Organizing Data
 Data when collected in original form is called “raw
data”.
 For example
Frequency Distribution
 We organize the raw data by using a frequency
distribution.
 The frequency is the number of values in a
specific class of data.
 A frequency distribution is the organizing of raw
data in table form, using classes and frequencies.
Frequency Distribution Cont.
 For the raw data set, a frequency distribution is
shown as follow:
Class limits
1-3
4-6
/////
/////
7-9
10-12
13-15
16-18
/////
/////
/////
/////
Tally
/////
/////
/////
/
////
Frequency
10
14
10
6
5
5
 We use a frequency distribution to construct a
histogram.
Histograms
Histograms are bar graphs in which
 The bars have the same width and always
touch (the edges of the bars are on class
boundaries which are described below).
 The width of a bar represents a quantity.
 The height of each bar indicates frequency.
To construct a histogram from
raw data:
 Decide on the number of classes (5 to 15 is
customary).
 Find a convenient class width.
 Organize the data into a frequency table.
 Find the class midpoints and the class boundaries.
 Sketch the histogram.
To find Class Width
 First compute:
Largest value - smallest value
Desired number of classes
 Increase the value computed to the next highest
whole number.
 The lower class limit of a class is the lowest data
that can fit into the class, the upper class limit is
the highest data value that can fit into the class.
The class width is the difference between lower
class limits of adjacent classes.
Class Width
Raw Data:
10.2 18.7
6.3
17.8
2.4
7.9
8.5 12.5
0.4 5.2
19.5 22.5
11.4
22.3
17.1
0.3
21.4
4.1
0.0
20.0
5.0
2.5
16.5
14.3
24.7
Use 5 classes.
24.7 – 0.0
5
= 4.94
Round class width up
to 5.
Frequency Table
 Determine class width.
 Create the classes. May use smallest data value
as lower limit of first class and add width to get
lower limit of next class.
 Tally data into classes.
 Compute midpoints for each class.
 Determine class boundaries.
Tallying the Data
tally
frequency
0.0 - 4.9
|||| |
6
5.0 - 9.9
||||
5
10.0 - 14.9
||||
4
15.0 - 19.9
||||
5
20.0 - 24.9
||||
5
Computing Class Width
 difference between the lower class
limit of one class and the lower class
limit of the next class
Finding Class Widths
f
class widths
0.0 - 4.9
6
5
5.0 - 9.9
6
5
10.0 - 14.9
4
5
15.0 - 19.9
5
5
20.0 - 24.9
5
5
Computing Class Midpoints
 lower class limit + upper class limit
2
Finding Class Midpoints
f
class midpoints
0.0 - 4.9
6
2.45
5.0 - 9.9
5
7.45
10.0 - 14.9
4
12.45
15.0 - 19.9
5
17.45
20.0 - 24.9
5
22.45
Class Boundaries
 Upper limit of one class + lower limit of next class
2
 Class boundaries cannot belong to any class.
 Class boundaries between adjacent classes are
the midpoint between the upper limit of the
first class, and the lower limit of the higher
class.
 Differences between upper and lower
boundaries of a given class is the class width.
Finding Class Boundaries
f
class boundaries
0.0 - 4.9
6
0.05 - 4.95
5.0 - 9.9
5
4.95 - 9.95
10.0 - 14.9
4
9.95 - 14.95
15.0 - 19.9
5
14.95 - 19.95
20.0 - 24.9
5
19.95 - 24.95
Constructing the Histogram
f
f
0.0 - 4.9
5.0 - 9.9
10.0 - 14.9
6
5
4
15.0 - 19.9
5
20.0 - 24.9
5
6
-
5
-
4
-
3
-
2
-
1
-
0
-
|
|
|
|
|
|
-0.05 4.95 9.95 14.95 19.95 24.95
Relative Frequencies
The relative frequency of a class is f/n where f
is the frequency of the class, and n is the
total of all frequencies.
Relative frequency tables are like frequency
tables except the relative frequency is given.
Relative frequency histograms are like
frequency histograms except the height of
the bars represent relative frequencies.
Relative Frequency
Histogram
f
relative
frequency
0.0 - 4.9
6
0.24
5.0 - 9.9
5
0.20
10.0 - 14.9
4
0.16
15.0 - 19.9
5
0.20
20.0 - 24.9
5
0.20
.24 .20 .16 .12 .08 .04 0
-
|
|
|
|
|
|
-0.05 4.95 9.95 14.95 19.95 24.95
Common Shapes of Histograms
Symmetrical
f
When folded vertically, both
sides are (more or less) the
same.
Common Shapes of Histograms
Also Symmetrical
f
Common Shapes of Histograms
Uniform
f
Common Shapes of Histograms
Skewed Histograms
Skewed left
Skewed right
Common Shapes of Histograms
Bimodal
f
The two largest rectangles are approximately equal
in height and are separated by at least one class.
Frequency Polygon
 A frequency polygon or
line graph emphasizes
the continuous rise or
fall of the frequencies.
 Dots are placed over
the midpoints of each
class.
 Dots are joined by line
segments.
 Zero frequency classes
are included at each
end.
Constructing the Frequency
Polygon
f
f
2-4
6
6
-
5-7
5
5
-
4
-
3
-
2
-
1
-
0
-
8 - 10
4
11 - 13
5
|
0
|
|
|
|
|
3
6
9
12 15
Cumulative Frequencies &
Ogives
 The cumulative frequency of a class is
the frequency of the class plus the
frequencies for all previous classes.
 An ogive is a cumulative frequency
polygon.
Cumulative Frequency
Table
f
f
2-4
6
5-7
5
8 - 10
11 - 13
Greater than 1.5
20
Greater than
14
4.5
Greater than 7.5
9
4
Greater than 10.5
5
5
20
Greater than 13.5
0
Constructing the Ogive
f
20
Greater than 4.5
14
Greater than 7.5
9
Greater than 10.5
5
Greater than 13.5
0
20 -
Cumulative frequency
Greater than 1.5
15 10 5 0 -
|
|
1.5 4.5
pounds
|
|
7.5
|
|
10.5 13.5