This unit covers classification of data, frequency distributions, and tabular, diagrammatic, and graphic presentation for exam use.
Data and collection
The aggregate of facts which can numerically be expressed are known as data.
Collection of data means the methods used for getting the necessary information from the units under investigation. The method depends upon the nature, objective, and scope of the enquiry.
The person who conducts the statistical enquiry for further analysis is known as an investigator. The persons from whom the information is collected are known as respondents or informants.
Types of data
Generally data are classified into two types:
Primary data: data originally collected by an investigator or an agent for the first time for the purpose of statistical enquiry.
Secondary data: data which were originally collected but are obtained from published or unpublished sources. This type of data is not original in character.
Methods of collecting primary data
- Direct personal interview
- Indirect oral interview
- Information through correspondents
- Mailed questionnaire method
- Schedule sent through enumerators
Factors when choosing a method
- Nature of enquiry
- Objective and scope of enquiry
- Financial resources
- Time factor
- Degree of accuracy desired
Sources of secondary data
- Published sources — reports and official publications; semi-official publications; private publications
- Unpublished sources
Classification of data
Data collected from the field are known as raw data. Classification and tabulation present data in condensed form so that important characteristics are highlighted. Classification changes voluminous heterogeneous data into homogeneous groups and makes data more useful for further processing such as tabulation, analysis, and interpretation.
Classification means arranging data into different groups or classes according to their similarities. It is the first step in tabulation because items with similarities must be brought together before data are presented in a table.
Objectives of classification
- Presents facts in a simpler form and pinpoints significant features at a glance
- Condenses a huge mass of data by eliminating irrelevant details so similarities and dissimilarities can be brought out
- Facilitates comparison
- Enables statistical treatment of the information collected
- Prepares the basis of tabulation
Types of classification
Geographical classification: data classified according to geographical or locational positions such as zones, districts, etc.
Chronological classification: data classified according to variation of time (for example, year-wise sales, month-wise rainfall).
Qualitative classification: data classified according to some quality such as religion, literacy, honesty, sex, etc.
Quantitative classification: data classified according to characteristics that can be measured quantitatively such as marks, heights, weights, incomes, etc.
Variable, frequency, and frequency distribution
The quantitative characteristic under study is known as a variable or variate. The values taken by the variable are known as variate values.
- Continuous — e.g. height, weight
- Discrete — e.g. family members, number of goals in a football match
Frequency is the number of times a variate value occurs. If the variate value 4 repeats 7 times, the frequency of 4 is 7.
A frequency distribution is a table in which collected and classified data are presented in different classes and the number of cases in each class is recorded. Kinds include univariate and bivariate frequency distributions.
Univariate series
Individual series
An individual series lists items singly after observations, as distinguished from listing them in groups.
| Day | SUN | MON | TUE | WED | THU | FRI | SAT |
|---|---|---|---|---|---|---|---|
| Temp (°C) | 32 | 31 | 35 | 26 | 25 | 30 | 25 |
Discrete series
A series formed from a discrete variable is a discrete series (discrete frequency distribution).
| Outcome of a die | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| No. of outcomes | 6 | 3 | 9 | 8 | 6 | 4 |
Continuous series
A series formed with class intervals and frequencies is a continuous series. Variate values lie between two numbers known as an interval.
| Wages (in Rs.) | 30–40 | 40–50 | 50–60 | 60–70 |
|---|---|---|---|---|
| No. of workers | 10 | 12 | 15 | 7 |
Class intervals
Methods of forming class intervals: exclusive method and inclusive method.
Exclusive type: the upper limit of each class interval is excluded from that class.
| Marks | 50–60 | 60–70 | 70–80 | 80–90 |
|---|---|---|---|---|
| No. of students | 40 | 20 | 5 | 2 |
Inclusive type: both the lower and upper limits of each class interval are included in that class. Width of the class interval = lower limit of next class − lower limit of previous class.
| Marks | 50–59 | 60–69 | 70–79 | 80–89 |
|---|---|---|---|---|
| No. of students | 10 | 20 | 5 | 2 |
To convert inclusive class intervals to exclusive class intervals, use a correction factor:
Example: \(CF = \dfrac{60 - 59}{2} = 0.5\); real limits \(50 - 0.5 = 49.5\) and \(59 + 0.5 = 59.5\).
| Marks | 49.5–59.5 | 59.5–69.5 | 69.5–79.5 | 79.5–89.5 |
|---|---|---|---|---|
| No. of students | 10 | 20 | 5 | 2 |
Open-end classes: if the lower limit of the first class or the upper limit of the last class (or both) are not specified, the class is open-ended (for example, “Below 10”, “Above 15°C”).
Cumulative frequency distribution
A frequency distribution in which frequencies are cumulated is a cumulative frequency distribution. Types: less than and more than.
| Simple frequency distribution | Less than cumulative frequency | ||
|---|---|---|---|
| Monthly income | No. of persons | Monthly income | No. of persons |
| 0–50 | 20 | Less than 50 | 20 |
| 50–100 | 30 | Less than 100 | 20+30=50 |
| 100–150 | 25 | Less than 150 | 50+25=75 |
| 150–200 | 33 | Less than 200 | 75+33=108 |
| 200–250 | 22 | Less than 250 | 108+22=130 |
To convert less-than cumulative frequencies to simple frequencies, subtract successive cumulative totals (for example, Below 20 → 8 for 0–20; Below 40 → 18 so 20–40 has \(18-8=10\)).
To convert more-than cumulative frequencies to simple frequencies, subtract successive more-than totals from the previous more-than total.
| More than cumulative | Simple frequency | ||
|---|---|---|---|
| Marks | No. of students | Marks | No. of students |
| Above 0 | 75 | 0–20 | 75−67=8 |
| Above 20 | 67 | 20–40 | 67−57=10 |
| Above 40 | 57 | 40–60 | 57−45=12 |
| Above 60 | 45 | 60–80 | 45−25=20 |
| Above 80 | 25 | 80–100 | 25 |
Tabular presentation
Tabulation is arranging data in an orderly manner into rows and columns. Rows are horizontal arrangements and columns are vertical arrangements.
Diagrammatic and graphic presentation
Diagrams and graphs give a bird’s-eye view of numerical data, are more attractive than raw figures, create a lasting impression, enable comparison of sets of data, and save time in grasping important features.
Types of diagrams
- Simple bar diagram
- Sub-divided or component bar diagram
- Percentage bar diagram
- Multiple bar diagram
- Pie (or angular) diagram
Graphs of frequency distributions
- Histogram
- Frequency polygon — with histogram or without histogram
- Ogive (cumulative frequency curve) — less than method; more than method
Histogram with unequal class widths
When class sizes are unequal, adjust frequencies before drawing the histogram. Take the smallest class size as base; adjusted frequency \(= f / k\) where \(k =\) class size ÷ smallest class size.
| Salaries (Rs.) | Frequency (f) | Class size (h) | k = h / 10 | Adjusted frequency (f/k) |
|---|---|---|---|---|
| 300–310 | 8 | 10 | 1 | 8 |
| 310–320 | 16 | 10 | 1 | 16 |
| 320–330 | 20 | 10 | 1 | 20 |
| 330–350 | 18 | 20 | 2 | 9 |
| 350–390 | 12 | 40 | 4 | 3 |
The last two class sizes are 2 and 4 times the smallest size 10, so divide those frequencies by 2 and 4 to get rectangle heights \(18/2 = 9\) and \(12/4 = 3\).
Pie diagram (angular)
Central angle for an item \(= \dfrac{\text{item value}}{\text{total}} \times 360^\circ\).
| Items | Expenditure (sons) | Angle | Expenditure (family) | Angle |
|---|---|---|---|---|
| Food | 40 | 72° | 30 | 108° |
| Clothing | 30 | 54° | 10 | 36° |
| Education | 50 | 90° | 20 | 72° |
| Fuel | 20 | 36° | 20 | 72° |
| Misc. | 60 | 108° | 20 | 72° |
| Total | 200 | 360° | 100 | 360° |
Ogive from a wage distribution
| Wages (′000 Rs.) | 50–60 | 60–70 | 70–80 | 80–90 | 90–100 | 100–110 | 110–120 |
|---|---|---|---|---|---|---|---|
| No. of employees | 8 | 10 | 16 | 14 | 10 | 5 | 2 |
| Less than ogive | More than ogive | ||
|---|---|---|---|
| Wages | c.f. | Wages | c.f. |
| Below 60 | 8 | More than 50 | 65 |
| Below 70 | 18 | More than 60 | 57 |
| Below 80 | 34 | More than 70 | 47 |
| Below 90 | 48 | More than 80 | 31 |
| Below 100 | 58 | More than 90 | 17 |
| Below 110 | 63 | More than 100 | 7 |
| Below 120 | 65 | More than 110 | 2 |
Important Questions
Board exam questions asked on this unit, by exam year (marks in brackets).
2079 Exam
What is chronological classification? Give an example. [2 marks]
2080 Exam
State any two principal objectives of classification. [2 marks]
-
Construct a frequency table for the following data regarding annual profit, in lakhs of rupees, in 50 firms, taking 25–34, 35–44, etc. as class intervals. [15 marks]
28 35 61 29 36 48 59 67 69 50 48 40 49 42 41 37 51 62 63 33 31 32 35 40 38 39 60 51 54 56 69 46 42 38 61 59 58 44 39 57 38 44 45 45 47 38 44 47 47 64 - Find the number of firms having profit between Rs. 37 lakhs and Rs. 58 lakhs.
- Find the profit above which 10% of the firms will have their profits.
- Find the middle 50% profit group.
2081 Exam
Define the qualitative classification of data with a suitable example. [2 marks]
2083 Exam
List different sources of secondary data. [2 marks]