:)

1. The More We Get Together
The more we get together,
Together, together,
The more we get together,
The happier we'll be.
For your friends are my friends,
And my friends are your
friends.
The more we get together,
The happier we'll be!

2. Graphical
Representation
&
Shapes of
Distribution

3. A graph adds life and beauty to
one’s work, but more than this, it
helps facilitate comparison and
interpretation without going
through the numerical data.

4. Kinds of Graphs
Bar Chart
Histogram
Frequency Polygon
Pie Chart
Ogive

5. Bar Chart of the Grouped Frequency Distribution
for the Entrance Examination Scores of 60
18 17Students
16 14
14
12 11
frequency
10 8
8 6
6
4 3
2 1
0
0
18-23 24-29 30-35 36-41 42-47 48-53 54-59 60- 65
class interval
A bar chart is a graph represented by
either vertical or horizontal rectangles
whose bases represent the class intervals
and whose heights represent the
frequencies.

6. The Histogram of the Grouped Frequency
Distribution for the Entrance Examination Scores
18
of 60 Students
16
14
Frequency
12
10
8
6
4
2
0
20.5 26.5 32.5 38.5 44.5 50.5 56.5
class mark
A histogram is a graph represented by
vertical or horizontal rectangles whose
bases are the class marks and whose
heights are the frequencies.

7. The Frequency Polygon of the Grouped Frequency
Distribution for the Entrance Examination Scores of 60
Students
18
16
14
Frequency
12
10
8
6
4
2
0
14.5 20.5 26.5 32.5 38.5 44.5 50.5 56.5 62.5
class mark
A frequency polygon is a line graph whose
bases are the class marks and whose heights are
the frequencies.

8. The Pie Chart of the Grouped Frequency
Distribution for the Entrance Examination
Scores of 60 Students
5.00% 1.67%
10.00%
13.33%
18.33%
23.33%
28.33%
A pie chart is a circle graph showing the
proportion of each class through either the
relative or percentage frequency.

9. A pie chart is drawn by dividing the circle according to
the number of classes. The size of each piece depends
on the relative or percentage frequency distribution.
How to compute for the
Relative Frequency?

10. The relative frequency of each class is obtained by
dividing the class frequency by the total frequency.
Relative Frequency Distribution for the Entrance
Examination Scores of 60 Students
Class Midpoint Frequency Relative
Interval (X) (f) Frequency
(ci) (rf)
18 - 23 20.5 6 0.1000
24 - 29 26.5 11 0.1833
30 - 35 32.5 17 0.2833
36 - 41 38.5 14 0.2333
42 - 47 44.5 8 0.1333
48 - 53 50.5 3 0.0500
54 - 59 56.5 1 0.0167
N = 60

11. The Less than and Greater than Ogives for the
Entrance Examination Scores of 60 Students
70
C F 60
u r 50 Less than ogive
m e
u q
40
l u 30
Greater than ogive
a e 20
t n
i c
10
v y 0
e 17.5 23.5 29.5 35.5 41.5 47.5 53.5 59.5
Class Boundaries
An ogive is a line graph where the bases
are the class boundaries and the heights
are the <cf for the less than ogive and
>cf for the greater than ogive.

12. Shapes of
Distribution
 Symmetrical
Asymmetrical

13. SYMMETRICAL
DISTRIBUTION
Normal Distribution
Each half or side of the
distribution is a mirror
image of the other side
(bell-shaped appearance)
Mean ,median ,and mode
coincides
(mean = median = mode)
Skewness is equal to
zero

15. ASYMMETRICAL
DISTRIBUTION
Negatively Skewed/Skewed
to the Left
In a negative skew the
tail extends far into the
negative side of the
Cartesian graph
mean < median
Skewness is less than 0.
the mass of the distribution
is concentrated on the right of
the figure

16. ASYMMETRICAL
DISTRIBUTION
Positively Skewed/Skewed to
the Right
In a positive skew the tail
on the right side of the
distribution exdends far
into the positive side of the
Cartesian graph.
mean > median
Skewness is greater than 0.
the mass of the distribution is
concentrated on the left of the
figure

17. Skewness refers to the degree of symmetry
or asymmetry of a distribution.
The extent of skewness can be obtained by
getting the coefficient of skewness using the
formula:
SK = 3(Mean – Median)
Standard deviation

18. Let us summarize the measurements from the 3 types of
distribution:
Normal Skewed to Skewed to
the left/ the right/
Negatively Positively
skewed skewed
Mean 4.00 5.58 2.40
Median 4.00 6.00 2.00
Mode 4.00 6.00 2.00
Standard 1.53 1.07 1.07
deviation

19. Using the formula to find the coefficient
of skewness, we have:
For normal
For skewed to the left
distribution: distribution: For skewed to the right
distribution:
SK= 3(Mean – Median)
SK= 3(Mean – Median)
Standard deviation Standard deviation SK= 3(Mean – Median)
= 3(5.6 – 6.0) Standard deviation
= 3(4.0 – 4.0) 1.07 = 3(2.4 – 2.0)
= - 1.12 1.07
1.53 = 1.12
=0
Notice that if
•SK = 0, distribution is normal
•SK < 0, distribution is skewed to the left
•SK > 0, distribution is skewed to the right

20. Exercise
Find the coefficient of skewness and indicate if the
distribution is normal, skewed to the left or skewed to the
right.
72, 81, 67, 83, 61, 75, 78, 82, 71, 67
Solution:
Find the mean : Mean = 73.7
Find the median: Median = 73.5
Find the SD: SD = 7.38
Find the SK: SK = 3(Mean – Median)/Standard deviation
= 3(73.7 – 73.5)/ 7.38
= 0.08
Interpretation: Since SK is positive, then it is skewed to the
right. But the value is too small, so we can say that the
distribution is almost normal.

21. FIN
Reporters:
Ando, Lilian
Dillo, Charlyn
Lapos, Emilia