Class 11 Commerce Statistics
First Internal Exam 2025 — Answers from the uploaded question-paper photos
SECTION B — One-sentence answers
Q11. State the empirical relation between mean, median and mode.
Answer: Mode = 3 Median − 2 Mean.
Q12. The 25th and 75th percentiles are 72.18 and 103.99. Find the quartile deviation.
Q1 = 72.18 and Q3 = 103.99.
Quartile deviation = (Q3 − Q1)/2 = (103.99 − 72.18)/2 = 15.905.
Answer: 15.905.
Q13. What is data?
Answer: Data are facts and figures collected for a specific purpose.
Q14. Define a discrete variable.
Answer: A discrete variable takes distinct, countable values.
Q15. Which is the best measure of dispersion?
Answer: Standard deviation is generally considered the best measure of dispersion.
SECTION C — Short answers
Q16. Prepare a ‘less than’ type cumulative frequency distribution.
Observation limit Frequency Cumulative frequency
Less than 10 8 8
Less than 20 82 90
Less than 30 33 123
Less than 40 37 160
Less than 50 44 204
The cumulative frequencies are found by successively adding the frequencies.
Q17. Explain two-dimensional diagrams in brief.
Answer: Two-dimensional diagrams represent data using areas, such as rectangles, squares and circles. The area of
each diagram is proportional to the value represented.
Q18. What is a questionnaire?
Answer: A questionnaire is a set of written questions prepared to collect information from respondents.
Q19. Mean = 5.5 and mode = 6.4. Find the median.
Mode = 3 Median − 2 Mean
6.4 = 3 Median − 11; therefore 3 Median = 17.4.
Median = 17.4/3 = 5.8.
Q20. Define secondary data.
Answer: Secondary data are data already collected by someone else and then used by a researcher.
Q21. Find range and coefficient of range for 6, 11, −3, 0, 8.
Largest value = 11; smallest value = −3.
Range = 11 − (−3) = 14.
Coefficient of range = [11 − (−3)]/[11 + (−3)] = 14/8 = 1.75.
Class 11 Commerce Statistics — First Internal Exam 2025 Page 1
, SECTION D — Detailed answers
Q22. Explain the use of mode as a measure of central tendency.
Definition: Mode is the value that occurs most frequently in a distribution.
• Easy to understand: It is simple to identify and explain.
• Most common value: It identifies the value occurring most often.
• Business use: It helps identify popular products, sizes and designs.
• Qualitative data: It can be used for categories such as colour, brand and type.
• Less affected by extreme values: Very large or small observations generally do not change the mode.
• Decision-making: It helps manufacturers and shopkeepers understand customer preferences.
Conclusion: Mode is useful for identifying the most common or popular value.
Q23. Obtain the original frequency distribution from the mid-values.
The difference between consecutive mid-values is 100, so the class width is 100. Half-width = 50. Each class is formed
by adding and subtracting 50 from its mid-value.
Class interval Frequency
200–300 20
300–400 80
400–500 80
500–600 40
600–700 60
700–800 20
Total 300
Q24. What is standard deviation? Why is it considered the best measure of dispersion?
Definition: Standard deviation is the positive square root of the mean of squared deviations from the arithmetic mean.
For individual observations: σ = √[Σ(x − x■)²/N].
• It is based on all observations.
• It has a precise mathematical definition.
• It is suitable for further algebraic and statistical calculations.
• It is generally relatively stable across samples.
• It helps compare variability when datasets are appropriately comparable.
• It is widely used in correlation, regression and other statistical methods.
Conclusion: Standard deviation is one of the most useful and widely used measures of dispersion.
Q25. Find the mean deviation of the number of goals.
Goals (x) Matches (f) fx |x − 3| f|x − 3|
1 1 1 2 2
2 4 8 1 4
3 6 18 0 0
4 4 16 1 4
5 1 5 2 2
Total 16 48 12
Class 11 Commerce Statistics — First Internal Exam 2025 Page 2
First Internal Exam 2025 — Answers from the uploaded question-paper photos
SECTION B — One-sentence answers
Q11. State the empirical relation between mean, median and mode.
Answer: Mode = 3 Median − 2 Mean.
Q12. The 25th and 75th percentiles are 72.18 and 103.99. Find the quartile deviation.
Q1 = 72.18 and Q3 = 103.99.
Quartile deviation = (Q3 − Q1)/2 = (103.99 − 72.18)/2 = 15.905.
Answer: 15.905.
Q13. What is data?
Answer: Data are facts and figures collected for a specific purpose.
Q14. Define a discrete variable.
Answer: A discrete variable takes distinct, countable values.
Q15. Which is the best measure of dispersion?
Answer: Standard deviation is generally considered the best measure of dispersion.
SECTION C — Short answers
Q16. Prepare a ‘less than’ type cumulative frequency distribution.
Observation limit Frequency Cumulative frequency
Less than 10 8 8
Less than 20 82 90
Less than 30 33 123
Less than 40 37 160
Less than 50 44 204
The cumulative frequencies are found by successively adding the frequencies.
Q17. Explain two-dimensional diagrams in brief.
Answer: Two-dimensional diagrams represent data using areas, such as rectangles, squares and circles. The area of
each diagram is proportional to the value represented.
Q18. What is a questionnaire?
Answer: A questionnaire is a set of written questions prepared to collect information from respondents.
Q19. Mean = 5.5 and mode = 6.4. Find the median.
Mode = 3 Median − 2 Mean
6.4 = 3 Median − 11; therefore 3 Median = 17.4.
Median = 17.4/3 = 5.8.
Q20. Define secondary data.
Answer: Secondary data are data already collected by someone else and then used by a researcher.
Q21. Find range and coefficient of range for 6, 11, −3, 0, 8.
Largest value = 11; smallest value = −3.
Range = 11 − (−3) = 14.
Coefficient of range = [11 − (−3)]/[11 + (−3)] = 14/8 = 1.75.
Class 11 Commerce Statistics — First Internal Exam 2025 Page 1
, SECTION D — Detailed answers
Q22. Explain the use of mode as a measure of central tendency.
Definition: Mode is the value that occurs most frequently in a distribution.
• Easy to understand: It is simple to identify and explain.
• Most common value: It identifies the value occurring most often.
• Business use: It helps identify popular products, sizes and designs.
• Qualitative data: It can be used for categories such as colour, brand and type.
• Less affected by extreme values: Very large or small observations generally do not change the mode.
• Decision-making: It helps manufacturers and shopkeepers understand customer preferences.
Conclusion: Mode is useful for identifying the most common or popular value.
Q23. Obtain the original frequency distribution from the mid-values.
The difference between consecutive mid-values is 100, so the class width is 100. Half-width = 50. Each class is formed
by adding and subtracting 50 from its mid-value.
Class interval Frequency
200–300 20
300–400 80
400–500 80
500–600 40
600–700 60
700–800 20
Total 300
Q24. What is standard deviation? Why is it considered the best measure of dispersion?
Definition: Standard deviation is the positive square root of the mean of squared deviations from the arithmetic mean.
For individual observations: σ = √[Σ(x − x■)²/N].
• It is based on all observations.
• It has a precise mathematical definition.
• It is suitable for further algebraic and statistical calculations.
• It is generally relatively stable across samples.
• It helps compare variability when datasets are appropriately comparable.
• It is widely used in correlation, regression and other statistical methods.
Conclusion: Standard deviation is one of the most useful and widely used measures of dispersion.
Q25. Find the mean deviation of the number of goals.
Goals (x) Matches (f) fx |x − 3| f|x − 3|
1 1 1 2 2
2 4 8 1 4
3 6 18 0 0
4 4 16 1 4
5 1 5 2 2
Total 16 48 12
Class 11 Commerce Statistics — First Internal Exam 2025 Page 2
