Chapter Overview

What You'll Learn

Key Highlights

Chapter introduces graphical data representation: bar graphs for discrete, histograms for continuous (uniform/varying widths), frequency polygons. Covers measures of central tendency: mean (\( \bar{x} = \frac{\sum x_i}{n} \)), median (middle value), mode (most frequent). Applications: Interpreting surveys, class marks, ogives for cumulative frequencies.

Comprehensive Chapter Summary

1. Graphical Representation of Data

• Tables discussed earlier; now graphical: one picture > thousand words; graphs show comparisons better than data.
• Study: (A) Bar graphs; (B) Histograms (uniform/varying widths); (C) Frequency polygons.
• Bar Graphs: Pictorial with uniform width bars, equal spacing on x-axis (variable), y-axis values; heights depict values.
• Example 1: Class IX 40 students' birth months bar graph (Fig. 12.1); Nov: 4 students; max Aug.
• Construction: Example 2 family ₹20,000 income expenditures (Table 12.1): Grocery 4k, Rent 5k, etc.; scale x:1 unit=head, y:1=1k ₹; draw bars (Fig. 12.2).
• Visualize: Education > double medicine; better than table.
• Activity 1: Represent four groups' data by bar graphs.
• Elaboration: Discrete/categorical variables; gaps between bars distinguish categories; scales chosen to fit max value.
• Application: Surveys, budgets; relative proportions at glance.
• Extension: Grouped data possible; vertical/horizontal bars.
• Verification: Heights proportional to values; no distortion.
• Histograms: For continuous class intervals; like bar but adjacent rectangles (no gaps).
• Example: 36 students weights Table 12.2: 30.5-35.5:9, etc.; x: weights scale 1cm=5kg (kink at 30.5), y: freq max 15; widths=class size, heights=freq (Fig. 12.3).
• Steps: Horizontal continuous intervals, vertical freq; rectangles width=class size, height=freq.
• Elaboration: Class marks for midpoints; uniform width first, then varying.
• Application: Heights, scores distributions.

2. Histograms of Varying Widths

• Varying: Heights proportional to freq density = freq / class width.
• Example: Table 12.3 marks 50 students; class 0-10:2 (width10), 10-30:8 (20), etc.; density= freq/width.
• Construction: x intervals, y density scale; rectangle area= freq (height×width=freq).
• Fig. 12.4: Heights 0.2, 0.4, 0.5, 0.3, 0.1 for densities.
• Elaboration: Ensures area proportional freq; for unequal intervals.
• Application: Income brackets, age groups varying sizes.
• Verification: Total area = total freq; convert to bar by freq.
• Extension: Less than ogive from histogram.

3. Frequency Polygons

• Line graph: Plot freq at class marks, join midpoints (add 0 freq ends).
• Example: From Table 12.2 marks 32.5,38,43,48,53,58; freq 9,6,15,3,1,2; plot join (Fig. 12.5).
• Steps: Class marks x, freq y; lines connect.
• Elaboration: Approximates histogram outline; for comparison multiple distributions.
• Application: Trends over intervals; cumulative polygons (ogives).
• Verification: Midpoints accurate; smooth curve for large data.
• Extension: Less than/more than ogive for median.

4. Measures of Central Tendency

• Mean: \( \bar{x} = \frac{\sum x_i f_i}{\sum f_i} \) or \( \frac{\sum x_i}{n} \); arithmetic average.
• Example: 7 obs 2,3,3,4,5,5,7; mean= \( \frac{29}{7} \approx 4.14 \).
• Assumed mean: For large, \( \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \), d=xi-a.
• Step deviation: \( \bar{x} = a + h \frac{\sum f_i u_i}{\sum f_i} \), u=(xi-a)/h.
• Median: Middle value n odd \( \frac{n+1}{2} \), even average; grouped \( l + \frac{(N/2 - cf)}{f} h \).
• Example: 7 obs median 4; grouped formula.
• Mode: Most frequent; grouped \( l + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} h \).
• Example: Mode 3,5 bimodal; grouped calc.
• Elaboration: Mean affected outliers, median central, mode peak.
• Application: Averages in scores, incomes.
• Exercise: Compute for tables, ogives median.
• Extension: Ogive intersect N/2 for median.
• Verification: Consistency across methods.

Key Concepts and Definitions

Important Facts

Questions and Answers from Chapter

Short Questions (1 Mark)

Medium Questions (3 Marks)

Long Questions (6 Marks)

Quick Revision Notes