Statistics

• (i) Class Boundaries:
Adjustment factor $d = \frac{20 - 19}{2} = 0.5$.
- $10-19 \implies \mathbf{9.5 - 19.5}$
- $20-29 \implies \mathbf{19.5 - 29.5}$
- $30-39 \implies \mathbf{29.5 - 39.5}$
- $40-49 \implies \mathbf{39.5 - 49.5}$
• (ii) Class Size ($h$):

$$h = 19.5 - 9.5 = \mathbf{10}$$

• (iii) Class Mark of 3rd Class ($30-39$):

$$x_3 = \frac{29.5 + 39.5}{2} = \frac{69}{2} = \mathbf{34.5}$$

• A kink (zig-zag mark) indicates that the scale on the horizontal axis does not start from zero, but rather begins at a higher non-zero value (e.g., $100$ or $150$).
• Drawing the kink prevents wasting unnecessary empty space on the graph paper between the origin $(0, 0)$ and the first class interval, while maintaining accuracy of scale for all subsequent intervals.

• Consider any single rectangle of the histogram over class $[x_1, x_2]$.
• When straight lines join the midpoint of this rectangle to the midpoints of adjacent rectangles, two small right-angled triangles are formed at the top corners: one triangle is included inside the polygon but outside the rectangle, and an identical congruent triangle is included in the rectangle but left outside the polygon.
• By RHS or ASA Congruence, these complementary triangular areas cancel out identically across every single bar of the distribution.
• Summing across all classes, the total area under the frequency polygon is exactly equal to the total area of all rectangles of the histogram. $\blacksquare$

• To close a frequency polygon into a closed geometric polygon, we imagine two additional classes with frequency zero ($f = 0$):
1. One imaginary class immediately preceding the first class interval.
2. One imaginary class immediately succeeding the last class interval.
• We calculate the class marks of these two imaginary intervals and join the first and last plotted points to these zero-frequency points on the horizontal axis ($y = 0$).

• Class Size ($h$): The difference between any two consecutive class marks:

$$h = 20 - 15 = \mathbf{5}$$

• Half the class size is $\frac{h}{2} = \frac{5}{2} = 2.5$.
• For each class mark $x_i$, the lower boundary is $x_i - 2.5$ and the upper boundary is $x_i + 2.5$:
- For $15$: $(15 - 2.5) - (15 + 2.5) \implies \mathbf{12.5 - 17.5}$
- For $20$: $(20 - 2.5) - (20 + 2.5) \implies \mathbf{17.5 - 22.5}$
- For $25$: $(25 - 2.5) - (25 + 2.5) \implies \mathbf{22.5 - 27.5}$
- For $30$: $(30 - 2.5) - (30 + 2.5) \implies \mathbf{27.5 - 32.5}$
- For $35$: $(35 - 2.5) - (35 + 2.5) \implies \mathbf{32.5 - 37.5}$

• No. A histogram cannot be drawn directly for inclusive class intervals.
• Because a histogram represents continuous data, the rectangles must be adjacent without any gaps between them.
• In an inclusive series (like $10-19, 20-29$), there is a gap of $1\text{ unit}$ between the upper limit of one class and the lower limit of the next. The distribution must first be converted into continuous class boundaries ($9.5-19.5, 19.5-29.5$) before constructing the histogram.

• Primary Data: Data collected directly by the investigator personally for a specific objective through direct field observation, surveys, or experiments (e.g., measuring student heights in a classroom). It is original and highly reliable.
• Secondary Data: Data obtained from previously published or compiled sources (such as census publications, government gazettes, newspapers, or websites) that was originally collected by another organization for a different purpose.