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Introduction to Statistics

Reading Time: 15 mins

Solved Examples: 13

Practice Questions: 3

What is Statistics?

Statistics is the branch of mathematics that deals with the collection, organisation, presentation, analysis and interpretation of data. It helps us understand large amounts of information by summarising it into meaningful forms such as tables, graphs and numerical measures.
Statistics is used to identify patterns, compare different sets of data and draw logical conclusions. For example, a school may use statistics to analyse students’ marks, a company may study monthly sales, or a weather department may examine rainfall records over several years.
In statistics, data may consist of numbers, measurements or observations collected for a specific purpose. Once the data is organised and analysed, it becomes easier to make informed decisions and understand trends.

Data and Its Types

Data is a collection of facts, figures or observations gathered for a particular purpose. These observations may be numerical or descriptive and are analysed to obtain meaningful information.
Data is broadly classified into the following types

Qualitative Data
- Describes qualities or characteristics.
- It is expressed in words rather than numbers.
- Example: Eye colour, blood group, favourite sport.
Quantitative Data
- Represents numerical values that can be measured or counted.
- Example: Height, weight, marks obtained, age.
Quantitative data is further divided into:
- Discrete Data
  - Consists of countable values.
  - Example: Number of students in a class, number of books on a shelf.
- Continuous Data
  - Consists of measurable values that can take any value within a range.
  - Example: Height, weight, temperature, distance.

Population and Sample

In statistics, it is often impossible or impractical to collect information from every member of a large group. Therefore, statisticians study a smaller representative group.

Population
- The complete collection of individuals, objects or observations being studied.
- Example: All students in a school.
Sample
- A part of the population selected for analysis.
- Example: 200 students chosen from the school for a survey.

A good sample should accurately represent the characteristics of the population so that reliable conclusions can be drawn.

Observation and Variable

An observation is a single piece of information collected during a statistical study.
Examples:

A student’s marks are 82.
A person’s height is 168 cm.

A variable is any characteristic that can take different values for different observations.

Examples:

Age
Height
Weight
Income
Marks obtained

Variables are generally classified as:

Categorical Variables
- Represent categories or labels.
- Example: Gender, blood group, favourite subject.
Numerical Variables
- Represent measurable quantities.
- Example: Age, salary, temperature.

Primary Data and Secondary Data

Depending on how the information is collected, data is classified into two types.

Primary Data
- Data collected directly by the investigator for a specific purpose.
- Usually obtained through surveys, interviews, questionnaires or experiments.
- Examples:
  - Conducting a survey to determine students’ favourite subject.
  - Measuring the heights of students in a class.
Secondary Data
- Data that has already been collected and published by someone else.
- Examples:
  - Census reports
  - Government publications
  - Research journals
  - School records

Primary data is generally more specific to the study, while secondary data is quicker and less expensive to obtain.

Organisation and Classification of Data

Raw data is often difficult to interpret because it is usually unorganised. Statistics organises data into a systematic form to make analysis easier.

Common methods of organising data include:

Arranging data in ascending or descending order.
Grouping similar observations together.
Preparing frequency tables.
Representing data using charts and graphs.

Proper organisation of data helps identify trends, compare values and draw accurate conclusions.

Summary of Introduction to Statistics

Concept	Summary
Statistics	The branch of mathematics concerned with the collection, organisation, presentation, analysis and interpretation of data.
Data	A collection of facts, figures or observations gathered for a specific purpose.
Qualitative Data	Descriptive data expressed in words or categories rather than numbers.
Quantitative Data	Numerical data that can be measured or counted.
Discrete Data	Countable numerical values, usually whole numbers.
Continuous Data	Measurable numerical values that can take any value within a given range.
Population	The complete set of individuals, objects or observations under study.
Sample	A representative subset selected from the population for analysis.
Observation	A single recorded value or piece of information collected during a statistical study.
Variable	A characteristic that can take different values for different observations.
Primary Data	Data collected first-hand by the investigator for a specific purpose.
Secondary Data	Data collected and published earlier by another individual or organisation.
Organisation of Data	The systematic arrangement of data into tables, groups or charts to simplify analysis and interpretation.

Solved Examples

Question 1: A teacher records the following marks obtained by 10 students in a Mathematics test:
48, 55, 62, 74, 68, 81, 59, 73, 65, 70
Identify the data and state the number of observations.

The marks obtained by the students represent the data.
Each student’s mark is one observation.
Number of observations = 10

Question 2: A weather station records the maximum temperature (in °C) for seven consecutive days as follows:
34, 36, 35, 37, 33, 38, 36
Identify the data and determine the total number of observations.

The recorded temperatures form the data.
Each day’s temperature is one observation.
Number of observations = 7

Question 3: Classify each of the following as Qualitative or Quantitative data.
(a) Blood group of patients
(b) Heights of students in a class
(c) Favourite sport of employees
(d) Monthly electricity consumption (in kWh)

(a) Blood group → Qualitative
(b) Heights of students → Quantitative
(c) Favourite sport → Qualitative
(d) Monthly electricity consumption → Quantitative

Question 4: Identify whether each of the following is Discrete or Continuous data.
(a) Number of vehicles passing a toll gate in one hour
(b) Weight of parcels in a courier office
(c) Number of goals scored in a football match
(d) Time taken by athletes to complete a race

(a) Number of vehicles → Discrete (countable)
(b) Weight of parcels → Continuous (measurable)
(c) Number of goals → Discrete (countable)
(d) Time taken → Continuous (measurable)

Question 5: Classify each of the following according to its type of data.
(a) Number of pages in a book
(b) Colour of a school bag
(c) Distance travelled by a train
(d) Number of mobile phones owned by a family
(e) Annual rainfall in a city
(f) Marital status of employees

(a) Number of pages in a book → Quantitative (Discrete)
(b) Colour of a school bag → Qualitative
(c) Distance travelled by a train → Quantitative (Continuous)
(d) Number of mobile phones owned by a family → Quantitative (Discrete)
(e) Annual rainfall in a city → Quantitative (Continuous)
(f) Marital status of employees → Qualitative

Question 6: A school has 1,250 students. To estimate the average daily study time of its students, a researcher selects 150 students at random and records the number of hours they study each day. Identify the Population and the Sample.

The population is the entire group about which information is required.
Population = All 1,250 students of the school.
The sample is the smaller group selected from the population for the study.
Sample = 150 students selected at random.

Question 7: The heights (in cm) of six students are recorded as 148, 152, 155, 149, 160, 154 Identify the variable and the number of observations.

The characteristic being measured is the height of the students, which is the variable.
Each recorded height represents one observation.
Number of observations = 6

Question 8: A researcher wants to study the daily exercise habits of students in a school. The researcher prepares a questionnaire and collects responses directly from 500 students.
Identify whether the data collected is Primary Data or Secondary Data.

The data is collected directly by the researcher for a specific purpose using a questionnaire.
Therefore, it is Primary Data.

Question 9: Classify the following sources of data as Primary Data or Secondary Data:
(a) A company conducts a customer satisfaction survey among its buyers.
(b) A researcher uses information from a published research journal.
(c) A teacher records the marks obtained by students in a class test.
(d) A business uses sales records maintained by another department

(a) Customer satisfaction survey conducted by the company → Primary Data
(b) Information from a published research journal → Secondary Data
(c) Marks recorded by the teacher from a class test → Primary Data
(d) Sales records maintained by another department → Secondary Data

Question 10: The marks obtained by 10 students in a test are given as 45, 62, 38, 55, 70, 62, 48, 55, 75, 40. Arrange the data in ascending order.

To organise the data, arrange the observations from the smallest value to the largest value.
Ascending order: 38, 40, 45, 48, 55, 55, 62, 62, 70, 75

Question 11: The number of books read by students in a month is given below 2, 3, 2, 5, 4, 3, 2, 6, 5, 4. Prepare a frequency table for the given data.

Number of Books	2	3	4	5	6
Frequency	3	2	2	2	1

Question 12: The ages of 15 participants in a survey are 12, 15, 18, 21, 14, 16, 19, 13, 17, 20, 15, 18, 14, 16, 22
Classify the data into the following age groups 12–14, 15–17, 18–20, 21–23

Age Group	12–14	15–17	18–20	21–23
Frequency	4	5	4	2

Question 13: The daily number of customers visiting a store for 12 days is 25, 18, 30, 22, 25, 35, 18, 28, 30, 22, 25, 40
Organise the data by preparing a frequency table.

Number of Customers	18	22	25	28	30	35	40
Frequency	2	2	3	1	2	1	1

Common Mistakes

Confusing the Meaning of Data and Information. Students often treat every number as data without understanding that data is a collection of facts, figures or observations collected for a specific purpose.
Confusing Qualitative and Quantitative Data. Students often classify descriptive information as numerical data. Remember that qualitative data represents categories or qualities, while quantitative data represents numbers that can be measured or counted.
Confusing Discrete and Continuous Data. Students sometimes consider all numerical data as discrete. Discrete data consists of countable values, whereas continuous data consists of measurable values that can take any value within a range.
Mixing Up Population and Sample. Students often consider the selected group as the population. Remember that the population is the complete group being studied, while the sample is a smaller group selected from the population for analysis.
Considering a Sample as the Entire Population. A sample represents only a part of the population. Conclusions drawn from a sample are used to understand the characteristics of the larger population.
Confusing Observations with Variables. Students often identify individual values as variables. A variable is the characteristic being measured, while observations are the actual recorded values of that variable.
Confusing Primary and Secondary Data. Students often assume that all collected information is primary data. Data collected directly by the investigator is primary data, while data obtained from existing sources is secondary data.
Ignoring the Source of Data. The same type of information can be primary or secondary depending on how it was obtained. For example, marks collected directly from students are primary data, while marks taken from school records are secondary data.
Incorrectly Organising Data. Students often arrange data randomly instead of following a systematic order. Data should be organised using ascending order, descending order, frequency tables or suitable classifications.
Incorrectly Counting Frequency. Students sometimes miss repeated values while preparing a frequency table. Each observation must be counted carefully to determine the correct frequency.
Creating Incorrect Class Intervals. While classifying data into groups, students often create overlapping or incomplete intervals. Each observation should belong to only one class interval, and all values should be included.
Ignoring the Total Number of Observations. The sum of all frequencies in a frequency table should always be equal to the total number of observations in the given data set.

Practice Questions

Question 1: Classify the following data as Qualitative or Quantitative:
(a) Favourite colour of students in a class
(b) Height of players in a football team
(c) Number of siblings of a student
(d) Blood group of patients

Question 2: A university has 15,000 students. A survey is conducted by selecting 500 students to study their preferred learning methods.
Identify:
(a) Population
(b) Sample

Question 3: The number of calls received by a customer service centre on 10 days is
12, 15, 12, 18, 15, 20, 12, 18, 15, 22
Prepare a frequency table.