Updated NCERT Solutions for Class 11 Statistics Chapter 4: Presentation of Data | Important Questions 2026-27

(a) one-dimensional diagrams

Explanation: Bar diagrams are considered one-dimensional because only one dimension, the height (or length) of the bars, is relevant for representing the value of the data. The width of the bars is kept uniform and has no statistical significance.

(b) mode

Explanation: The mode can be determined graphically from a histogram. The process involves identifying the tallest rectangle (modal class), drawing two lines diagonally from its upper corners to the corners of adjacent bars, and dropping a perpendicular from the intersection point to the X-axis to find the modal value.

(c) median

Explanation: The median can be located graphically where the 'less than' and 'more than' ogives intersect. The x-coordinate of this intersection point is the median.

(a) long-term trend

Explanation: An arithmetic line graph (time-series graph) is very effective in showing the long-term trend (increase, decrease, or stability) of data over a period. While it can show other patterns, its primary use is to understand the overall trend.

False.

Explanation: A fundamental rule is that the width of all bars must be equal in a bar diagram. This ensures only the height represents the data's magnitude, preventing visual misinterpretation.

False.

Explanation: While common, it is not essential. For distributions with unequal class intervals, rectangles will have unequal widths. In such cases, frequencies must be adjusted to frequency density so the area of each rectangle remains proportional to the class frequency.

True.

Explanation: A histogram's adjacent rectangles with no gaps represent a continuous frequency distribution. Discrete data must be converted to a continuous series before a histogram can be drawn.

False.

Explanation: A histogram is for continuous data (no gaps, 2-D), while a column/bar diagram is for discrete data (gaps, 1-D).

True.

Explanation: The mode can be located graphically by identifying the tallest bar in the histogram and using diagonal lines to find the modal value on the X-axis.

True.

Explanation: The median is the value of the x-coordinate where the 'less than' and 'more than' ogives intersect.

Advantages:

1. Simple and Natural: It presents a small amount of data as part of a descriptive narrative.
2. Provides Context: Data is presented with explanations, helping to emphasize specific points.
3. No Special Skill Needed: It's easy to write.

Disadvantages:

1. Not Suitable for Large Data: Becomes cumbersome and difficult to understand with a large volume of data.
2. Difficult to Compare: Hard for a reader to quickly compare values or identify trends.
3. Less Appealing: A dense paragraph of numbers is less engaging than a chart.

Tabular presentation is superior to textual presentation in these ways:

• Systematic and Organized: Data is in logical rows and columns, making it easy to read.
• Facilitates Comparison: Makes it easy to compare values across categories.
• Saves Space and Time: Presents large amounts of data compactly.
• Foundation for Analysis: Tables are the basis for creating graphs and charts.
• Reduces Errors: Clear layout reduces chances of misinterpretation.

Diagrammatic presentation simplifies data in the following ways:

• Visually Attractive: Diagrams are engaging and hold the reader's attention.
• Quick Understanding: Complex data can be understood at a glance, even by a layman.
• Easy Comparison: Visual comparisons are quick and intuitive (e.g., tallest bar).
• Reveals Trends: Can easily reveal patterns and trends hidden in tables.
• Memory Aid: Visual information is easier to remember.

The essential requirements of a good statistical table are:

1. Table Number: For easy reference.
2. Title: A clear, concise title explaining the contents.
3. Headnote: Clarifies units of measurement (e.g., "in ₹ Crores").
4. Stubs: Headings for the horizontal rows.
5. Captions: Headings for the vertical columns.
6. Body: The main part containing the numerical data.
7. Footnote: Clarifies specific items within the table (marked with *).
8. Source: Mentions the source of the data for credibility.

If the 'less than' and 'more than' ogives intersect at a point, the value on the x-axis corresponding to that point represents the Median of the frequency distribution.

A blank table for students by Faculty (Arts, Science, Commerce), Sex (Boys, Girls), and Year (2024-25, 2025-26) is shown below.

A 'less than' ogive is a cumulative frequency curve that shows the number of observations less than a certain value. It is an upward-sloping curve.

Construction Steps:

1. Convert the given frequency distribution into a 'less than' cumulative frequency distribution.
2. Take the upper class limits of the class intervals on the X-axis.
3. Take the corresponding 'less than' cumulative frequencies on the Y-axis.
4. Plot the points and join them with a smooth, freehand curve.

Data: Food (₹4000), Rent (₹2000), Education (₹1500), Clothing (₹1000), Miscellaneous (₹1500).

Step 1: Calculate the total expenditure.

Total Expenditure = 4000 + 2000 + 1500 + 1000 + 1500 = ₹10,000

Step 2: Calculate the angle for each component.

The formula is: (Value of Component / Total Value) * 360°

• Food: (4000 / 10000) * 360° = 144°
• Rent: (2000 / 10000) * 360° = 72°
• Education: (1500 / 10000) * 360° = 54°
• Clothing: (1000 / 10000) * 360° = 36°
• Miscellaneous: (1500 / 10000) * 360° = 54°

Step 3: Draw the Pie Chart.

(Draw a circle and use a protractor to create sectors with the calculated angles. Label each sector clearly.)

[Image: A colorful pie chart showing five sectors for Food, Rent, Education, Clothing, and Miscellaneous, labeled with their respective percentages or values.]

Data: Marks (0-10, 10-20, 20-30, 30-40, 40-50) and No. of Students (5, 12, 15, 8, 4).

To draw the Histogram:

1. Represent Marks (Class Intervals) on the X-axis and Number of Students (Frequency) on the Y-axis.
2. Choose an appropriate scale.
3. Draw adjacent rectangles for each class interval with height corresponding to its frequency.

To draw the Frequency Polygon:

1. Find the mid-point of each class interval (5, 15, 25, 35, 45).
2. Plot points with mid-points on the X-axis and frequencies on the Y-axis: (5, 5), (15, 12), (25, 15), (35, 8), (45, 4).
3. Join these points with straight lines.
4. Close the polygon by joining the first point to (-5, 0) and the last point to (55, 0) on the X-axis.

[Image: A histogram with 5 adjacent bars, and a frequency polygon line connecting the mid-points of the tops of these bars.]

1. What is the most appropriate diagram to represent this data?

   A Simple Bar Diagram is most appropriate because the data is categorical (names of sports) and we are comparing the preferences.

2. Calculate the percentage of students who prefer Football.

   Percentage = (60 / 200) * 100 = 30%.

3. What is the central angle for Cricket in a pie chart?

   Central Angle = (80 / 200) * 360° = 0.4 * 360° = 144°.

1. How many students scored less than 30 marks?

   Looking at the graph, the cumulative frequency for 30 marks is 45 students.

2. How many students scored between 20 and 40 marks?

   (Students scoring < 40) - (Students scoring < 20) = 55 - 25 = 30 students.

3. What can you infer from N/2 = 30?

   N/2 = 60/2 = 30 is used to find the median. By locating 30 on the Y-axis and finding the corresponding X-axis value from the curve, we can graphically determine the median marks.