in a distribution
does not indicate the variability
between scores in a distribution
simply indicates the value(s) that occur
most frequently
The Median
In a frequency distribution


scores are placed in order from lowest to
highest
the median is the middle of the
distribution.
It is the 50th percentile
 50% of the scores in the frequency
distribution fall below and above the median

Properties of the Median
Attributes of the median

stability



the median is unaffected by extreme
scores
it is calculated by counting the number
of cases
it does not consider the value of the
case
Calculating the Median
The median can be calculated easily
and determined by inspection


In the table, N (the number of cases) =
50 - the number of state
determine where the middle case lies

one half of 50 is 25
The Median
Example:



The first number in the distribution,
zero, has a frequency of 25
Therefore, the median is zero
Half of the states executed no one
during the time period, 1977 to 1995
The Mean
The mean is
the average score in a distribution
 calculated by adding all the scores
in a distribution and dividing the
total by the number of cases

Calculating the Mean
Example:




a total of 303 inmates (fx = 303) were
executed between 1977 and 1995
there were 50 states (N = 50)
The mean is 6.06 (303/50)
An average of six inmates were
executed by each jurisdiction during the
period 1977 – 1995
Characteristics of the Mean
The mean is


unlike the mode and median
the mean is sensitive to extreme scores
Example:


Texas executed 104 inmates between
1977 and 1995
The next closest jurisdiction executed
36 inmates
104 executions (Texas) were an
extreme score in this distribution. The
median for this distribution was zero.
Half of the jurisdictions executed one
or no inmates during this time period.
Yet, our mean was 6.06 – over six
points above the median. The Texas
executions drove up the average
number for the time period.
This attribute of the mean occurs
because it is computed by using the
value of each score in the
distribution.
The mode and median fail to use the
value of each score in a distribution.
The mode is derived from the
frequency of the scores. The median
is based on the position of the
The mean is amenable to statistical
analysis and comparisons between
distributions while the mode and
median are not.
Also, the sum of the deviations from
the mean (how far each score stands
in relation to the mean) is zero.
Symmetric Distribution
zero skewness
mode = median = mean
Positively Skewed
Distribution
Positively
skewed:
Mean and
Median are
to the right of
the Mode
Negatively Skewed
Distribution
Negatively
Skewed:
Mean and
Median are
to the left of
the Mode
Levels of Measurement
Numbers are used to measure
concepts
like fear of crime
support for the police or capital
punishment.

The numbers are used as a code
Question?
Statistically, the question is


can we use mathematics to now
analyze this code that we have
established?
does it make sense to treat the
numbers as such and perform
arithmetic operations on them?
This code is called the level of
measurement. It involves converting
the concepts to numerical data.
There are four categories and each
have different attributes. However,
the levels of measurement are
cumulative, kind of like the steps on a
ladder. You have to step on the first
step to reach the second, and so on.
Each succeeding level
automatically possesses the
attributes of the level preceding it,
plus another distinct one.
Levels of Measurement
Nominal level:involves the process
of classifying data into categories.
When we classify respondents by
race or sex, we are using nominal
measurement (i.e., 1 – Male, 2 –
Female).
Nominal level measurement follows
three basic rules:
1.
The list of categories must be
exhaustive and cover all the types
of observations made.
2. The categories must be mutually
exclusive. Each observation
can only
be classified in one
way.
3. No ordering (>) is present in the
list of
categories. The order is
arbitrary and no one classification is
It does not make sense to discuss the
mean (average) or median (midpoint)
with nominal data.
It cannot be summed and divided, nor
can it be ranked in order from highest
to lowest.
Example: Table 3.2 from NCVS.
Table 3.2: Sex of Respondents in the National Crime Survey
Sex of Respondents
Frequency
Percentage
Male
524
52.1
Female
481
47.9
TOTALS
1005
100
Here we see that the majority of the
respondents are Male (52.1%).
Although everyone knows that women
are smarter, we cannot say that the
mutually exclusive categories of sex
are in rank order, or can we say that
one sex is “average.”
LEVELS OF
MEASUREMENT
Ordinal level: Exists when we can also
detect degrees of difference between
the categories on the scale. The
values of the variable indicate order
or ranking. EXAMPLE:“Do you favor
or oppose the death penalty for
persons convicted of murder?”
Choices:(1) Favor (2) Oppose (3)
Neither (4) Don’t Know.
Transitivity
Ordinal level measurement requires
transitivity.
If A is > B and B is > C, A must be
greater than C or ordinal level
measurement is not present.
“Favor” is > “Oppose,” “Oppose” is >
“Neither,” “Neither is > “Don’t Know,”
and “Favor” is > “Don’t Know.”
“Do you favor the death penalty for
persons convicted of murder?”
Favor
727
72.6%
Oppose
177
17.6%
Neither
79
7.9%
Don’t
Know
19
1.9%
LEVELS OF
MEASUREMENT
Interval level: assumes that the
difference between each item on the
scale have equal units (or intervals)
of measurement between them. It
also assumes that this unit has a
common recognized meaning.
LEVELS OF
MEASUREMENT
Ratio level: Data possessing a natural
zero point and organized into
measures for which differences are
meaningful.
Examples:A year is a common,
constant unit of measurement. Before
birth, a person is considered to have
zero years of age. with ratio level
measurement.
For example, analysis of the age of
the respondents to the National
Crime Survey revealed that the
mean was 45.6 years.
The median or midpoint was 42.
The mode was also 42.
We can also compare groups of
respondents according to their age.
The fourteen survey respondents were
44 (lets call them “Group A”) and
twenty one respondents were 22
(Group B).
We draw the following conclusions
about Respondents from Groups A
and B:
They have different ages (Nominal
Measurement).
Members of Group A are 22 years
older than members of Group B
(Ordinal and Interval Measurement).
Members of Group A are twice as old
as members of Group B (Ratio
Measurement).