Statistics - Collection and Presentation
Statistics is the branch of mathematics that deals with the collection, organisation, analysis, interpretation, and presentation of data.
In Class 9, statistics focuses on understanding primary and secondary data, organising data into frequency distribution tables, and presenting data using graphical methods such as bar graphs, histograms, and frequency polygons.
The word "statistics" comes from the Latin word status, meaning a political state. Originally, it referred to the collection of data useful to the state, such as population, income, and military strength.
What is Statistics - Collection and Presentation?
Definition: Statistics is the science of collecting, organising, analysing, and interpreting numerical data to make informed decisions.
Data: Facts or figures collected for a specific purpose. Each piece of data is called an observation.
Types of Data:
- Primary Data: Data collected directly by the investigator for a specific purpose. It is original and first-hand.
- Secondary Data: Data that has already been collected by someone else and is being reused. Sources include books, websites, government records.
Types of Presentation:
- Tabular form: Frequency distribution tables
- Graphical form: Bar graphs, histograms, frequency polygons
- Numerical measures: Mean, median, mode
Statistics - Collection and Presentation Formula
Key Formulas and Definitions:
1. Range of Data:
Range = Maximum value − Minimum value
2. Class Width (or Class Size):
- Class width = Upper class limit − Lower class limit
3. Class Mark (Mid-value):
Class mark = (Upper limit + Lower limit) / 2
4. Mean of ungrouped data:
- Mean = Sum of all observations / Number of observations
Derivation and Proof
Steps in Statistical Analysis:
Step 1: Collection of Data
- Decide the purpose of the study.
- Collect data through surveys, experiments, or observations.
- Primary data is collected directly; secondary data is obtained from existing sources.
Step 2: Organisation of Data
- Arrange raw data in ascending or descending order.
- Create a frequency distribution table (ungrouped or grouped).
Step 3: Presentation of Data
- Ungrouped frequency table: Lists each value and its frequency.
- Grouped frequency table: Groups data into class intervals with frequencies.
- Represent graphically using bar graphs, histograms, or frequency polygons.
Step 4: Analysis of Data
- Calculate measures of central tendency: mean, median, mode.
- Draw conclusions from the analysis.
Types and Properties
Types of Data and Frequency Distributions:
1. Raw Data
- Data in its original, unorganised form as collected.
- Example: Test marks of 30 students listed in any order.
2. Ungrouped Frequency Distribution
- Each distinct value is listed with its frequency (number of times it occurs).
- Suitable when the range of data is small.
3. Grouped Frequency Distribution
- Data is divided into class intervals (groups).
- Each class interval has a corresponding frequency.
- Suitable when the range of data is large.
4. Cumulative Frequency Distribution
- Shows the cumulative total of frequencies up to each class boundary.
- Two types: "less than" and "more than" cumulative frequency.
5. Continuous vs Discrete Data
- Discrete data: Takes only specific values (e.g., number of students).
- Continuous data: Can take any value in a range (e.g., height, weight).
Solved Examples
Example 1: Example 1: Identify primary and secondary data
Problem: Classify the following as primary or secondary data:
- (a) A teacher records the attendance of students in her class.
- (b) A student uses census data from the government website for a project.
Solution:
- (a) Primary data — collected directly by the teacher for her own purpose.
- (b) Secondary data — already collected by the government and reused by the student.
Example 2: Example 2: Create an ungrouped frequency table
Problem: The blood groups of 20 students are: A, B, O, A, AB, O, A, B, O, A, B, A, O, B, AB, A, O, B, A, O.
Prepare a frequency distribution table.
Solution:
Frequency Table:
- A → 7
- B → 5
- O → 6
- AB → 2
- Total = 20 ✓
Answer: Blood group A is the most common (frequency 7).
Example 3: Example 3: Create a grouped frequency table
Problem: The marks of 25 students (out of 50) are: 12, 25, 38, 42, 17, 30, 22, 48, 35, 19, 40, 27, 33, 15, 45, 21, 36, 28, 44, 10, 31, 23, 39, 46, 14. Prepare a grouped frequency table with class intervals of width 10.
Solution:
- 10 – 20: 12, 17, 19, 15, 10, 14 → Frequency = 6
- 20 – 30: 25, 22, 27, 21, 28, 23 → Frequency = 6
- 30 – 40: 38, 30, 35, 33, 36, 31, 39 → Frequency = 7
- 40 – 50: 42, 48, 40, 45, 44, 46 → Frequency = 6
- Total = 25 ✓
Answer: The class 30–40 has the highest frequency (7 students).
Example 4: Example 4: Find range and class mark
Problem: For the class interval 20 – 30, find the class mark, lower limit, upper limit, and class width.
Solution:
- Lower limit = 20
- Upper limit = 30
- Class width = 30 − 20 = 10
- Class mark = (20 + 30) / 2 = 25
Answer: Class mark = 25, class width = 10.
Example 5: Example 5: Convert discontinuous to continuous class intervals
Problem: Convert the following discontinuous class intervals to continuous: 1–5, 6–10, 11–15, 16–20.
Solution:
Adjustment: Gap between classes = 6 − 5 = 1. Adjustment factor = 1/2 = 0.5.
- Subtract 0.5 from lower limits and add 0.5 to upper limits.
- 0.5 – 5.5, 5.5 – 10.5, 10.5 – 15.5, 15.5 – 20.5
Answer: Continuous classes are 0.5–5.5, 5.5–10.5, 10.5–15.5, 15.5–20.5.
Example 6: Example 6: Find the range of data
Problem: The heights (in cm) of 10 students are: 135, 142, 150, 128, 147, 155, 132, 148, 140, 138. Find the range.
Solution:
- Maximum = 155 cm
- Minimum = 128 cm
- Range = 155 − 128 = 27 cm
Answer: The range is 27 cm.
Example 7: Example 7: Tally marks and frequency
Problem: The number of goals scored in 20 football matches are: 0, 1, 2, 1, 3, 0, 2, 1, 0, 4, 1, 2, 3, 1, 0, 2, 1, 3, 0, 1. Prepare a frequency table using tally marks.
Solution:
- 0 goals: |||| → 5 matches
- 1 goal: |||| || → 7 matches
- 2 goals: |||| → 4 matches
- 3 goals: ||| → 3 matches
- 4 goals: | → 1 match
- Total = 20 ✓
Answer: The most common score is 1 goal (7 matches).
Example 8: Example 8: Identify data collection method
Problem: A company wants to know customer satisfaction. Which is better: sending a survey (primary) or reading online reviews (secondary)?
Solution:
- Survey (Primary data): More reliable, specific to the company's questions, but time-consuming and costly.
- Online reviews (Secondary data): Quick and free, but may be biased, not specific to the company's needs, and not verifiable.
Answer: A survey provides primary data that is more reliable and directly relevant. Online reviews provide secondary data that is quicker to access but less reliable.
Real-World Applications
Applications of Statistics:
- Census: Government collects population data for planning public services, budgets, and infrastructure.
- Weather Forecasting: Meteorologists analyse temperature, rainfall, and wind data to predict weather patterns.
- Sports: Player statistics (batting average, goals scored, assists) are used for team selection and strategy.
- Medicine: Clinical trials collect data on drug effectiveness and side effects.
- Economics: GDP, inflation, employment data guide government policy and business decisions.
- Education: Exam results are analysed to evaluate teaching methods and student performance.
Key Points to Remember
- Statistics deals with collection, organisation, analysis, and interpretation of data.
- Primary data is collected first-hand; secondary data is obtained from existing sources.
- Range = Maximum value − Minimum value.
- Data is organised into ungrouped or grouped frequency distribution tables.
- Class mark = (Upper limit + Lower limit) / 2.
- Class width = Upper limit − Lower limit.
- Discontinuous class intervals must be converted to continuous before drawing histograms.
- Graphical representations include bar graphs, histograms, and frequency polygons.
- Measures of central tendency are mean, median, and mode.
- This is covered in NCERT Class 9, Chapter 14 (Statistics).
Practice Problems
- Classify the following as primary or secondary data: (a) Recording rainfall daily at your school. (b) Using data from a newspaper article for a project.
- The ages of 15 players in a team are: 22, 25, 30, 22, 28, 35, 22, 30, 28, 25, 35, 22, 30, 28, 25. Prepare an ungrouped frequency distribution table.
- Prepare a grouped frequency table for the following data (class width 10): 5, 12, 18, 25, 32, 8, 15, 22, 29, 35, 42, 48, 11, 27, 38.
- Find the range, class mark of the class 40-50, and verify that the sum of frequencies equals the total number of observations for your table.
- The weights (in kg) of 20 students are given in the class intervals 30-35, 35-40, 40-45, 45-50, 50-55. If the frequencies are 2, 5, 8, 3, 2 respectively, find the class marks for each interval.
- Give two examples each of primary data and secondary data from everyday life.
Frequently Asked Questions
Q1. What is statistics in mathematics?
Statistics is the branch of mathematics that deals with collecting, organising, analysing, and interpreting numerical data to draw conclusions and make decisions.
Q2. What is the difference between primary and secondary data?
Primary data is collected directly by the investigator for a specific purpose. Secondary data has already been collected by someone else and is reused from sources like books, websites, or government records.
Q3. What is raw data?
Raw data is data in its original, unorganised form as it was collected. It has not been arranged, sorted, or grouped.
Q4. What is a frequency distribution table?
A frequency distribution table organises data by listing each value (or class interval) along with the number of times it occurs (frequency). It can be ungrouped (individual values) or grouped (class intervals).
Q5. What is the range of data?
The range is the difference between the maximum and minimum values in a data set. It gives a measure of how spread out the data is.
Q6. What is a class mark?
The class mark is the midpoint of a class interval, calculated as (upper limit + lower limit) / 2. It represents the central value of the class.
Q7. Why is statistics important?
Statistics helps in making informed decisions by analysing data. It is used in government planning, medical research, business strategy, weather forecasting, sports analysis, and education.
Q8. Is this topic in the CBSE Class 9 syllabus?
Yes. Statistics is covered in Chapter 14 of NCERT Class 9 Mathematics. It includes data collection, presentation, grouped frequency, histograms, frequency polygons, and measures of central tendency.
Related Topics
- Frequency Distribution Table
- Histogram of Grouped Data
- Frequency Polygon
- Mean of Grouped Data
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Median of Grouped Data
- Mode of Grouped Data
- Cumulative Frequency Distribution
- Ogive (Cumulative Frequency Curve)
- Empirical Relationship Between Mean, Median, Mode
- Mean by Assumed Mean Method
- Mean by Step-Deviation Method
- Statistics in Real Life
