Transcript EPSY 439 - Texas A&M University

EPSY
439
Central Tendency and Shape
Mean
Median
Mode
Skewness
Kurtosis
Questionnaire Personal Control Scale
For each item, indicate the extent to which
the statement applies to you by using the
following scale. (Paulhus & Van Selst, 1990)
1
Disagree
strongly
2
3
4
5
6
Disagree
Disagree
slightly
Neither
agree or
disagree
Agree
slightly
Agree
7
Agree
strongly
1. I can usually achieve what I want when I work hard for it.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
2
Questionnaire Personal Control Scale
1
Disagree
strongly
2
3
4
5
6
Disagree
Disagree
slightly
Neither
agree or
disagree
Agree
slightly
Agree
7
Agree
strongly
2. Once I make plans I am almost certain to make them
work.
3. I prefer games involving some luck over games of pure
skill.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
3
Questionnaire Personal Control Scale
1
Disagree
strongly
2
3
4
5
6
Disagree
Disagree
slightly
Neither
agree or
disagree
Agree
slightly
Agree
7
Agree
strongly
4. I can learn almost anything if I set my mind to it.
5. My major accomplishments are entirely due to my hard
work and ability.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
4
Questionnaire Personal Control Scale
1
Disagree
strongly
2
3
4
5
6
Disagree
Disagree
slightly
Neither
agree or
disagree
Agree
slightly
Agree
7
Agree
strongly
6. I usually do not set goals because I have a hard time
following through on them.
7. Bad luck has sometimes prevented me from achieving
things.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
5
Questionnaire Personal Control Scale
1
Disagree
strongly
2
3
4
5
6
Disagree
Disagree
slightly
Neither
agree or
disagree
Agree
slightly
Agree
7
Agree
strongly
8. Almost anything is possible for me if I really want it.
9. Most of what will happen in my career is beyond my
control.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
6
Questionnaire Personal Control Scale
1
Disagree
strongly
2
3
4
5
6
Disagree
Disagree
slightly
Neither
agree or
disagree
Agree
slightly
Agree
7
Agree
strongly
10. I find it pointless to keep working on something that is too
difficult for me.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
7
Demonstration Problem
Personal Control Questionnaire
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
8
Scores of 100 College Students Personal Control Scale
61
52
55
53
51
31
66
52
53
49
55
52
50
70
53
52
49
46
64
56
July 20, 2015
52
39
45
47
58
55
55
56
45
45
47
55
66
47
62
46
55
61
49
52
49
57
57
52
54
59
58
47
42
55
42
44
51
42
57
47
48
32
54
39
44
53
54
50
46
69
41
57
61
47
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
50
53
55
56
47
53
52
55
52
56
58
37
48
48
59
46
42
49
37
44
49
46
44
44
41
50
67
52
48
44
9
Frequency Distributions Personal Control Scale
P
e
r
1. Find highest and lowest scores.
m a
u
l
l
i
a
d
t
e
r
u
r
r
c
c
c
e
e
e
e
n
n
n
n
c
V
7
0
a
l
i
1
.
0
0
0
6
9
1
.
0
0
0
6
7
1
.
0
0
0
6
6
2
.
0
0
0
6
4
1
.
0
0
0
6
2
1
.
0
0
0
6
1
3
.
0
0
0
5
9
2
.
0
0
0
5
8
3
.
0
0
0
5
7
4
.
0
0
0
5
6
4
.
0
0
0
5
5
9
.
0
0
0
5
4
3
.
0
0
0
5
3
6
.
0
0
0
5
2
0
.
0
0
0
5
1
2
.
0
0
0
5
0
4
.
0
0
0
4
9
6
.
0
0
0
4
8
4
.
0
0
0
4
7
7
.
0
0
0
4
6
5
.
0
0
0
4
5
3
.
0
0
0
4
4
6
.
0
0
0
4
2
4
.
0
0
0
4
1
2
.
0
0
0
3
9
2
.
0
0
0
3
7
2
.
0
0
0
3
2
1
.
0
0
0
3
1
1
.
0
0
0
T
o
t
a
0
0
0
2. Write numbers in descending
order.
3. For each score, tally the number
of times it appears in the
unorganized group and record
that value in the frequency
column.
4. Include columns for cf, %, and c%
to round out your frequency table.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
10
Frequency Distributions Personal Control Scale
Class Interval
69 - 71
66 - 68
63 - 65
60 - 62
57 - 59
54 - 56
51 - 53
48 - 50
45 - 47
42 - 44
39 - 41
36 - 38
33 - 35
30 - 32
July 20, 2015
Exact Limits
Midpoint
68.5 - 71.5
70
65.5 - 68.5
% is equal
to the f  N. 67
62.5 - 65.5
64
59.5 - 62.5(16  100)61
For example,
X
56.5 - 59.5
58
100 = 53.5
16%- 56.5
55
50.5 - 53.5
52
47.5 - 50.5
49
44.5 - 47.5
46
41.5 - 44.5
43
38.5 - 41.5
40
35.5 - 38.5
37
32.5 - 35.5
34
29.5 - 32.5
31
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
f
2
3
1
4
9
16
18
14
15
10
4
2
0
2
cf
100
98
95
94
90
81
65
47
33
18
8
4
2
2
%
2
3
1
4
9
16
18
14
15
10
4
2
0
2
c%
100
98
95
94
90
81
65
47
33
18
8
4
2
2
11
Histogram Personal Control Scale
20
18
18
16
16
15
14
frequency
14
12
10
10
9
8
6
4
4
Std. Dev = 7.37
4
0
Mean = 51.0
3
2
2
2
2
1
N = 100.00
31.0 37.0 43.0 49.0 55.0 61.0 67.0
34.0 40.0 46.0 52.0 58.0 64.0 70.0
Pers onal Control Scale Score
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
12
Central Tendency
Describing Distributions:
Skewness
Kurtosis
Mean
Median
Mode
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
13
Shape of a Frequency
Distribution - Symmetrical

Technically, the shape of a distribution is
defined by an equation that prescribes
the exact relation between each X and Y
value on the graph.
 A few
less precise terms will serve to
describe the shape of most distributions.
– In a symmetrical distribution, it is possible to
draw a vertical line through the middle so that
one side of the distribution is the exact mirror of
the other.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
14
Shape of a Frequency
Distribution - Skewed (Positive)
– In a skewed distribution, the scores tend to pile
up toward one end of the scale and taper off
gradually at the other end.
– The section where the scores taper off toward
one end of a distribution is called the tail of the
distribution.
– A skewed distribution with the tail to the righthand side is said to be positively skewed
because the tail points toward the positive
(above-zero) end of the X-axis.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
15
Shape of a Frequency
Distribution - Skewed (Negative)
– If the tail points to the left, the distribution is said
to be negatively skewed.
Symmetrical Distribution:
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
16
Shape of a Frequency
Distribution - Examples
Skewed Distributions:
Positive skew
Negative skew
For a very difficult exam, most scores
will tend to be low, with only a few
individuals earning high scores. This
will produce a positively skewed distribution.
July 20, 2015
Similarly, a very easy exam will tend to
produce a negatively skewed distribution,
with most of the students earning high
scores and only a few with low values.
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
17
Measures of Central Tendency:
Mode - Nominal Data
Mode:
Bar Chart
Predominant Religious Affiliation for 109 Countries
50
40
Frequency
The mode is
defined as the most
frequent score in a
distribution and is
determined by
41
30
27
20
16
10
inspecting the data.
8
7
4
0
Muslim
O rt hodox
B uddhist
T aoist
Hin du
Cat holic
P rot est ant
A nim is t
T ribal
Jewish
Religion
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
18
Measures of Central Tendency:
Mode - Grouped Data
Mode: Grouped Frequency Distribution
Interval
100 - 104
95 - 99
90 - 94
85 - 89
80 - 84
75 - 79
70 - 74
65 - 69
Exact Limits
99.5 - 104.5
94.5 - 99.5
89.5 - 94.5
84.5 - 89.5
79.5 - 84.5
74.5 - 79.5
69.5 - 74.5
64.5 - 69.5
Midpoint
102
97
92
87
82
77
72
67
f
1
2
0
7
9
14
15
4
cf
52
51
49
49
42
33
19
4
The mode of a frequency distribution of class intervals is the
midpoint of the class interval with the largest frequency.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
19
Measures of Central Tendency:
Median - Simple Frequency
Histogram
Median:
20
50 % of the cases
50 % of the cases
X
Frequency
The median is
defined as the 50th
percentile; that is,
the point on the
scale of
measurement
below which 50
percent of the
scores fall.
Literacy Rates for Males Across 109 Countries
median = 87
10
S td. Dev = 20. 45
Mean = 78. 7
N = 85. 00
0
30. 0
40. 0 50. 0 60. 0
70. 0 80. 0 90. 0 100. 0
35. 0
45. 0 55. 0 65. 0
75. 0 85. 0 95. 0
Males Who Read (%)
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
20
Measures of Central Tendency:
Median - Grouped Frequencies
Median: Grouped Frequency Distribution
Interval
100 - 104
95 - 99
90 - 94
85 - 89
80 - 84
75 - 79
70 - 74
65 - 69
Exact Limits
99.5 - 104.5
94.5 - 99.5
89.5 - 94.5
84.5 - 89.5
79.5 - 84.5
74.5 - 79.5
69.5 - 74.5
64.5 - 69.5
Midpoint
102
97
92
87
82
77
72
67
f
1
2
0
7
9
14
15
4
cf
52
51
49
49
42
33
19
4
%
2
4
0
13
17
27
29
8
c%
100
98
95
95
81
64
37
8
To find the median of a grouped frequency distribution first locate the
interval containing the score that separates the distribution into halves.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
21
Measures of Central Tendency:
Median - Grouped Frequencies
Median: Grouped Frequency Distribution
Interval
75 - 79
70 - 74
65 - 69
Exact Limits
74.5 - 79.5
69.5 - 74.5
64.5 - 69.5
Midpoint
77
72
67
f
14
15
4
cf
33
19
4
%
27
29
8
c%
64
37
8
Step 2 - Identify the upper and lower limits of the interval.
Step 3 - Identify the c% values associated with upper and lower limits of
the interval.
79.5
64
Step 4 - Determine the corresponding interval
widths.
5
?
50
27
Step 5 - Indicate percentile that you want to
74.5
37
translate into a percentile rank.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
22
Measures of Central Tendency:
Median - Group Frequencies
Median: Grouped Frequency Distribution
5
79.5
64
76.9
?
50
74.5
37
.48 * 5  2.4
74.5  2.4  76.9
27
Step 6 - Compute the proportion of distance
that 50 is into the interval.
50  37 13

.48
27
27
Step 7 - Multiply the proportion computed in
Step 6 by interval width and add
that value to the lower lower limit of
the interval.
The percentile rank for the 50th percentile (median) in this grouped
frequency distribution is the score 76.9
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
23
Measures of Central Tendency:
Median - SFD (Example)
Median: Simple Frequency Distribution
Here’s a simple example:
Suppose a college freshman arrives at school in
the fall with a promise of a monthly allowance for
spending money. Sure enough, on the first of each
month, there is money to spend. However, three
months into the school term, our student discovers
a recurring problem: There is too much month left
at the end of the money.
July 20, 2015Bob
2015
Hall
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
24
Measures of Central Tendency:
Median - SFD (Example)
Median: Simple Frequency Distribution
Here’s a simple example:
In pondering his problem, it occurs to our student
that lots of money escapes from his pocket at the
Student Center. So, for a two-week period, he
keeps a careful accounting of every cent he spends
at the center on soft drinks, snacks, video games,
coffee, and so forth. His data are presented in the
following table.
July 20, 2015Bob
2015
Hall
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
25
Measures of Central Tendency:
Median - SFD (Example)
Median: Simple Frequency Distribution
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Amount Spent
$2.56
$0.47
$1.25
$0.00
$3.25
$1.15
$0.00
$0.00
$6.78
$2.12
$0.00
$0.00
$3.78
$3.62
July 20, 2015Bob
2015
Hall
Arrange the data in
a descending order
Median = $1.20
halfway between
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
Amount Spent
$6.78
$3.78
$3.62
$3.25
$2.56
$2.12
$1.25
$1.15
$0.47
$0.00
$0.00
$0.00
$0.00
$0.00
7 scores
7 scores
26
Measures of Central Tendency:
Mean - Definition
Mean:
e
The mean is
defined as the
7
7
2 arithmetic average
8 of the scores in the
0
distribution; that is
2
the sum (S) of all
4
the scores divided
8
by the number (n)
0
of scores.
l
l
i
e
d
d
w
July 20, 2015
N
X
R
M

n
M
8382
S

M
107
S
78.34
D
V
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
27
Measures of Central Tendency:
Mean - Properties
Properties of the Mean:
The sum of deviations of all scores from
the mean is zero.
 The sum of squares of the deviations
from the mean is smaller than the sum
of squares of the deviations from any
other value in the distribution.

July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
28
Measures of Central Tendency:
Mean - Sum of Deviations
Properties of the Mean:
1. The sum of
deviations of
all scores from
the mean is
zero.
If X i is a given score, xi is the deviation of
that score from the mean; that is,

xi  X i  X

Stated symbolically, the first property of the
mean is
  X i  X     xi   0
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
29
Measures of Central Tendency:
Mean - Balance Point
Properties of the Mean:
The analogy between the mean and the
2. If scores are
balance point of a seesaw illustrates another
ordered along
important characteristic of the mean: It is very
a scale of
sensitive to extreme scores at one or the other
values, the
end of the range.
mean will fall
-9 = 2(-4)
-1
+2 +3 +4= +9
directly at the
“balance point”
or “center of
gravity” of the
5
2
3 4
7
8
9
10
distribution.
X 6
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
30
Measures of Central Tendency:
Mean - Problem With Outliers
Comparison of means
of two distributions,
one of which contains
an extreme score
July 20, 2015
X
X*
10
38
9
9
8
8
6
6
5
5
2
2
2
2
S X = 42 S X * = 70
X = 6 X * = 10
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
31
Measures of Central Tendency:
Averaging Means - Problem
Suppose a teaching assistant gives his
instructor the mean final exam scores for
each of her two statistics classes. Suppose
further that the instructor wishes to obtain a
measure of the average performance of all
her students, regardless of class. She could,
of course, obtain the final examination
scores for students in both classes and
calculate a mean over all these scores.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
32
Measures of Central Tendency:
Averaging Means - Procedure
Suppose, however, that she is looking for a
simpler way or that she doesn’t have all the
scores handy. Could she use the mean of
the two means to obtain a measure of the
average performance of her students?
• This procedure is only appropriate when
the means are based on samples of
equal size.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
33
Measures of Central Tendency:
Averaging Means - Example
Suppose, for instance, that the mean final
examination score was 77 in class 1 and 83
in class 2, and that each class had the same
number of students (say, 25). In this case,
the mean for all students in both classes is
simply the mean of the two separate class
means:
77  83
 80
2
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
34
Measures of Central Tendency:
Averaging Means - Weighted
Suppose, however, that this instructor’s
classes were unequal in size. In this more
common case, a weighted mean must be
computed -- that is, means based on a larger
sample are weighted more heavily than
those based on a smaller number of cases.
With 30 students and 20 students and
means of 77 and 83 the mean for all
students in both classes can be obtained by:
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
35
Measures of Central Tendency:
Averaging Means - Calculation
•
Calculating the sum of all scores for class 1 and for
class 2.
•
Adding the two sums and dividing by the total number
of students.
X
X
Class 1
 NX1  30  77  2310
Class 2
 NX 2  20  83  1660
2310  1660 3970

 79.4
30  20
50
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
36
Relationship Among Measures
of Central Tendency
• There is no one best measure of the
central tendency of a distribution.
• The preferred measure depends on the
shape of the distribution and on what you
are trying to communicate about the
distribution.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
37
Relationship Among Measures
of Central Tendency
• With symmetric or nearly symmetric
distributions, often the mean or median is
preferred to the mode since the mode is
the least stable measure of central
tendency.
• Most often in practice, the mean is used,
because the researcher wants to draw
inferences from the sample to the
population.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
38
Relationship Among Measures
of Central Tendency
• The mean is used because it is more
mathematically tractable than the median.
• In some cases, however, the median or
mode may be preferred to the mean as a
measure of central tendency.
July 20, 2015
Copyright
Copyright
©©
2000
2000
Robert
Robert
J. J.
Hall
Hall
39