Organisation of DataClass 11 Statistics For Economics NCERT Solutions
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Q1EXERCISES
Which of the following alternatives is true?
(i)
The class midpoint is equal to:
(a)
The average of the upper class limit and the lower class limit.
(b)
The product of upper class limit and the lower class limit.
(c)
The ratio of the upper class limit and the lower class limit.
(d)
None of the above.
(ii)
The frequency distribution of two variables is known as
(a)
Univariate Distribution
(b)
Bivariate Distribution
(c)
Multivariate Distribution
(d)
None of the above
(iii)
Statistical calculations in classified data are based on
(a)
the actual values of observations
(b)
the upper class limits
(c)
the lower class limits
(d)
the class midpoints
(iv)
Range is the
(a)
difference between the largest and the smallest observations
(b)
difference between the smallest and the largest observations
(c)
average of the largest and the smallest observations
(d)
ratio of the largest to the smallest observation
Solution
(i) The correct alternative is (a) The average of the upper class limit and the lower class limit.
As stated in the chapter, the formula for the class midpoint is: Class Mid-Point = (Upper Class Limit + Lower Class Limit) / 2.
(ii) The correct alternative is (b) Bivariate Distribution.
The chapter defines a Bivariate Frequency Distribution as the frequency distribution of two variables, such as sales and advertisement expenditure.
(iii) The correct alternative is (d) the class midpoints.
The chapter explains under the 'Loss of Information' section that once data is grouped into classes, further statistical calculations are based on the class mark (midpoint), not the individual observations.
(iv) The correct alternative is (a) difference between the largest and the smallest observations.
The chapter defines the range as the difference between the largest and the smallest values of a variable when discussing how to determine the number of classes.
Q2EXERCISES
Can there be any advantage in classifying things? Explain with an example from your daily life.
Solution
Yes, there are significant advantages in classifying things. Classification is the process of arranging or organising things into groups or classes based on some common criteria. This process brings order to disorganised data or items, making them easier to manage and understand.
The main advantages of classification are:
- It saves time and effort: When things are properly classified, it becomes easy to locate a specific item. This saves the valuable time and effort that would otherwise be spent searching through a large, unorganised collection.
- It makes analysis easier: Classification condenses raw data into a summary format, which can then be easily subjected to statistical analysis to draw meaningful conclusions.
- It facilitates comparison: By grouping similar items together, classification allows for easy comparison between different groups.
An example from daily life, as mentioned in the chapter, is arranging schoolbooks. If you classify your books according to subjects like 'History', 'Mathematics', and 'Science', each subject becomes a class. When you need your history textbook, you can go directly to the 'History' group and find it easily. Without this classification, you would have to search through your entire collection of books, which would be a tedious and time-consuming task.
Q3EXERCISES
What is a variable? Distinguish between a discrete and a continuous variable.
Solution
A variable is a characteristic or phenomenon which is capable of being measured and changes its value over time. For example, the marks of students in an examination are a variable because they differ from one student to another.
Variables can be distinguished into two types: discrete and continuous.
Discrete Variable:
- A discrete variable can only take certain specific values, usually integers or whole numbers. Its value changes in finite 'jumps' from one value to the next.
- It cannot take any intermediate or fractional values between two consecutive values.
- For example, the 'number of students in a class' is a discrete variable. A class can have 25 or 26 students, but it cannot have 25.5 students. The variable jumps from 25 to 26.
Continuous Variable:
- A continuous variable can take any numerical value within a given range. This includes integral values, fractional values, and values that are not exact fractions.
- The values of a continuous variable can be broken down into infinite gradations.
- For example, the 'height of a student' is a continuous variable. As a student grows, their height can take any value between, say, 150 cm and 151 cm, such as 150.25 cm or 150.8 cm. Other examples include weight, time, and distance.
Q4EXERCISES
Explain the 'exclusive' and 'inclusive' methods used in classification of data.
Solution
The 'exclusive' and 'inclusive' methods are two ways of forming class intervals in a frequency distribution.
Exclusive Method:
- In this method, the upper limit of one class interval is the same as the lower limit of the next class interval (e.g., 10-20, 20-30, 30-40).
- An item with a value equal to the upper limit of a class is excluded from that class and included in the next class where it is the lower limit. For example, a value of 20 would be included in the class 20-30, not in 10-20.
- This method is particularly useful for continuous variables because it ensures there are no gaps in the data classification, maintaining the continuity of the series.
- Example of exclusive class intervals: 0-10, 10-20, 20-30.
Inclusive Method:
- In this method, both the lower and upper limits of a class interval are included in that class itself. There is a gap between the upper limit of one class and the lower limit of the next class.
- For example, if the classes are 0-10 and 11-20, an observation with a value of 10 is included in the 0-10 class, and an observation with a value of 11 is included in the 11-20 class.
- This method is often used for discrete variables where fractional values are not possible. For continuous variables, this method creates a discontinuity (e.g., a gap between 10 and 11) which may need adjustment before further analysis.
- Example of inclusive class intervals: 0-10, 11-20, 21-30.
Q5EXERCISES
Use the data in Table 3.2 that relate to monthly household expenditure (in Rs) on food of 50 households and
(i)
Obtain the range of monthly household expenditure on food.
(ii)
Divide the range into appropriate number of class intervals and obtain the frequency distribution of expenditure.
(iii)
Find the number of households whose monthly expenditure on food is
(a)
less than Rs 2000
(b)
more than Rs 3000
(c)
between Rs 1500 and Rs 2500
Solution
(i) Range of monthly household expenditure on food:
First, we need to find the highest and lowest values from Table 3.2.
- Highest value (Maximum expenditure) = Rs 5090
- Lowest value (Minimum expenditure) = Rs 1007
The range is the difference between the highest and the lowest value.
Range = Highest Value - Lowest Value
Range = 5090 - 1007 = Rs 4083
(ii) Frequency distribution of expenditure:
Let us divide the range into 9 class intervals of size 500 each, using the exclusive method.
Frequency Distribution of Monthly Household Expenditure
| Expenditure (in Rs) | Tally Marks | Frequency (No. of Households) |
|---|---|---|
| 1000 - 1500 | NN NN NN NN | 20 |
| 1500 - 2000 | NN NN | |
| 2000 - 2500 | NN | |
| 2500 - 3000 | NN | 5 |
| 3000 - 3500 | ||
| 3500 - 4000 | ||
| 4000 - 4500 | ||
| 4500 - 5000 | 0 | |
| 5000 - 5500 | ||
| Total | 50 |
(iii) Number of households based on expenditure:
(a) less than Rs 2000:
This includes households in the classes '1000 - 1500' and '1500 - 2000'.
Number of households = 20 + 13 = 33.
(b) more than Rs 3000:
This includes households in the classes '3000 - 3500', '3500 - 4000', '4000 - 4500', and '5000 - 5500'.
Number of households = 2 + 1 + 2 + 1 = 6.
(c) between Rs 1500 and Rs 2500:
This includes households in the classes '1500 - 2000' and '2000 - 2500'.
Number of households = 13 + 6 = 19.
Q6EXERCISES
In a city 45 families were surveyed for the number of Cell phones they used. Prepare a frequency array based on their replies as recorded below. 1 3 2 2 2 2 1 2 1 2 2 3 3 3 3 3 3 2 3 2 2 6 1 6 2 1 5 1 5 3 2 4 2 7 4 2 4 3 4 2 0 3 1 4 3
Solution
A frequency array is used to classify the data of a discrete variable. Here, the variable is the 'number of cell phones', which is a discrete variable. The frequency array is prepared by counting the number of times each value occurs in the data.
Note: The question states that 45 families were surveyed, but the provided data contains only 41 observations. The frequency array is prepared based on the 41 observations given.
Frequency Array for the Number of Cell Phones Used by Families
| Number of Cell Phones (Variable) | Tally Marks | Frequency (Number of Families) |
|---|---|---|
| 0 | ||
| 1 | NN | |
| 2 | NN NN | |
| 3 | NN NN | |
| 4 | ||
| 5 | ||
| 6 | ||
| 7 | ||
| Total | 41 |
Q7EXERCISES
What is 'loss of information' in classified data?
Solution
'Loss of information' refers to an inherent shortcoming of classifying raw data into a frequency distribution. While classification summarises the data to make it concise and comprehensible, it does so at the cost of losing the details found in the original raw data.
When data is grouped into classes, the individual observations within a class lose their specific identity. For example, if the class '20-30' has a frequency of 6, we only know that six observations fall within this range. We do not know their actual values (e.g., 20, 22, 25, 25, 25, 28). For the purpose of further statistical calculations, all values in a class are assumed to be equal to the class mark (the midpoint of the class interval). In this case, all six values would be treated as 25. This use of a representative value instead of the actual values is the 'loss of information'. While much is gained by summarising data, this loss of precision is a key limitation.
Q8EXERCISES
Do you agree that classified data is better than raw data? Why?
Solution
Yes, I agree that for the purpose of analysis and drawing conclusions, classified data is better than raw data.
Raw data, especially when it is large, is highly disorganised and cumbersome to handle. It is difficult to draw any meaningful conclusions from a large set of unorganised numbers. Classification organises and summarises this data, making it comprehensible and manageable.
Here are the reasons why classified data is better than raw data:
- Brings Order: Classification brings order to the data, arranging it into logical groups. As in the example of the kabadiwallah sorting his junk, classification makes it easy to manage and locate information.
- Enables Analysis: Raw data does not easily yield to statistical methods. Classification, by creating frequency distributions, prepares the data for further statistical analysis, allowing us to calculate averages, identify patterns, and understand the structure of the data.
- Facilitates Comparison: By grouping data into classes, we can easily compare the characteristics of different groups. For example, census data, when classified by gender, education, or occupation, allows for a clear understanding and comparison of different segments of the population.
- Saves Time and Effort: It is a tedious task to pull specific information from large unclassified data. A frequency distribution presents the information in a condensed form, making it easy to see key features like the concentration of data in certain ranges.
While there is a 'loss of information' in terms of individual data points, the benefit of being able to make sense of the data and draw meaningful conclusions far outweighs this limitation.
Q9EXERCISES
Distinguish between univariate and bivariate frequency distribution.
Solution
The main distinction between a univariate and a bivariate frequency distribution lies in the number of variables they describe.
Univariate Frequency Distribution:
- A univariate frequency distribution deals with a single variable.
- It shows how the different values of that one variable are distributed across different classes.
- For example, the frequency distribution of marks obtained by 100 students (Table 3.6 in the chapter) is a univariate distribution because it only considers the variable 'marks'.
- It is typically presented in a two-column table, with one column for the class intervals of the variable and another for the corresponding frequencies.
Bivariate Frequency Distribution:
- A bivariate frequency distribution deals with two variables simultaneously.
- It is designed to show the joint frequency distribution of two variables. It helps in understanding the relationship between them.
- For example, the frequency distribution showing the 'sales' and 'advertisement expenditure' of 20 firms (Table 3.9 in the chapter) is a bivariate distribution.
- It is presented in a two-way table, or a contingency table, where the rows represent the classes of one variable and the columns represent the classes of the second variable. Each cell in the table shows the frequency of observations that fall into that specific combination of row and column classes.
Q10EXERCISES
Prepare a frequency distribution by inclusive method taking class interval of 7 from the following data. 28 17 15 22 29 21 23 27 18 12 7 2 9 4 1 8 3 10 5 20 16 12 8 4 33 27 21 15 3 36 27 18 9 2 4 6 32 31 29 18 14 13 15 11 9 7 1 5 37 32 28 26 24 20 19 25 19 20 6 9
Solution
To prepare a frequency distribution using the inclusive method with a class interval of 7, we first need to determine the range of the data and then form the classes.
- Smallest Value = 1
- Largest Value = 37
The class intervals will be formed in an inclusive manner (e.g., 1-7, 8-14, etc.).
Frequency Distribution (Inclusive Method)
| Class Interval | Tally Marks | Frequency |
|---|---|---|
| 1 - 7 | NN NN NN | 15 |
| 8 - 14 | NN NN | |
| 15 - 21 | NN NN NN | 15 |
| 22 - 28 | NN NN | 10 |
| 29 - 35 | NN | |
| 36 - 42 | ||
| Total | 60 |
Q11EXERCISES
"The quick brown fox jumps over the lazy dog" Examine the above sentence carefully and note the numbers of letters in each word. Treating the number of letters as a variable, prepare a frequency array for this data.
Solution
First, we list each word in the sentence and count the number of letters in it. The number of letters is our discrete variable.
- The: 3 letters
- quick: 5 letters
- brown: 5 letters
- fox: 3 letters
- jumps: 5 letters
- over: 4 letters
- the: 3 letters
- lazy: 4 letters
- dog: 3 letters
The data set for the variable 'number of letters' is: 3, 5, 5, 3, 5, 4, 3, 4, 3.
Now, we can prepare a frequency array by counting how many times each value appears.
Frequency Array for Number of Letters in Each Word
| Number of Letters (Variable) | Tally Marks | Frequency (Number of Words) |
|---|---|---|
| 3 | ||
| 4 | ||
| 5 | ||
| Total | 9 |