Lesson 13 - 1

Year 1

CS113/0401/v1

LESSON 13TYPES OF DATA

Qualitative Not usually numeric

No particular order

Examples:

– Colour, Types of Materials

Quantitative Numeric

Ordered

Measurable

Continuous

– E.g. Length, Age, Weight

Discrete

– E.g. Shoe size, Number of people

Lesson 13 - 2

Year 1

CS113/0401/v1

First stage in making raw data understandable

RAW DATA

Number of sheets of listing paper used by each of 120 jobs

Not easily digested!

17

24 11

14

18

17

7

5

21

6 11 18 22 14 6 17

14

8

12132712 189

14

18 14

13

21

8

27

9

11

16 27 21 14 11 19 7

10

29

17121419 129

23

17 24

7

13

14

17

21

8

17 19 24 26 2 5 18

14

16

7162813 148

19

27 9

18

8

24

19

7

13

14 16 19 11 17 23 12

25

16

15102118 1411

9

14 28

20

12

16

10

8

9

11 22 10 17 9 18 12

24

8

716520 710

DATA TABULATION (1)

Lesson 13 - 3

Year 1

CS113/0401/v1

Category (No of sheets

used)Tally Frequency

0 - 11

5 - 261111 1111 1111 1111 1111 1

10 - 371111 1111 1111 1111 11111111 1111 11

15- 311111 1111 1111 1111 11111111 1

20 - 161111 1111 1111 1

25 - 91111 1111

120Total

Frequency distribution table

DATA TABULATION (2)

Tabulate in (discrete) categories

Lesson 13 - 4

Year 1

CS113/0401/v1

FREQUENCY DISTRIBUTION (1)

Raw data

Raw data are collected data which have been organized numerically

Array

An array is an arrangement of raw numerical data in ascending or descending order of magnitude. The difference between the largest and smallest number is called the range of the data

Lesson 13 - 5

Year 1

CS113/0401/v1

FREQUENCY DISTRIBUTION (2)

Frequency distribution

When summarizing a large number of raw data it is often useful to distribute the data into classes or categories and to determine the number of individuals belonging to each class, called the class frequency

Lesson 13 - 6

Year 1

CS113/0401/v1

EXAMPLE

A set of 100 students obtained from an alphabetical listing of an university record.

Their weights ranging from 60kg to 74kg are tabulated.

Lesson 13 - 7

Year 1

CS113/0401/v1

Mass ( kilograms) Number of Students

60 - 6263 - 6566 - 6869 - 7172 - 74

51842278

Total 100

EXAMPLE

The first class or category, for example consists of masses from 60 to 62 kg and is indicated by the symbol 60 - 62. Since 5 students have masses belonging to this class, the corresponding class frequency is 5.

Data organized and summarized in the above frequency distribution are often called grouped data

Lesson 13 - 8

Year 1

CS113/0401/v1

CLASS INTERVAL

A symbol defining a class such as 60 - 62 is called a class interval. The end numbers 60 and 62, are called the class limits.

The smaller number 60 is the lower class limit and the larger number 62 is the upper class limit.

Lesson 13 - 9

Year 1

CS113/0401/v1

CLASS MARK

A class mark is the midpoint of the class interval and is obtained by adding the lower and upper class limits and dividing by two

In the previous examples, the class mark of the interval 60 - 62 is (60 + 62) / 2 = 61

Lesson 13 - 10

Year 1

CS113/0401/v1

MEDIAN (1)

The median of a set of numbers arranged in order of magnitude is the middle value or the arithmetic mean of the two middle values.

Example 1 The set of numbers

3, 4, 4, 5, 6, 8, 8, 8, 10

For an odd number of data the median occurs at position

(N + 1) / 2

= 10 / 2

= 5th position

Therefore the median = 6

Lesson 13 - 11

Year 1

CS113/0401/v1

MEDIAN (2)

Example 2 The set of numbers

5, 5, 7, 9, 11, 12, 15, 18

For even number of data the median is the average of the two middle values

The median

= (Pos 4 + Pos 5) / 2

= (9 + 11) / 2

= 10

Lesson 13 - 12

Year 1

CS113/0401/v1

For grouped data the median, obtained by interpolation is given by

MEDIAN = L1 + C

Where

L1 = lower class boundary of the median class(I.e. the class

containing the median).

N = number of items in the data

(I.e. total frequency)

ƒ median

- ƒ 1N

2

MEDIAN (1)

Lesson 13 - 13

Year 1

CS113/0401/v1

MEDIAN (2)

ƒ 1 = sum of frequencies of all classes lower than the median

class

median = frequency of median class

c = size of median class interval

Lesson 13 - 14

Year 1

CS113/0401/v1

MEDIAN OF A GROUPED FREQUENCY DISTRIBUTION

Draw a Cumulative Frequency Diagram

Search for the middle value on the c axis and read off the corresponding value on the x axis

This is the median

Lesson 13 - 15

Year 1

CS113/0401/v1

MEDIAN FROM A CUMULATIVE FREQUENCY

DIAGRAM

Lesson 13 - 16

Year 1

CS113/0401/v1

MODE (1)

The mode of a set of numbers is that value which occurs with the greatest frequency, I.e. it is the most common value. The mode may not exit, and even of it does exists it may not be unique

Example The set

2, 2, 5, 7, 9, 9, 9, 10, 11, 12, 18 has mode 9

Example The set

3, 5, 8, 10, 12, 15, 16

has no mode

Lesson 13 - 17

Year 1

CS113/0401/v1

MODE (2)

Example

The set

2, 3, 4, 4, 4, 5, 5, 7, 7, 7, 9

has mode 4 and 7 and is called bimodal

A distribution having only one mode is called unimodal

Lesson 13 - 18

Year 1

CS113/0401/v1

MODE OF A FREQUENCY DISTRICUTION

Ungrouped data

Mode is the x value which has the highest value of

Grouped data

Can’t find mode, only the modal class

Lesson 13 - 19

Year 1

CS113/0401/v1

x f

51 - 55

55 - 60

61 - 65

121610

MODAL CLASS

55 - 60 is the modal class

We don’t know x values before grouping, so we can’t find the mode exactly

N.B.

Actual mode might not even be in this class

Lesson 13 - 20

Year 1

CS113/0401/v1

In cases where grouped data where frequency curve has been constructed to fit the data, the mode will be the value (or values) of x corresponding to the maximum point (or points) on the curve, From a frequency distribution or histogram the mode can be obtained from the following formula,

Mode = L1 + ( (

1 + 2

1* c

MODE (1)

Lesson 13 - 21

Year 1

CS113/0401/v1

Where

L1 = lower class boundary of modal class(i.e. class containing the mode).

1 = excess of modal frequency over frequency of next lower class

2 = excess of modal frequency over frequency of the next higher class

c = size of modal class interval

MODE (2)

Lesson 13 - 22

Year 1

CS113/0401/v1

GROUPED MODE FROM HISTOGRAM (1)

Can only ESTIMATE

Assume mode is in Modal Class

Lesson 13 - 23

Year 1

CS113/0401/v1

Calculation

Mode Estimate

= 25 + 5 x

= 25 + 5 x

= 25 + 1.9

= 26.9

40

40 + 64

40

104

GROUPED MODE FROM HISTOGRAM (2)

Lesson 13 - 24

Year 1

CS113/0401/v1

X =

X1 + X2 + X3 + ….. + Xn

N

=

n

i=1

Xi

N

ARITHMETIC MEAN (1)

The arithmetic mean or the mean of a set of N numbers X1, X2, X3, ..., Xn is donoted by X is defined as

Lesson 13 - 25

Year 1

CS113/0401/v1

ARITHMETIC MEAN (2)

Eight numbers:7, 21, 13, 17, 23, 18, 9, 20

Add them = 128

Divide by 8 = 16

This is the arithmetic mean

It is the the most common definition of “average”

It only works with quantitative data

Lesson 13 - 26

Year 1

CS113/0401/v1

X =

1X1 + 2X2 + ….. + nXn

1 + 2 + …. n

=

n

i=1

iXi

in

i=1

X

ARITHMETIC MEAN (3)

If the number X1, X2, X3, ..., Xn occurs 1, 2, 3, ..., n times respectively, the arithmetic mean is

Lesson 13 - 27

Year 1

CS113/0401/v1

MEAN OF A FREQUENCY DISTRIBUTION

Mean age = = 20.77

(rounded to nearest integer, 21)

2077100

Age (x) xFrequency ()

17

18

19

20

21

22

23

24

25

26

3

8

14

21

24

13

7

6

3

1

51

144

266

420

504

286

161

144

75

26

= 100 x = 2077

Lesson 13 - 28

Year 1

CS113/0401/v1

HISTOGRAMS (1)

Only used for quantitative data

Histogram is like a bar chart, but with no gaps between bars and calibrated horizontal axis

Order of bars depends on value and on horizontal scale

Lesson 13 - 29

Year 1

CS113/0401/v1

HISTOGRAMS (2)

Lesson 13 - 30

Year 1

CS113/0401/v1

HISTOGRAMS (3)

Lesson 13 - 31

Year 1

CS113/0401/v1

AREA IN HISTOGRAMS

Lesson 13 - 32

Year 1

CS113/0401/v1

Line of Code No of Programs100 -

150 -

125 -

39

51

42

24

12

3

325 - 349

300 -

21275 -

30250 -

200 -

175 -

225 -

12

6

CUMULATIVE FREQUENCY DIAGRAMS (1)

Table 1:

Lesson 13 - 33

Year 1

CS113/0401/v1

Line of Code(less than)

CumulativeFrequency

100

150

125

132

81

39

15

3

0

325

300

201275

171250

200

175

225

222

234

240350

CUMULATIVE FREQUENCY DIAGRAMS (2)

Table 2:

Lesson 13 - 34

Year 1

CS113/0401/v1

020406080

100120140160180200220240

0 50 100 150 200 250 300 350

Lines of code (less than)

Cu

mm

ula

tiv

e F

req

ue

nc

y

CUMULATIVE FREQUENCY DIAGRAMS(3)

Cumulative FrequencyCurve

Lesson 13 - 35

Year 1

CS113/0401/v1

n

i=1

(Xi - X) 2

N

STANDARD DEVIATION (1)

The Standard Deviation of a set of N numbers X1, X2, ..., Xn is denoted by S.D. and is defined by

S.D. =

Where

X = Arithmetic Mean

N = Total Number of element in the set

Lesson 13 - 36

Year 1

CS113/0401/v1

n

j=1[ j (Xj - X) 2 ]

n

j=1

i Xi2

i

i Xi2

- i

( )

S.D.

or

S.D. =

STANDARD DEVIATION (2) (GROUPED DATA)

If X1, X2, ..., Xn occurs with frequencies 1, 2, ..., n respectively, the standard deviation can be written as

Lesson 13 - 37

Year 1

CS113/0401/v1

Question 6 c) NCC 1/93

On test the actual access times for 50 hard disc drives were distributed as follows:

Calculate the mean access time and the standard deviation.

Time (ms)

No. of Drives

22.6

3

22.7

1

23.022.9

106

22.8 23.223.1

914 25

23.3

Lesson 13 - 38

Year 1

CS113/0401/v1

Alternative Question 6cx

22.6

22.7

22.8

22.9

23.0

23.1

23.2

23.3

f fx fx2

1

3

6

10

14

9

5

2

22.6

68.1

136.8

229.0

322.0

207.9

116.0

46.6

510.76

1545.87

3119.04

5244.10

7406.00

4802.49

2691.20

1085.78

1149.0 26405.24 (1 mark for each total) 2

2[1] [1]

Mean = 114950

= 22.98 [1]

S.D =fx2

f( X )2

= 26405.2450 (22.98)2

= 0.156

[1]

Lesson 13 - 39

Year 1

CS113/0401/v1

The variance of a set of data is defined as the square of the standard deviation and is thus given by (S.D.)

Variance =

i.e.

Variance = (S.D.)2

n

j=1[ j (Xj - X) 2 ]

n

j=1 j

VARIANCE