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DESCRIPTIVE
STATISTICS
Nothing new!! You are already
using it!!
SIMPLE STATISTICS
• Mode
• Median
• Mean
• Standard deviation
• Frequency
Mode
• The score attained by more students than any other scores.
• E.g. 25, 20, 18, 18, 18, 16, 15, 14, 14, 10, 10
• What is the mode in this distribution?
• 18
Median
• The point which separates the higher half of a sample from
the lower half (the midpoint)
• E.g.
• Uneven number of scores: 5, 4, 3, 2, 1
• What is the median?
• 3
• Even number of scores: 10, 8, 6, 4
• What is the median?
• 7
Mean
• It is adding up all the scores and then dividing the sum by the
total number of scores.
• 10, 15, 18, 23, 29= mean is 19 [(10+15+18+23+29)/5)]
Exercise for mode, median, &
mean
Raw
score
Frequency
78
75
70
67
66
62
59
55
53
51
49
40
32
1
2
2
3
2
4
1
1
2
1
4
6
3
____
N=32
• Mode?
• = 40 (6 students get this score)
Exercise for mode, median, &
mean
Raw
score
Frequency
78
75
70
67
66
62
59
55
-------53
51
49
40
32
1
2
2
3
2
Top 16
4
1
1
-------2
1
4
Bottom 16
6
3
____
N=32
• Median?
• = 54 (sum of the middle scores
divided by 2)
• Total of 32 students
• middle score for the top 16 is 55
• middle score for the bottom 16 is
53
• 55+53/2= 54
Exercise for mode, median, &
mean
Raw
score
Frequency
78
75
70
67
66
62
59
55
53
51
49
40
32
1
2
2
3
2
4
1
1
2
1
4
6
3
____
N=32
• Mean?
• = 54.75 ≈ 55
• (total number of scores for all
subjects is 1752 divided by 32 equals
55)
STANDARD DEVIATION
• Averages are useful statistics but not sufficient.
• Eg.
• A: 19, 20, 25, 32, 39 (mean= 25; median=25)
• B: 2, 3, 25, 30, 75 (mean= 25; median=25)
• In both the mean and the median is 25; however, in A the
scores are clustered around the mean and in B they are spread
out.
• Thus, there is a need to describe the spread (variability) within
a distribution.
• There are various ways to measure the spread but the most
useful one is standard deviation.
• The more spread out the scores are, the larger the deviation
is, and the closer the scores are to the mean, the smaller the
deviation.
• Thus, a sd. of 2.7 means there is less variability than a sd. of
8.3.
Why is standard deviation useful?
• E.g. If you are comparing test scores for different schools, the standard
deviation will tell you how diverse the test scores are for each school.
• Imagine School A has a higher test score mean than School B. Your first
reaction might be to say that the kids at School A are smarter.
• But a bigger standard deviation for one school tells you that there are
relatively more kids at that school scoring toward one extreme or the
other. Then, it is possible that lots of gifted students were sent to School
A.
• In this way, looking at the standard deviation can point you in the right
direction when asking why the information is the way it is.
FREQUENCY
• Giving raw scores and percentages.
• Quantitative data can be summarized using frequency tables.
Sample frequency tables
Raw score
Frequency
78
75
70
67
66
62
59
55
53
51
49
40
32
1
2
2
3
2
4
1
5
2
1
4
5
3
____
N=35
Raw scores
(intervals of 5)
Frequency
76-80
71-75
66-70
61-65
56-60
51-55
46-50
41-45
36-40
31-35
1
2
7
4
1
8
4
0
5
3
--N=35
SPSS outcome of frequency data
• 191 people completed the item
• Most thought the course was a little too hard.
SPSS Outcome of frequency data
Sample Frequency Table (APA)
Male
Female
Total
Junior High School Teachers
40
60
100
High School Teachers
60
40
100
TOTAL
100
100
200
More Complex Table (APA)
Table 1. Position, Gender, and Ethnicity of School Leaders
Administrators
Teachers
Total
Experienced
Inexp.
Experienced
Inexp.
Male
50
20
150
80
300
Female
20
10
150
120
300
Total
70
30
300
200
600
Another Sample Frequency Table
Relationship between Self-Esteem and Gender
Self-Esteem
Gender
Low
Middle
High
Male
10
15
5
Female
5
10
15
• It is also possible to present this data in a graph, frequency
polygon, bar chart, pie chart, etc.
76-80
71-75
66-70
61-65
56-60
51-55
46-50
41-45
36-40
31-35
FREQUENCY POLYGON
9
8
7
6
5
4
3
2
1
0
Bar Graph
Pie Chart
18
16
14
2
12
6
10
8
17
17
14
6
11
11
4
6
14
2
2
0
Exercises
.
1. Twenty-five randomly selected students were asked the number of
novels they have read this semester. The results are as follows:
Number of books
0
1
2
3
4
frequency
5
9
6
4
1
N= 25
.
A) Find the mode
Number of books
0
1
2
3
4
1 (9 people read one book)
frequency
5
9
6
4
1
N= 25
.
B) Find the median
Number of books
0
1
2
3
4
1 (mid point is 13; 5+9= 14)
frequency
5
9
6
4
1
N= 25
.
C) Find the mean
Number of books
0
1
2
3
4
frequency
5
9
6
4
1
N= 25
1.48 (9+12+12+4= 42÷25= 1.48)
.
D) What percentage of the students read fewer than 3 books
Number of books
0
1
2
3
4
80% (20x100÷25= 80)
Or
20/25=0.8
frequency
5
9
6
4
1
N= 25
2. Following 3 students are applying to the same school. They came from different schoolswith different
grading systems.
Which student has the best Grade Point Average(GPA) when compared to his school?
St.
GPA
School. Av.
GPA
s.d.
A
2.7
3.2
0.8
B
87
75
20
C
8.6
8
0.4
• C