Presentation of DataClass 11 Statistics For Economics NCERT Solutions
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Q1EXERCISES
Bar diagram is a
(i)
one-dimensional diagram
(ii)
two-dimensional diagram
(iii)
diagram with no dimension
(iv)
none of the above
Solution
The correct answer is (i) one-dimensional diagram.
In a bar diagram, the magnitude of the data is represented only by one dimension, which is the height or length of the bars. The width of the bars is kept uniform and does not represent any value; it is only for visual appeal. Therefore, it is considered a one-dimensional diagram.
Q2EXERCISES
Data represented through a histogram can help in finding graphically the
(i)
mean
(ii)
mode
(iii)
median
(iv)
all the above
Solution
The correct answer is (ii) mode.
A histogram can be used to determine the mode of a frequency distribution graphically. This is done by identifying the tallest rectangle (representing the modal class) and drawing lines from its top corners to the top corners of the adjacent rectangles. The x-coordinate of the intersection point of these lines gives the value of the mode.
Q3EXERCISES
Ogives can be helpful in locating graphically the
(i)
mode
(ii)
mean
(iii)
median
(iv)
none of the above
Solution
The correct answer is (iii) median.
Ogives, or cumulative frequency curves, are used to locate the median of a frequency distribution graphically. The median is the value on the x-axis that corresponds to the intersection point of the 'less than' ogive and the 'more than' ogive.
Q4EXERCISES
Data represented through arithmetic line graph help in understanding
(i)
long term trend
(ii)
cyclicity in data
(iii)
seasonality in data
(iv)
all the above
Solution
The correct answer is (iv) all the above.
An arithmetic line graph, also known as a time series graph, plots data over a period of time. This form of representation is very useful for analyzing and understanding various time-related patterns in the data, including the long-term trend, as well as cyclical and seasonal variations.
Q5EXERCISES
Width of bars in a bar diagram need not be equal (True/False).
Solution
False.
For a proper and non-misleading visual comparison, the bars in a bar diagram should be of equal width. The chapter defines a bar diagram as comprising "a group of equispaced and equiwidth rectangular bars". Unequal widths can distort the visual impression of the data.
Q6EXERCISES
Width of rectangles in a histogram should essentially be equal (True/False).
Solution
False.
While histograms are often drawn with rectangles of equal width for equal class intervals, it is not an essential condition. Histograms can be constructed for data with unequal class intervals. In such cases, the height of the rectangles is adjusted to represent frequency density (class frequency divided by class width) so that the area of each rectangle remains proportional to its class frequency.
Q7EXERCISES
Histogram can only be formed with continuous classification of data (True/False).
Solution
True.
A histogram is used to represent a continuous frequency distribution. The rectangles in a histogram are drawn adjacent to each other without any gaps, which signifies the continuous nature of the variable. If the original data is not in a continuous classification, it must be converted into one before a histogram can be drawn.
Q8EXERCISES
Histogram and column diagram are the same method of presentation of data. (True/False)
Solution
False.
A column diagram is another name for a vertical bar diagram. A histogram and a bar/column diagram are different.
- A histogram represents continuous data, has no gaps between rectangles, and is a two-dimensional diagram where area represents frequency.
- A bar diagram represents discrete data or attributes, has gaps between bars, and is a one-dimensional diagram where only the height represents the value.
Q9EXERCISES
Mode of a frequency distribution can be known graphically with the help of histogram. (True/False)
Solution
True.
The mode can be located graphically using a histogram. The process involves identifying the tallest bar (the modal class) and drawing lines from its top corners to the top corners of the adjacent bars. The x-coordinate of the point where these two lines intersect gives the modal value.
Q10EXERCISES
Median of a frequency distribution cannot be known from the ogives. (True/False)
Solution
False.
The median of a frequency distribution can be determined graphically by drawing 'less than' and 'more than' ogives on the same graph. The x-coordinate of the intersection point of the two ogives represents the median value.
Q11EXERCISES
What kind of diagrams are more effective in representing the following?
(i)
Monthly rainfall in a year
(ii)
Composition of the population of Delhi by religion
(iii)
Components of cost in a factory
Solution
The diagrams that are more effective for representing the given data are:
-
(i) Monthly rainfall in a year: A simple bar diagram is effective. Each month can be represented by a bar, and the height of the bar can represent the amount of rainfall, allowing for easy comparison across months.
-
(ii) Composition of the population of Delhi by religion: A pie diagram is most effective. It visually represents the proportion of the total population that belongs to each religion, clearly showing the composition as parts of a whole.
-
(iii) Components of cost in a factory: A component bar diagram or a pie diagram would be effective. Both diagrams can show the breakdown of the total cost into its various components (e.g., raw materials, labor, overheads) and illustrate the relative share of each component.
Q12EXERCISES
Suppose you want to emphasise the increase in the share of urban non-workers and lower level of urbanisation in India as shown in Example 4.2. How would you do it in the tabular form?
Solution
To emphasise the high share of urban non-workers and the low level of urbanisation, the data can be presented in a table that includes calculated percentages. This makes the comparisons more direct and impactful.
Table: Share of Non-Workers and Level of Urbanisation in India (2001)
(Figures in crores)
| Category | Rural | Urban | Total |
|---|---|---|---|
| Population | |||
| Workers | 31 | 9 | 40 |
| Non-workers | 43 | 19 | 62 |
| Total Population | 74 | 28 | 102 |
| Percentage Distribution | |||
| Share of Non-workers (%) | |||
| (within Rural/Urban) | 58.1% | 67.9% | 60.8% |
| Share of Population (%) | |||
| (of Total Population) | 72.5% | 27.5% | 100% |
Emphasis is achieved by:
- Calculating Shares: The table explicitly calculates the 'Share of Non-workers'. The higher figure for urban non-workers (67.9% compared to 58.1% for rural) is immediately apparent. This row highlights the high share of non-workers in urban areas.
- Showing Urbanisation Level: The last row clearly shows the 'Share of Population', highlighting that only 27.5% of the population is urban, thus emphasising the lower level of urbanisation.
- Clear Title and Headings: A specific title and clear row headings direct the reader's attention to the key points being presented.
Q13EXERCISES
How does the procedure of drawing a histogram differ when class intervals are unequal in comparison to equal class intervals in a frequency table?
Solution
The procedure for drawing a histogram differs significantly when class intervals are unequal compared to when they are equal.
-
Equal Class Intervals: When all class intervals have the same width, the frequency of each class is directly represented by the height of its corresponding rectangle. The width of all rectangles is the same, and the area of each rectangle is directly proportional to the frequency of that class.
-
Unequal Class Intervals: When the class intervals are of unequal width, using the frequency as the height of the rectangle would be misleading because it would make wider bars seem more significant. To correct this, the heights of the rectangles must be adjusted.
- Calculate Frequency Density: For each class, a 'frequency density' is calculated. The formula is: Frequency Density = Class Frequency / Width of the Class Interval
- Plot Adjusted Heights: The histogram is then drawn using these frequency densities as the heights of the rectangles on the y-axis. The x-axis still represents the class intervals.
- Area Represents Frequency: By making this adjustment, the area of each rectangle (Height × Width = Frequency Density × Class Width = Frequency) becomes proportional to the actual frequency of the class, ensuring an accurate visual representation of the data distribution.
Q14EXERCISES
The Indian Sugar Mills Association reported that, 'Sugar production during the first fortnight of December 2001 was about 3,87,000 tonnes, as against 3,78,000 tonnes during the same fortnight last year (2000). The off-take of sugar from factories during the first fortnight of December 2001 was 2,83,000 tonnes for internal consumption and 41,000 tonnes for exports as against 1,54,000 tonnes for internal consumption and nil for exports during the same fortnight last season.'
(i)
Present the data in tabular form.
(ii)
Suppose you were to present these data in diagrammatic form which of the diagrams would you use and why?
(iii)
Present these data diagrammatically.
Solution
(i) Present the data in tabular form.
Table: Sugar Production and Off-take in India
(During the first fortnight of December, in '000 tonnes)
| Item | 2000 | 2001 |
|---|---|---|
| Production | 378 | 387 |
| Off-take | ||
| - Internal Consumption | 154 | 283 |
| - Exports | 0 | 41 |
| Total Off-take | 154 | 324 |
(ii) Suppose you were to present these data in diagrammatic form which of the diagrams would you use and why?
A multiple bar diagram would be the most effective diagram to represent this data.
Reason: This diagram is specifically designed for comparing two or more sets of data across different categories. Here, we need to compare the values for production, internal consumption, and exports for two different years (2000 and 2001). A multiple bar diagram would place the bars for 2000 and 2001 side-by-side for each category, making the year-on-year comparison clear and straightforward.
(iii) Present these data diagrammatically.
To present this data using a multiple bar diagram:
- The X-axis would represent the categories: 'Production', 'Internal Consumption', and 'Exports'.
- The Y-axis would represent the quantity in '000 tonnes, with a suitable scale (e.g., from 0 to 400).
- For each category on the X-axis, two adjacent bars would be drawn: one representing the year 2000 and the other representing the year 2001.
- A legend (or key) would be used to distinguish the bars for each year (e.g., blue bars for 2000 and red bars for 2001).
Description of the resulting diagram:
- For 'Production', there would be two bars: a bar of height 378 units for 2000 and an adjacent bar of height 387 units for 2001.
- For 'Internal Consumption', there would be two bars: a bar of height 154 units for 2000 and an adjacent bar of height 283 units for 2001.
- For 'Exports', there would be two bars: a bar of height 0 units for 2000 and an adjacent bar of height 41 units for 2001.
Q15EXERCISES
The following table shows the estimated sectoral real growth rates (percentage change over the previous year) in GDP at factor cost.
Year Agriculture and allied sectors Industry Services 1994-95 5.0 9.2 7.0 1995-96 -0.9 11.8 10.3 1996-97 9.6 6.0 7.1 1997-98 -1.9 5.9 9.0 1998-99 7.2 4.0 8.3 1999-2000 0.8 6.9 8.2
Represent the data as multiple time series graphs.
Solution
To represent the given data as multiple time series graphs (also known as multiple arithmetic line graphs), the following procedure should be followed:
-
Set up the Axes:
- The X-axis will represent the 'Year', with markings for 1994-95, 1995-96, 1996-97, 1997-98, 1998-99, and 1999-2000.
- The Y-axis will represent the 'Growth Rate (in percent)'. Since the data includes negative values, the scale on the Y-axis must extend below zero (e.g., from -2% to 12%).
-
Plot the Data for Each Sector:
- Agriculture and Allied Sectors: Plot the points corresponding to the growth rates for each year: (1994-95, 5.0), (1995-96, -0.9), (1996-97, 9.6), (1997-98, -1.9), (1998-99, 7.2), (1999-2000, 0.8). Join these points sequentially with a specific type of line (e.g., a solid line).
- Industry: On the same graph, plot the points for the industry sector: (1994-95, 9.2), (1995-96, 11.8), (1996-97, 6.0), (1997-98, 5.9), (1998-99, 4.0), (1999-2000, 6.9). Join these points with a different type of line (e.g., a dashed line).
- Services: Similarly, plot the points for the services sector: (1994-95, 7.0), (1995-96, 10.3), (1996-97, 7.1), (1997-98, 9.0), (1998-99, 8.3), (1999-2000, 8.2). Join these points with a third type of line (e.g., a dotted line).
-
Add a Legend and Title:
- A Title should be given to the graph, for example, "Sectoral Real Growth Rates in GDP at Factor Cost (1994-2000)".
- A Legend (or key) is essential to identify which line corresponds to which sector. For example:
—Agriculture and Allied Sectors---Industry...Services
This multiple line graph will allow for easy comparison of the performance and fluctuations of the three different sectors of the economy over the given period.