Data & Freq Dist

Data o o

Data are numerical statement of facts. When information are represented by numeric values, they become data.

Types of data 1. Quantitative 2. Primary

& qualitative: – on the basis of nature of facts

& Secondary: – on the basis of sources

Sources of primary data:  survey, lab & field experiments, etc. Source secondary data:  published sources: government & international publications, reports of organizations, publications of research institutes, magazines, etc.  unpublished sources: organization files & records, etc.

Variable o o o o

A variable is the quantitative characteristic which can be measured or counted. It is called variable since its numerical value varies in magnitude from one unit to other unit. They are generally denoted by: X, Y, Z etc. and their corresponding values by: x, y, z, etc. Variate values: different values taken by a variable For example: height, weight, income, expenditure etc.

Types of variable 1. Discrete variable o takes at most a countable number (whole number, exact value) of values o such as: family members, no. of children, no. of students, etc. 2. Continuous variable o takes all possible values within a certain range, o such as: weight, size, height

Arrangement of Data Data Array o o

It is the simplest ways to present data. It arranges values in ascending or descending order.

The advantages data arrays o can quickly notice the lowest & highest values in data o can

easily divide the data into classes (sections)

o can

see whether any values appear more than once in the array

o can

observe the distance (difference) between succeeding values in the data

However, it is a cumbersome form for displaying large number of data.

Frequency Distribution (Table) o

One way of compressing data

o

It is a table that organizes data into classes, i.e., into groups of values describing one characteristic of the data.

Types of frequency distribution o

Discrete Frequency Distribution

o

Continuous Frequency Distribution

Discrete Frequency Distribution o

It is the distribution formed by a discrete variable. No. of children

0

1

2

3

4

5

No. of families

1

3

6

6

3

1

Continuous Frequency Distribution o o

It is the distribution formed by a continuous variable. Data are divided into class intervals and are presented along with the corresponding frequencies.

Marks No. of students

10-20

20-30

30-40

40-50

7

12

19

3

Relative Frequency Distribution o

o

The frequency of each value can be expressed as a fraction or a percentage of the total number of observations. The fractions or percentages so obtained is called relative frequencies and the arrangement is called relative distribution. No. of children

0

1

2

3

4

5

Total

No. of families

1

3

6

6

3

1

20

0.05

0.15

0.30

0.30

0.15

0.05

1.00

Relative Frequency

Closed-end classes o

o

In closed-end classes, the lower limit of the first class and the upper limit of the highest class are clearly specified or defined. For example: the classes 0-10, …….., 50-60 are closed end classes. Marks

Wages

10-20

500-1000

20-30

1000-1500

30-40

1500-2000

40-50

2000-2500

50-60

2500-3000

Open-end Class o

o

In open-end classes, the lower limit of the first class and the upper limit of the highest class or both are not specified. For example: Below 20, 50 and above. Class

Class

Class

Below 20

10-20

Below 20

20-30

20-30

20-30

30-40

30-40

30-40

40-50

40-50

40-50

50-60

50 and above

50 and above

Methods of classifying data 1. Exclusive method Income ('000 Rs.)

No. of employees

10-20 20-30 30-40 40-50 50-60

25 20 16 8 5

2. Inclusive method

 

Income ('000 Rs.)

No. of employees

10-19 20-29 30-39 40-49 50-59

25 20 16 8 5

Conversion of inclusive classes into exclusive classes Correction factor, (C.F.) = (Lower limit of 2nd class – Upper limit of 1st class)/2 (C.F.) = (20 – 19)/2 = 0.5 o o

Subtract C.F. from lower limits of classes Real lower limit = lower limit - C.F. Add C.F. to upper limits of classes Real upper limit = Upper limit + C.F. Income ('000 Rs.)

No. of employees

9.5-19.5 19.5-29.5 29.5-39.5 39.5-49.5 49.5-59.5

25 20 16 8 5

Constructing a Frequency Distribution 1.

Number of classes

o

should neither be too small nor too large should be preferably between 5 and 15 If too few classes are formed, the necessary details may be lost and if too many classes are formed, further processing of data would become difficult and tiresome. The number of classes.

o o

2. Size of class intervals o o o

depends on the range of data and number of classes. The size of class interval of 2, 5, 10, 15, 20, 25, 50, 100, etc. are preferred rather than figures like 1, 3, 7, 11, 23, etc. The multiples of 2, 5 and 10 are in common use.

3. The starting point or the lower limit of the first class should be 0, 5 or multiples of 5. The selection of the starting point is based on the smallest value of data. The starting point of the first class should not necessarily be the lower value of data. 4. Finally, class frequencies are obtained by using tally bars.

Cumulative Frequency Distribution The frequency obtained on successively adding frequencies of classes of variable according to a certain law is called cumulative

frequency.

The

distribution

made

by

such

cumulation of frequencies is called the cumulative frequency distribution. There are two types of cumulative frequency distribution.

1. Less than Cumulative Frequency Distribution 2. More than Cumulative Frequency Distribution

Less than cumulative frequency distribution  

Upper limits of the classes are listed in ascending order of magnitude. Frequencies are added successively from the lowest class (top) to the highest class (bottom). Income ('000 Rs.)

No. of employees

Less than 20 Less than 30 Less than 40 Less than 50 Less than 60

25 25+20=45 45+16=61 61+8=69 69+5=74

More than Cumulative Frequency Distribution  Lower limits of the classes are listed in ascending order of magnitude.  Frequencies are added successively from the highest class (bottom) to the lowest class (top). Income ('000 Rs.)

No. of employees

More than 10 More than 20 More than 30 More than 40 More than 50

49+25=74 29+20=49 13+16=29 5+8=13 5

Stem-and-leaf Diagram Stem-and-leaf Diagram is the visual representation of the data. It is the arrangement of data that gives some information about the patterns within the data. It is constructed at least for two digit data. Each data is splitted into two parts, a stem, consisting of one or more of leading digits and a leaf, which consist of the remaining or following digits. The stem and leaf parts are separated by a vertical line. The stems are placed to the left of a vertical line and the leaves to the right of the line.

Example of Stem-and-leaf diagram 26 37 40 18 14 45 32 68 31 37 20 32 15 27 46 44 62 58 38 42 22 26 44 41 34 55 50 63 29 22 1 2 3 4 5 6

5 6 7 0 5 2

8 0 2 4 0 8

4 2 4 1 8 3

1 2 3 4 5 6

4 0 0 0 0 2

5 2 1 1 5 3

8 2 2 2 8 8

1

4

means 14

6 2 6

7 1 5

9 0 4

2 7 2

6 2 4

6 4 4

7 7 5

9 7 6