Maharashtra Board Class 11 Economics Chapter 3 Partition Values Solutions

Choose the Correct Option:

 

Question 1. Statements that do not apply to Quartiles.
(a) First arrange the values in ascending or descending order.
(b) Observation can be divided into 4 parts.
(c) They are represented as \( Q_1 \), \( Q_2 \), and \( Q_3 \).
(d) None of the options
Answer: (d) None of the options
In simple words: All the statements (a, b, and c) are actually true for quartiles, which means none of them are incorrect.

🎯 Exam Tip: Remember that quartiles divide a dataset into four equal parts using three partition points: \( Q_1 \), \( Q_2 \), and \( Q_3 \). Since all these statements are correct, none of them "do not apply".

Question 1. Choose the correct option based on the statements:
(d) \( Q_2 \) is also known as the median.
Options:
(a) a
(b) b and c
(c) a, b and c
(d) None of the options
Answer: (d) None of the options
In simple words: None of the given combinations of statements represent the correct answer for this question.

🎯 Exam Tip: Carefully read all the statements before choosing 'None of the options' to ensure no correct combination is missed.

 

Question 2. Find \( D_7 \) from the given data:
Data: 4, 5, 6, 7, 8, 9, 10, 11, 12
Options:
(a) 7
(b) 9
(c) 10
(d) 12
Answer: (a) 7
In simple words: The 7th decile (\( D_7 \)) represents the value below which 70% of the data falls. By calculating its position in the ordered list of 9 numbers, we find the corresponding value.

🎯 Exam Tip: Always arrange the data in ascending order first before calculating any partition values like deciles or quartiles.

 

Question 3. Statements related to partition values that are correct:
(a) Exact divisions of percentiles into 100 parts gives 99 points
(b) Deciles have total of 9 parts
(c) Quartiles are shown by \( Q_1 \), \( Q_2 \), and \( Q_3 \)
(d) Symbolically, Percentiles and Deciles are shown by P and D
Options:
(a) a and c
(b) a and b
(c) a, b and c
(d) a, c and d
Answer: (d) a, c and d
In simple words: Percentiles divide data into 100 parts using 99 points, quartiles are represented by \( Q_1, Q_2, Q_3 \), and percentiles/deciles are denoted by P and D. Deciles actually divide data into 10 parts using 9 points, making statement (b) incorrect.

🎯 Exam Tip: Remember that partition values always have one less point than the number of parts they divide the data into (e.g., 4 parts need 3 quartiles, 10 parts need 9 deciles).

 

Identify the Correct Pairs from the Given Options:

Question 1. Match the following:
Group A
1) Quartiles
2) Deciles
3) Percentiles

Group B
a) \( D_j = \text{size of } j \left( \frac{n+1}{10} \right)^{\text{th Observation}} \)
b) \( P_k = l + \left( \frac{\frac{kn}{100} - cf}{f} \right) \times h \)
c) \( Q_i = l + \left( \frac{\frac{in}{4} - cf}{f} \right) \times h \)

Options:
(a) 1-b, 2-c, 3-a
(b) 1-c, 2-a, 3-b
(c) 1-c, 2-b, 3-a
(d) 1-a, 2-b, 3-c
Answer: (b) 1-c, 2-a, 3-b
In simple words: Quartiles divide data into 4 parts (formula c), deciles divide data into 10 parts (formula a), and percentiles divide data into 100 parts (formula b).

🎯 Exam Tip: Remember the division factors: Quartiles use 4, Deciles use 10, and Percentiles use 100 in their respective formulas to easily match them.

Give Economic Terms:

 

Question 1. Procedure for dividing the data into equal parts.
Answer: Partitioning. This mathematical process helps economists and statisticians analyze specific segments of a large dataset more effectively.
In simple words: Partitioning means cutting a big group of numbers into smaller, equal-sized groups so they are easier to study.

🎯 Exam Tip: Write the exact term 'Partitioning' clearly, as spelling mistakes in economic terms can lead to a loss of marks.

 

Question 2. The value that divides the series into ten equal parts.
Answer: Deciles. These partition values are widely used in economic studies to measure income distribution and inequality across populations.
In simple words: Deciles are the specific values that split a long list of numbers into ten equal sections.

🎯 Exam Tip: Remember that 'Deci' stands for ten, which will help you instantly connect this term to ten equal parts.

 

Question 3. The value that divides the whole set of observations into four equal parts.
Answer: Quartiles. There are three quartiles, denoted as \( Q_1 \), \( Q_2 \), and \( Q_3 \), which split the data into four equal quarters.
In simple words: Quartiles are the values that divide a set of data into four equal quarters, just like four quarters make a whole dollar.

🎯 Exam Tip: Do not confuse quartiles with deciles or percentiles; always associate 'Quartile' with division into four parts.

Solve the Following:

 

Question 1. Calculate \( Q_1 \), \( D_4 \), and \( P_{26} \) for the following data:
18, 24, 45, 29, 4, 7, 28, 49, 16, 26, 25, 12, 10, 9, 8
Answer: First, arrange the given data in ascending order:
4, 7, 8, 9, 10, 12, 16, 18, 24, 25, 26, 28, 29, 45, 49
Here, the total number of observations \( N = 15 \).

1) Calculation of First Quartile (\( Q_1 \)):
\( Q_1 = \text{size of } \left( \frac{N+1}{4} \right)^{\text{th}} \text{ observation} \)

\( \implies Q_1 = \text{size of } \left( \frac{15+1}{4} \right)^{\text{th}} \text{ observation} \)

\( \implies Q_1 = \text{size of } \left( \frac{16}{4} \right)^{\text{th}} \text{ observation} \)

\( \implies Q_1 = \text{size of } 4^{\text{th}} \text{ observation} \)
Since the \( 4^{\text{th}} \) observation in the arranged data is 9,

\( \implies Q_1 = 9 \)

2) Calculation of Fourth Decile (\( D_4 \)):
\( D_4 = \text{size of } 4 \left( \frac{N+1}{10} \right)^{\text{th}} \text{ observation} \)

\( \implies D_4 = \text{size of } 4 \left( \frac{15+1}{10} \right)^{\text{th}} \text{ observation} \)

\( \implies D_4 = \text{size of } 4 \left( \frac{16}{10} \right)^{\text{th}} \text{ observation} \)

\( \implies D_4 = \text{size of } 4 \times 1.6^{\text{th}} \text{ observation} \)

\( \implies D_4 = \text{size of } 6.4^{\text{th}} \text{ observation} \)

\( \implies D_4 = 6^{\text{th}} \text{ observation} + 0.4 \times (7^{\text{th}} \text{ observation} - 6^{\text{th}} \text{ observation}) \)
Since the \( 6^{\text{th}} \) observation is 12 and the \( 7^{\text{th}} \) observation is 16,

\( \implies D_4 = 12 + 0.4 \times (16 - 12) \)

\( \implies D_4 = 12 + 0.4 \times 4 \)

\( \implies D_4 = 12 + 1.6 \)

\( \implies D_4 = 13.6 \)

3) Calculation of Twenty-Sixth Percentile (\( P_{26} \)):
\( P_{26} = \text{size of } 26 \left( \frac{N+1}{100} \right)^{\text{th}} \text{ observation} \)

\( \implies P_{26} = \text{size of } 26 \left( \frac{15+1}{100} \right)^{\text{th}} \text{ observation} \)

\( \implies P_{26} = \text{size of } 26 \left( \frac{16}{100} \right)^{\text{th}} \text{ observation} \)

\( \implies P_{26} = \text{size of } 26 \times 0.16^{\text{th}} \text{ observation} \)

\( \implies P_{26} = \text{size of } 4.16^{\text{th}} \text{ observation} \)

\( \implies P_{26} = 4^{\text{th}} \text{ observation} + 0.16 \times (5^{\text{th}} \text{ observation} - 4^{\text{th}} \text{ observation}) \)
Since the \( 4^{\text{th}} \) observation is 9 and the \( 5^{\text{th}} \) observation is 10,

\( \implies P_{26} = 9 + 0.16 \times (10 - 9) \)

\( \implies P_{26} = 9 + 0.16 \times 1 \)

\( \implies P_{26} = 9.16 \)

Thus, the calculated values are \( Q_1 = 9 \), \( D_4 = 13.6 \), and \( P_{26} = 9.16 \).
In simple words: To find these partition values, we first arrange the numbers from smallest to largest, and then use the formulas to find the exact positions of the first quarter, the fourth tenth, and the twenty-sixth hundredth parts.

🎯 Exam Tip: Always arrange the data in ascending order first; forgetting this step is the most common mistake students make in partition value problems.

 

Question 2. Calculate \( Q_3 \), \( D_5 \), and \( P_{35} \) for the given data.

Income (in lakhs Rs.)	1	2	3	4	5	6
No. of family	2	5	20	25	15	12

Answer: \( Q_3 \) = Rs. 5 Lakhs, \( D_5 \) = Rs. 4 Lakhs, \( P_{35} \) = Rs. 4 Lakhs. These partition values help us understand the distribution of income across different families.
In simple words: These values divide the family income data into different parts. For example, \( Q_3 \) shows the income level below which 75% of the families fall.

🎯 Exam Tip: Always arrange the data in ascending order and calculate cumulative frequencies carefully before applying partition formulas.

 

Question 3. Find out \( P_{50} \) for the following data:

Wages (in Rs.) (x)	Number of workers
0-20	4
20-40	6
40-60	10
60-80	25
80-100	15

Answer: \( P_{50} \) = Rs. 68 Wages. This indicates that half of the workers earn less than this amount while the other half earn more.
In simple words: \( P_{50} \) is the 50th percentile, which is exactly the same as the median. It tells us that 50% of the workers earn Rs. 68 or less.

🎯 Exam Tip: Remember that the 50th percentile (\( P_{50} \)), 5th decile (\( D_5 \)), and 2nd quartile (\( Q_2 \)) are all equal to the median.

 

Question 4. Calculate \( Q_3 \) for the following data:

Sales (in lakhs Rs.)	10-20	20-30	30-40	40-50	50-60	60-70
No. of firms	20	30	70	48	32	50

Answer: \( Q_3 \) = Rs. 56.09 Lakhs. This value represents the upper quartile boundary for the sales of the firms.
In simple words: \( Q_3 \) is the third quartile, meaning 75% of the firms have sales of Rs. 56.09 Lakhs or less.

🎯 Exam Tip: For continuous grouped data, always use the interpolation formula \( Q_3 = L + \left(\frac{\frac{3N}{4} - cf}{f}\right) \times h \) and clearly define each term.

 

Question 5. Calculate \( D_7 \) for the following data.

Profit (in crores Rs.)	10-20	20-30	30-40	40-50	50-60	60-70
No. of firms	20	30	70	48	32	50

Answer: \( D_7 = \text{Rs. } 52.1875 \) crores. This value indicates that 70% of the firms have a profit of Rs. 52.1875 crores or less.
In simple words: \( D_7 \) represents the 7th decile, which divides the data into ten equal parts. It tells us the profit level below which 70% of the firms lie.

🎯 Exam Tip: Always write the final unit (like crores or lakhs) in your final answer to avoid losing marks.

 

Question 6. Calculate \( P_{15} \) for the following data.

Investment (in lakhs Rs.)	0-10	10-20	20-30	30-40	40-50	50-60
No. of firms	5	10	25	30	20	10

Answer: \( P_{15} = \text{Rs. } 20 \) lakhs. This indicates that 15% of the firms have an investment of Rs. 20 lakhs or less.
In simple words: \( P_{15} \) is the 15th percentile, which means 15% of the firms have investments below this value, while the remaining 85% have higher investments.

🎯 Exam Tip: Double-check your cumulative frequency calculations before applying the percentile formula.

5. State with Reasons Whether You Agree or Disagree with the Following Statements:

 

Question 1. Partition values have application only in theory but not in practice.
OR
Partition values are not useful in economics.
Answer: No, I do not agree with the statement. Partition values like quartiles are useful to economists to know the final information related to income, sales, stock data, etc. Also, deciles and percentiles help economists to measure in detail the poverty line, inequality of income, household, wealth, etc. Hence, partition values are highly useful in practical Economics.
In simple words: Partition values are not just theoretical ideas; they are used in real life to study things like poverty, wealth gaps, and business sales.

🎯 Exam Tip: Clearly state "I agree" or "I disagree" at the very beginning of your answer before writing the reasons.

 

Question 2. Average can misinterpret the representative value.
Answer: Yes, I agree with the statement. An average (like the arithmetic mean) can misinterpret the representative value because it is highly affected by extreme values or outliers in the data. For instance, a single extremely high income can make the average income of a group look very high, even if most people in that group earn very little.
In simple words: A simple average can sometimes give a misleading picture if there are extremely high or low numbers in the group.

🎯 Exam Tip: Use a simple numerical example to illustrate how extreme values distort the average.

Question 3. Median is also known as the second quartile.
Answer: Yes, I agree with the statement. Median is the middlemost value in the arranged data. It is the value that divides the series into two equal parts, so that the number of items above it is equal to the number of items below it. It is not affected by extreme values. Since the median divides the data into two equal halves, we can write Median = \( \frac{n}{2} \). Similarly, the second quartile (\( Q_2 \)) divides the data into four equal parts and represents two of those parts, which means \( Q_2 = 2 \left(\frac{n}{4}\right) = \frac{n}{2} \). Therefore, the Median is equal to the second quartile.
In simple words: The median is the exact middle point of a dataset, which splits it into two equal halves. Since the second quartile (Q2) also splits the data exactly at the 50% mark, they are the same thing.

🎯 Exam Tip: Remember that the median, the 5th decile (D5), the 50th percentile (P50), and the second quartile (Q2) all represent the exact same middle value of a dataset.

 

Answer the Following Questions on the Basis of the Given Table:

Marks	30	10	20	40	50
No. of Students	13	4	7	8	6

 

Question 1. Write the formula of \( Q_1 \) and \( Q_3 \).
Answer: The formulas for the first quartile (\( Q_1 \)) and the third quartile (\( Q_3 \)) for a discrete series are calculated based on the total number of observations. These formulas help us locate the exact position of the quartile values in an ordered dataset:
\( Q_1 = \text{size of } \left(\frac{n+1}{4}\right)^{\text{th}} \text{ observation} \)
\( Q_3 = \text{size of } 3\left(\frac{n+1}{4}\right)^{\text{th}} \text;{ observation} \)
In simple words: To find the first quartile (Q1), we look for the value at one-quarter of the way through the sorted data. For the third quartile (Q3), we look for the value three-quarters of the way through.

🎯 Exam Tip: Always remember to add 1 to 'n' before dividing by 4 when working with discrete or individual series formulas.

 

Question 2. Find out the cumulative frequency of the last value in the above data.
Answer: To find the cumulative frequency, we first arrange the data in ascending order of marks and then accumulate the frequencies step-by-step:

Marks (x)	No. of Students (f)	Cumulative Frequency (cf)
10	4	4
20	7	11 (4 + 7)
30	13	24 (11 + 13)
40	8	32 (24 + 8)
50	6	38 (32 + 6)

The last value in the arranged data is 50, and its cumulative frequency is 38.
In simple words: We arrange the marks from lowest to highest and keep adding up the number of students as we go. By the time we reach the last mark (50), the total running sum of students is 38.

🎯 Exam Tip: Always arrange the variable (Marks) in ascending order before calculating cumulative frequencies, otherwise your cumulative totals will be incorrect.

 

Question 3. Find out the value of ‘n’ in the above data.
Answer: n = 38. The value of 'n' represents the total number of observations in the given frequency distribution.
In simple words: 'n' is the total number of people or items we are looking at in the data.

🎯 Exam Tip: Always double-check your calculations by summing up all individual frequencies to find the correct value of 'n'.

 

Question 4. Find out the median of the above data?
Answer: Median = 30. This value represents the middlemost point of the dataset when arranged in ascending order.
In simple words: The median is the middle number in a list of numbers sorted from smallest to largest.

🎯 Exam Tip: Remember to arrange the data in ascending order before finding the median value.

11th Economics Digest Chapter 3 Partition Values Intext Questions And Answers

Do You Know? (Text Book Page No. 14)

Statistics Day: Prof. Prasanta Chandra Mahalanobis, an Indian Statistician was instrumental in formulating India’s strategy for industrialization in the Second Five Year Plan (1956-61) which later came to be known as Mahalanobis Model.

Mahalanobis devised a measure of comparison between two data sets that are known as the Mahalanobis distance. He also devised a statistical method called ‘fractile graphical analysis’ which could be used to compare the socio-economic conditions of different groups of people. In recognition of the notable contributions made by P. C. Mahalanobis in the field of economic planning and statistical development, the Government of India has designated the 29th of June every year, coinciding with his birth anniversary as ‘Statistics Day’, in the category of Special day to be celebrated at the national level.

 

Question 1. Who formulated India’s strategy for industrialization?
Answer: Prof. Prasanta Chandra Mahalanobis, an Indian Statistician formulated India’s strategy for industrialization. His contributions laid a strong foundation for the country's economic planning and development.
In simple words: A famous Indian scientist named Prof. Prasanta Chandra Mahalanobis created the plan to build big industries in India.

🎯 Exam Tip: Mention both his full name and his profession as an Indian Statistician to secure full marks.

 

Question 2. What is the Mahalanobis distance?
Answer: A measure of comparison between two data sets is called Mahalanobis distance. It helps in identifying how similar or different the groups of data are from each other.
In simple words: This distance measures how far apart two different groups of data are. It is very useful in statistics for comparing complex datasets.

🎯 Exam Tip: Remember that Mahalanobis distance is used for comparing datasets, not just single points. Mentioning "data sets" is key to scoring full marks.

 

Question 3. What is fractile graphical analysis?
Answer: Fractile graphical analysis is a statistical method that can be used to compare the socio-economic; conditions of different groups of people. This method was devised by Prof. P.C. Mahalanobis. It provides a visual representation of economic inequalities across various sections of society.
In simple words: This is a visual way to compare the living standards and wealth of different groups of people. It was created by the famous statistician Prof. P.C. Mahalanobis.

🎯 Exam Tip: Always mention the name of Prof. P.C. Mahalanobis as the creator of this method to secure full marks.

 

Question 4. When and why is ‘Statistics Day’ celebrate?
Answer: Statistics Day is celebrated on 29th June every year in recognition of the notable contributions made by Prof. P.C. Mahalanobis in the field of economic planning and statistical development. This special day highlights the critical role of statistics in shaping national policies.
In simple words: We celebrate Statistics Day on June 29th to honor Prof. P.C. Mahalanobis. He made huge contributions to how our country plans its economy using data.

🎯 Exam Tip: Be sure to write both the exact date (29th June) and the reason (honoring Prof. P.C. Mahalanobis) to get complete marks.

 

Question 5. What is the Mahalanobis model?
Answer: Mahalanobis model is India’s strategy for industrialization in the second five-year plan (1956-61), which was formulated by Prof. P.C. Mahalanobis. This model emphasized the development of heavy industries to boost long-term economic growth.
In simple words: This was India's plan to build big industries and factories during the second five-year plan. It was designed to help the country become self-reliant.

🎯 Exam Tip: Clearly state the time period of the second five-year plan (1956-61) as it is a crucial detail examiners look for.