1. The document discusses different methods for presenting numerical data, including tables, graphs, and numerical techniques. 2. It provides examples of tabular data arranged in classes with frequencies, class marks, cumulative frequencies, and relative frequencies. 3. The document also describes histograms and bar graphs as ways to graphically display class data, with histograms using rectangle heights and bar graphs using points at the class midpoint and frequency coordinates.

1. Tabular Presentation
Graphical Data
Numerical Techniques describing
Data

3. It is a table which sorts data
according to a certain pattern.
It involves separating a number
into two parts.

4. 48 35 26 18 10
46 33 24 16
44 30 22 14
41 28 22 12

5. STEM LEAVES
4 8 6 4 1
3 5 3 0
2 8 6 4 2 2
1 8 6 4 2 0

6. A table shows the data
arranged into different class and
the number of cases which fall
into each class.

7. 1. CLASSES/SCORES
2. CLASS INTERVAL
# Classes = 𝒏 + 𝟏
C =
𝒓𝒂𝒏𝒈𝒆
# 𝒐𝒇 𝑪𝒍𝒂𝒔𝒔𝒆𝒔
Range = Highest Score – Lowest Score

8. 48 35 26 18 10
46 33 24 16
44 30 22 14
41 28 22 12

9. CLASSE
S
10-17
18-25
26-33
34-41
42-49

10. 3. FREQUENCY – tells us the number of
observations in each class.
4. CLASS MARK/ CLASS MIDPOINT (cm)
cm =
𝒍𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕+𝒖𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕
𝟐

11. CLASSE
S
10-17
18-25
26-33
34-41
42-49
UPPERLIMIT
LOWERLIMIT

12. CLASSES TALLY f cm
10-17 llll 4 13.5
18-25 llll 4 21.5
26-33 llll 4 29.5
34-41 ll 2 37.5
42-49 lll 3 45.5
N = 17

13. 5. CLASS BOUNDARIES
Lower Limit: minus 0.5
Upper limit: add 0.5
6. CUMULATIVE FREQUENCY – tells us the
sum of frequencies in a particular class of
interest.
• Less than Cumulative frequency (<cf)
• Greater then Cumulative frequency (>cf)

14. N = 17
CLASS
ES
f cm cb <cf >cf
10-17 4 13.5 9.5-17.5 4 17
18-25 4 21.5 17.5-25.5 8 13
26-33 4 29.5 25.5-33.5 12 9
34-41 2 37.5 33.5-41.5 14 5
42-49 3 45.5 41.5-49.5 17 3

15. 7. RELATIVE FREQUENCY (rf) –
gives us the percentage of
observations in a particular class of
interest.
rf =
𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏

16. CLASSE
S f cm cb <cf >cf rf
10-17 4 13.5 9.5-17.5 4 17 23.5
18-25 4 21.5
17.5-
25.5
8 13 23.5
26-33 4 29.5
25.5-
33.5
12 9 23.5
34-41 2 37.5
33.5-
41.5
14 5 11.8
42-49 3 45.5
41.5-
49.5
17 3 17.7
N = 17 100%

18. A type of graph where the classes
are labeled along the horizontal line
while the class frequencies are
long the vertical line.

19. Similar to histogram. Only that is
instead of the height of rectangles
depicting the frequencies of the
classes, we will be using points or
coordinates, which is the intersection of
the frequency and class midpoint of a
class.