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S TAT I S T I C S
CHAPTER 6
is the science of the collection, organization, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.
STATISTICS
Class intervals
a set of numerical data that is grouped into several classes and the range of each class is known as class interval
There should be between 5 and 20 classes. The class width should be an odd number. This will
guarantee that the class midpoints are integers instead of decimals.
The classes must be mutually exclusive. This means that no data value can fall into two different classes
The classes must be all inclusive. This means that all data values must be included.
The classes must be continuous. There are no gaps in a frequency distribution. Classes that have no values in them must be included (unless it's the first or last class which are dropped).
The classes must be equal in width.
Class intervals
Find the largest and smallest values Compute the Range = Maximum - Minimum Select the number of classes desired. This is usually
between 5 and 20. Find the class width by dividing the range by the number
of classes and rounding up. Pick a suitable starting point less than or equal to the
minimum value.
STATISTICS
The data below shows the marks obtained by 40 students in a monthly test.
99 88 75 92 58 75 80 70
64 42 70 58 90 68 50 78
43 89 45 93 61 81 58 65
69 76 88 58 91 67 71 52
55 40 80 80 78 46 61 69
STATISTICS
The lowest value : 40
The highest value : 99
Difference : 99-40 = 59
Width of class : 5
Number of class intervals =
12
5
59
5
59
The lowest value : 40
The highest value : 99
Difference : 99-40 = 59
Width of class : 10
Number of class intervals =
610
59
Class intervals
40-49
50-59
60-69
70-79
80-89
90-99
Class Limit Lower limit : the lowest value of the class interval Upper limit: the highest value of the class interval
Class Limit
Class intervals Lower limit Upper limit
40-49 40 49
50-59 50 59
60-69 60 69
70-79 70 79
80-89 80 89
90-99 90 99
Class Boundary
-lower boundary is the midpoint between the lower limit of the class interval and the upper limit of the previous class interval
- Upper boundary is the midpoint between the upper limit of the class interval and the lower limit of the succeeding class interval
Class boundary
40 - 49 50 – 59 60 – 69
Lower boundary
5.49
)5049(2
1
5.59
)6059(2
1
Upper boundary
Class Boundary
Class intervals
Lower limit Upper limit Lower Boundary
Upper boundary
40-49 40 49 39.5 49.5
50-59 50 59 49.5 59.5
60-69 60 69 59.5 69.5
70-79 70 79 69.5 79.5
80-89 80 89 79.5 89.5
90-99 90 99 89.5 99.5
Class intervals
Lower limit
Upper limit
Lower Boundary
Upper boundary
Midpoint Frequency
40-49 40 49 39.5 49.5 44.5 5
50-59 50 59 49.5 59.5 54.5 7
60-69 60 69 59.5 69.5 64.5 8
70-79 70 79 69.5 79.5 74.5 9
80-89 80 89 79.5 89.5 84.5 6
90-99 90 99 89.5 99.5 94.5 5
40
HISTOGRAM
bar chart with
i. horizontal axis represented by the upper boundary and the vertical axis represented by the frequency
Or
ii. frequency versus midpoint
Histogram
29.5-39.5 39.5-49.5 49.5-59.5 59.5-69.5 69.5-79.5 79.5-89.5 89.5-99.50
1
2
3
4
5
6
7
8
9
10
Histogram of marks
Upper boundary(marks)
freq
uen
cy
HISTOGRAM
44.5 54.5 64.5 .74.5 84.5 94.50
1
2
3
4
5
6
7
8
9
10
Histogram of Marks
midpoints ( marks )
Fre
qu
ency
Frequency polygon
is a closed line graphTwo methods :i. From histogramii. frequency versus midpoint
29.5-39.5 39.5-49.5 49.5-59.5 59.5-69.5 69.5-79.5 79.5-89.5 89.5-99.5 99.5-109.50
1
2
3
4
5
6
7
8
9
10
Histogram and Frequency polygon
Upper boundary (marks )
Fre
quen
cy
Frequency polygon
34.5 44.5 54.5 64.5 .74.5 84.5 94.5 104.50123456789
10
Frequency polygon of marks
Midpoint (marks)
Fre
quen
cy
Cumulative Frequency Table
Class intervals
Upper boundary
Frequency Cumulative frequency
30-39 39.5 0 0
40-49 49.5 5 5
50-59 59.5 7 12
60-69 69.5 8 20
70-79 79.5 9 29
80-89 89.5 6 35
90-99 99.5 5 40
40
Must add a class interval with 0 frequency
Ogive : Cumulative frequency curve
Cumulative frequency versus upper boundaries
29.5 39.5 49.5 59.5 69.5 79.5 89.5 99.50
5
10
15
20
25
30
35
40
45
Ogive of marks
Upper boundary ( marks )
Cu
mu
lati
ve f
req
uen
cy
Measures of Dispersion
- the amount or distances the values are spread out in a set of data
i. Range :midpoint of the highest class – midpoint of the lowest class
ii. Median : the value at half of the distributioniii. First quartile(Q1): the value at the first quarter iv. Third quartile (Q3): the value at the third quarterv. Interquartile range : Q3 – Q1
29.5 39.5 49.5 59.5 69.5 79.5 89.5 99.50
5
10
15
20
25
30
35
40
45
Ogive of marks
Upper boundary ( marks )
Cu
mu
lati
ve f
req
uen
cy
Exercise 1: SPM JUN’09
a) Complete the table below
Marks Frequency Midpoint Upper boundary Cumulative frequency
60-64 0 62 64.5 0
65-69 2 67 69.5 2
70-74 6