MAT114, 217 Lecture Note

8/2/2019 MAT114, 217 Lecture Note.

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STATISTICS

Statistics is concerned with Scientific methods for collecting, organizing, summarizing,

presenting, and analyzing data as well as with drawing valid conclusions and making

reasonable decisions on the basis of such analysis.

POPULATION AND SAMPLE

It is often impossible or impractical to observe the entire group especially if it is large.

Instead of examining the entire group, called the population, of universe, one examines asmall part of the group, called a sample.

FINITE POPULATION: all bolts produced in a factory.

INFINITE POPULATION: heads, tails in successive tosses of a coin.

INDUCTIVE STATISTICS: important conclusions about the population.

DESCRIPTIVE, OR DEDUCTIVE STATISTICS: it only seek to describe and analyze a

given group without drawing any conclusions or inferences.

DATA: numerical facts, information or series of observations that can be measured or

quantified.

RAW DATA: are collected data that have not been organized numerically. An example is

the set of heights of 100 male students obtained from an alphabetical listing of universityrecords.

ARRAYS: an array is an arrangement of raw numerical data in ascending or descendingorder of magnitude. The difference between the largest and smallest numbers is called therange of the data.

FREQUENCY DISTRIBUTIONS

a tabular arrangement of data by classes together with the corresponding class frequencies

is called a frequency distribution or frequency table.

CLASS INTERVALS AND CLASS LIMITS

A symbol defining a class, such as 60 62 is called a class interval. The end numbers, 60

and 62 are called class limits; the smaller number 60 is the lower class limit, and the larger

number 62 is the upper class limit.

CLASS BOUNDARIES

The class boundaries are obtained by adding the upper limit of one class interval to thelower limit of the next-higher class interval and divided by 2.

THE SIZE, OR WIDTH OF A CLASS INTERVAL


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The size, or width, of a class interval is the difference between the lower and upper class

boundaries and is also referred to as the class width, class size, or class length.

THE CLASS MARK

The class mark is the midpoint of the class interval and is obtained by adding the lower and

upper class limits and dividing by 2. Thus the class mark of the interval 60 62 is (60 62)/2 = 61. The class mark is also called the class midpoint.

Example 1The following is a record of the total number of goals scored in thirty different matches all

over the federation, in one week-end of a football season.

3 5 1 7 2 2 4 3 0 5

0 8 3 2 6 7 5 2 1 1

11 2 2 10 9 9 6 3 4 6

(a) Prepare a frequency table to represent the data(b) What is the least number of goal scored

(c) What is the greatest number of goal scored.

Solution

Total number ofgoals scored

Tally Frequency

01

2

3

45

67

8

9

1011

/////

//// /

////

/////

/////

/

//

//

23

6

4

23

32

1

2

11

30

The least number of goals scored is zero.

The greatest number of goals scored is 11.

GROUPED FREQUENCY DISTRIBUTION

Below are the masses to the nearest kilogram of 50 students in a class.10 35 34 40 20 44 29 34 23 36

32 45 33 45 37 39 46 31 40 24

34 40 45 48 41 32 35 42 33 2546 43 25 49 44 17 20 46 38 27


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30 44 31 40 43 46 45 40 41 37

Construct a grouped frequency distribution table using the class interval 10-13, 14-17, 18-

21, etc.Solution

Class interval tally frequency

10 1314 1718 21

22 25

26 29

30 3334 37

38 41

42 4546 49

////

////

///

//// ////// ////

//// ///

//// //////// /

112

4

3

79

8

96

50

DIAGRAMATIC AND GRAPHICAL PRESENTATION OF

DATA

PICTOGRAM

Pictorial representation involves the use of pictures to convey the information which we

wish to pass on.

Example

The total exports of coffee in metric tonnes in country X are given in the table below forthree consecutive years.

Year Export in metric tonne

1975

1976

1977

300,000

400,000

500,000

Draw a pictogram to show the total exports for each of the three consecutive years.

Solution

Coffee cup will be an appropriate representation

1975

1976

1977

BAR CHART


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In a bar-chart, rectangular bars of equal widths but heights proportional to the size of the

items we want to display are used.

Example

The number of crates of soft drinks sold to students of a high school in city A during one

break-time is as follows:Brands of soft drinks Number of crates

Coca-cola

Pepsi-cola

Fanta

Seven-upCrush

sprite

5

3

4

21

2

Draw a bar-chart to display the data in the table above.

0

0.5

1

1.52

2.5

3

3.5

44.5

5

coca

cola

pepsi

cola

fanta seven

up

crush sprite

North

PIE - CHART

A Pie-chart is a circular chart divided into sectors whose angles are proportional to themagnitudes of their corresponding constituents of the total we wish to display.

ExampleThe table below is a break-down of monthly expenditure of a salary earner.

Expenditure Amount

Rent

FoodTransportClothes

Savings

60

802040

100

Draw a pie-chart to display the data.

Solution.

Expenditure Amount Angle of sector


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Rent 60 072360300

60=

Food 80 096360

300

80=

Transport 20 024360300

20=

Clothes 40 048360300

40=

Savings 100 0120360300

100=

Total 300 360

A pie-chart showing the expenditure pattern of a salary earner every month.

HISTOGRAM

A histogram is a graphic display of a frequency distrubution. It consists of rectangular bars

placed side by side with the horizontal axis as the variable axis, while the vertical axis isthe frequency axis.

The following are the lengths in cm of fifty planks cut by a machine in a sawmill.

33 49 60 58 59 71 42 88 68 9154 32 81 59 41 55 38 56 86 62

50 69 50 84 77 33 71 42 69 9361 51 23 76 63 96 26 70 66 80

44 52 46 33 68 39 61 71 48 66

(a) Using class interval of 21-30, 31-40, ---(i) Construct the frequency table.


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(ii) Draw the histogram for the distribution

(b)

(i) Identify the modal class(ii) Use your histogram to estimate the mode of the distribution.

Solution

Class interval Tally Frequency Classboundries

21 30

31 40

41 50

51 6061 70

71 80

//

//// /

//// ////

//// //////// //// /

//// /

2

6

9

911

6

20.5 30.5

30.5 40.5

40.5 50.5

50.5 60.560.5 70.5

70.5 80.5

20.5 30.5 40.5 50.5 60.5 70.5

2

4

6

8

10

12

80.5

modal class = 61 70

mode = 63.5

FREQUENCY POLYGON

The mid-point of the top of each bar of a histogram.

Example

The age distribution of 77 people in a farm settlement is shown in the table below.

Age (years) 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50

frequency 5 8 10 12 3 14 9 16

Construct a histogram and hence draw a frequency polygon of the above distribution

Solution

Age(years) Class mark Frequency Class boundaries

11 15

16 20

13

18

5

8

10.5 15.5

15.5 20.5


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21 25

26 3031 35

36 40

41 45

46 50

23

2833

38

43

48

10

123

14

9

16

20.5 25.5

25.5 30.530.5 35.5

35.5 40.5

40.5 45.5

45.5 50.5

1 0.5 1 5.5 2 0.5 2 5.5 3 0.5 3 5.5

2

4

6

8

1 0

1 2

40.5

1 4

1 6

45.5 50.5

CUMULATIVE FREQUENCY CURVE

The sum of all the frequencies from the first to that of a particular class is called thecumulative frequency of that class in a frequency distribution. The graph of the cumulative

frequency curve against the upper class boundary is called a cumulative frequency curve or

Ogive.

ExampleThe table below shows the weekly profit in Naira from a mini- market.

Weekly

profit

1- 10 11-20 21- 30 31- 40 41- 50 51- 60

frequency 6 6 12 11 10 5

(a) Draw the cumulative frequency graph of the data

(b) From your graph, estimate the


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(i) Lower quartile

(ii) Median

(iii) Upper quartile(iv) Semi-interquartile range

(v) 80th percentile

Solution

Classinterval

frequency Classboundaries

Cumulativefrequency

1 10

11 2021 30

31 40

41 5051 60

6

612

11

105

0.5 10.5

10.5 20.520.5 30.5

30.5 40.5

40.5 50.550.5 60.5

6

1224

35

4550

0.5

5

10.5 20.5 30.5 40.5 50.5 60.5

10

15

20

25

30

35

40

45

50

Q2

(a)

(i) 2

150,)1(2

1 +=+= itemnMedian

th

= 25.25th item

Q2 = 31.50


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If the minimum mark for distinction is 75%, how many candidates passed with

distinction?

The table below shows how a man spends his income in a month.

Items Amount Spent

Food

House rent

Provisions

Electricity

Transportation

others

4500

3000

2500

2000

5000

3000

Represent the information on a pie chart.

What percentage of his income is spent on transportation?

The ages, in years, of 50 teachers in a school are given below:

21 37 49 27 49 42 26 33 46 40

50 29 23 24 29 31 36 22 27 38

30 26 42 39 34 23 21 32 41 46

46 31 33 29 28 43 47 40 34 44

26 38 34 49 45 27 25 33 39 40

Form a frequency distribution of the data using the intervals: 21 25, 26 30,

31 35, etc.

Draw the histogram of the distribution.

Use your histogram to estimate the mode.


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Calculate the mean age.

The table shows the marks obtained by a group of students in a class test.

Marks 40 -44

45 -49

50 -54

55 -59

60 -64

65 -69

No. of

students

4 9 18 23 10 6

Draw a histogram for the distribution.

Use your histogram to estimate the median of the distribution.

The table gives the distribution of marks of 60 candidates in a test.

Marks 23 - 25 26 -

28

29 - 31 32 - 34 35 - 37 38 -

40

frequenc

y

3 7 15 21 10 4

Draw a cumulative frequency curve of the distribution.

From your curve, estimate the

80th percentile;

Median;

Semi-interquartile range.

The distribution of the lives (in days) of 40 transistor batteries is shown in the

table.

Battery life (in

days)

26 -

30

31 - 35 36 - 40 41 - 45 46

50

51 - 55

frequency 4 7 13 8 6 2


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Draw a histogram for the distribution.

Use your graph to determine the mode for the distribution.

The table shows the frequency distribution of the scores obtained by 100students in an examination.

Marks 30 -

39

40 -

49

50 - 59 60 -

69

70 -

79

80 - 89 90 - 99

Frequen

cy

9 14 32 20 15 7 3

Draw a cumulative frequency curve for the distribution.

Use your curve to determine the:

Median;

Lower quartile;

Lowest mark for distinction if 5% of the students passed with distinction.

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MAT114, 217 Lecture Note