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