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Introduction to Statistics

Key Concepts

Solved Examples: 40

Practice Questions: 3

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

Question 14: A school wants to study the daily number of hours spent studying by its students. The researcher records the study time of 80 randomly selected students. What is the statistical variable being studied?

The statistical variable is the daily study time of the students, measured in hours.
∴ Statistical variable = Daily study time (in hours).

Question 15: A library records the number of books borrowed by each member during a month. What is the statistical variable being studied?

The statistical variable is the number of books borrowed by each member.
∴ Statistical variable = Number of books borrowed per month.

Question 16: A university has 8,000 students. A researcher wants to study the average time students spend using the university library and selects 400 students from different departments for the study. Identify the population and the sample.

The population is the complete group about which the researcher wants information.
∴ Population = All 8,000 students of the university
The sample is the smaller group selected for the study.
∴ Sample = 400 selected students

Question 17: A school district has 50 schools and wants to estimate the average mathematics score of all Grade 10 students. The district selects 10 schools and uses the scores of all Grade 10 students in those schools. Identify the population and the sample

The population consists of all Grade 10 students in the school district.
∴ Population = All Grade 10 students in the 50 schools
The sample consists of the Grade 10 students from the 10 selected schools.
∴ Sample = All Grade 10 students in the 10 selected schools

Question 18: A researcher wants to study the average daily screen time of teenagers. She asks 300 teenagers to record their screen time for seven days and collects their records. Is the data primary or secondary?

The researcher collects the information directly from the teenagers for her specific study.
∴ Primary Data

Question 19: A company wants to study changes in the population of cities where it operates. It uses population figures published in government census reports. Is the data primary or secondary?

The population figures were already collected and published by the government and are being used by the company for its study.
∴ Secondary Data

Question 20: A school wants to compare the attendance of its students over the past five years. It uses attendance records maintained by the school during those years. Is the data primary or secondary?

The school is using records that were already collected and maintained before the current study.
∴ Secondary Data

Question 21: A researcher studying road safety visits several busy roads and personally counts the number of vehicles passing each road during selected one-hour periods. Is the data primary or secondary?

The researcher collects the information directly through observation for the study.
∴ Primary Data

Question 22: A survey records the blood groups of 500 students using the labels A, B, AB and O. Is the data qualitative or quantitative?

Blood groups represent categories or labels, even though they may be written using letters.
∴ Qualitative Data

Question 23: A school assigns each student a unique identification number and records these numbers for a statistical study. Is the data qualitative or quantitative?

The identification numbers are used only as labels to identify students. They do not represent a numerical quantity.
∴ Qualitative Data

Question 24: A survey records the ratings given by customers to a product as 1, 2, 3, 4 or 5 stars. Is the data qualitative or quantitative?

The ratings are numerical values representing the level of satisfaction and can be compared numerically.
∴ Quantitative Data

Question 25: A hospital records the number of patients admitted to its emergency department each day. Is the data discrete or continuous?

The number of patients is obtained by counting and can take only whole-number values.
∴ Discrete Data

Question 26: A laboratory records the amount of water in different containers, measured in millilitres. Is the data discrete or continuous?

The amount of water is obtained by measurement and can take different values within a range.
∴ Continuous Data

Question 27: A teacher records the number of questions answered correctly by each student in a test containing 50 questions. Is the data discrete or continuous?

The number of correctly answered questions is obtained by counting and can take only whole-number values from 0 to 50.
∴ Discrete Data

Question 28: A school records the heights, weights and ages of 100 students to study their physical characteristics. What is the main purpose of collecting this data?

The data is being collected to study and understand the physical characteristics of the students.
∴ Purpose = To study the physical characteristics of the students

Question 29: A transport department records the daily number of buses arriving at a station and the average waiting time of passengers. What is the purpose of collecting this data?

The data is being collected to study bus service patterns and passenger waiting times.
∴ Purpose = To study bus service patterns and passenger waiting times

Question 30: A software platform asks users to rate their experience after a system update by selecting one of the following options: Poor, Satisfactory, Good, or Excellent. Identify the data classification type of the feedback variable.

The responses represent categories rather than numerical quantities.
∴ Data type = Qualitative Data
The responses are categories or labels.
∴ Variable type = Categorical Variable
The categories have a natural order, but they are still categorical rather than numerical.
∴ Answer = Qualitative Data and Categorical Variable

Question 31: Student identification numbers such as 1001, 1002 and 1003 are assigned to students participating in an examination. Classify this data as Qualitative or Quantitative and explain your answer.

Although student identification numbers are written using digits, they are used only to identify individual students. They do not represent a quantity that can be measured or counted.
For example, ID 1002 does not represent twice the quantity represented by ID 501.
∴ Qualitative Data
The identification numbers function as labels for the students.
∴ Variable type = Categorical Variable
The student identification numbers represent Qualitative Data and form a Categorical Variable because they are used as identification labels rather than numerical quantities.

Question 32: A researcher wants to study the average daily screen time of all high school teenagers in a large city. The researcher surveys 100 students entering a local video game arcade on a Saturday afternoon. Identify:
(a) The population
(b) The sample

(a) Population = All high school teenagers in the city
(b) Sample = The 100 students surveyed at the video game arcade

Question 33: A quality control inspector wants to study the quality of toys produced at a factory. Instead of examining every toy produced, the inspector selects 50 toys from the factory’s production line for testing. Identify:
(a) The population
(b) The sample

(a) Population = All toys produced by the factory
(b) Sample = The 50 toys selected for testing

Question 34: An agricultural investigator visits 15 farms to directly measure soil moisture levels using a digital sensor. Later, the investigator uses historical annual rainfall data published by the national meteorological department. Classify the two sources of data.

Soil moisture measurements → Primary Data
Published historical rainfall data → Secondary Data

Question 35: A retail corporation records customer checkout waiting times using timestamps from its cash registers. A university student later uses a summary of these waiting times published in the corporation’s annual report. Classify the data as Primary or Secondary from the perspective of each user.

For the corporation → Primary Data
For the student → Secondary Data

Question 36: A town council wants to study traffic congestion. It hires a consulting firm that collects vehicle counts at major intersections and also obtains historical accident records from the local police. Classify each source of data.

(a) Vehicle counts collected by the consulting firm’s staff → Primary Data
(b) Historical accident records obtained from the police → Secondary Data

Question 37: Classify each of the following quantitative variables as Discrete or Continuous data:
(a) Number of pages in a textbook
(b) Weight of a bag of rice
(c) Number of goals scored by a football player in a tournament

(a) Number of pages → Discrete Data
(b) Weight of a bag of rice → Continuous Data
(c) Number of goals scored → Discrete Data

Question 38: A laboratory records the exact time taken by a chemical solution to change colour during an experiment. The computer displays the time rounded to the nearest whole second. Is the underlying data Discrete or Continuous?

The time taken is obtained by measurement and can take fractional values.
∴ Continuous Data
Rounding the displayed value does not change the underlying nature of the variable.

Question 39: Classify each of the following as Discrete or Continuous data:
(a) Daily cash revenue of a shop, recorded to the nearest ¤0.01
(b) Total number of points scored by a team in a quiz tournament
(c) Age of a living tree measured from a fixed reference date

(a) Daily cash revenue → Discrete Data
(b) Total points scored → Discrete Data
(c) Age of a tree → Continuous Data

Question 40: A student wants to quickly distinguish between Discrete and Continuous data. What is the basic difference between counting and measuring?

Counting produces distinct, countable values and usually answers “How many?” → Discrete Data
Measuring produces values that can vary continuously within a range and usually answers “How much?”, “How long?” or “How heavy?” → Continuous Data
∴ Counting → Discrete Data; Measuring → Continuous Data

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. It may consist of numerical values, measurements or descriptive information and can be analysed to obtain meaningful information.
Data is broadly classified into the following types

Qualitative Data
- Describes qualities or characteristics rather than numerical quantities.
- 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 numerical values, usually obtained by counting.
  - 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 population. In such cases, statisticians study a smaller representative group called a sample.

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 can be classified as Primary Data or Secondary Data.

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 by someone else and is used for a different study or purpose.
- Examples:
  - Census reports
  - Government publications
  - Research journals
  - School records

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

Organisation and Classification of Data

Raw data is often difficult to interpret because it is usually unorganised. Statistics helps organise 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 patterns, compare values and draw meaningful 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.

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.