(a) Compute Quartile deviation from the following data.

| C.I | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| f | 12 | 19 | 5 | 10 | 9 | 6 | 6 |

OR

(b) Find out the coefficient of correlation and interpret.

| Heigh

Question

(a) Compute Quartile deviation from the following data.

| C.I | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| f | 12 | 19 | 5 | 10 | 9 | 6 | 6 |

OR

(b) Find out the coefficient of correlation and interpret.

| Height of father (in inches) | 65 | 66 | 67 | 67 | 68 | 69 | 71 | 73 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Height of son (in inches) | 67 | 68 | 64 | 68 | 72 | 70 | 69 | 70 |

Accepted Answer

>

> [!TLDR]
> (a) $Q_1=22.5,\ Q_3\approx54.72\Rightarrow QD=\dfrac{Q_3-Q_1}{2}\approx16.11$. (b) Karl Pearson $r\approx+0.47$ (moderate positive).
>
> **Part (a) — Quartile deviation from grouped data.**
>
> **Step 1 — Cumulative frequencies ($N=\Sigma f=67$).**
>
> | C.I | $f$ | c.f. |
> | --- | --- | --- |
> | 10-20 | 12 | 12 |
> | 20-30 | 19 | 31 |
> | 30-40 | 5 | 36 |
> | 40-50 | 10 | 46 |
> | 50-60 | 9 | 55 |
> | 60-70 | 6 | 61 |
> | 70-80 | 6 | 67 |
>
> **Step 2 — $Q_1$ (size of $N/4=16.75$th item).** Falls in class 20-30 ($L=20,\ f=19,\ 	ext{c.f.}=12,\ h=10$).
> $$Q_1=L+\dfrac{\frac{N}{4}-	ext{c.f.}}{f}	imes h=20+\dfrac{16.75-12}{19}	imes10=20+\dfrac{47.5}{19}=20+2.5=22.5.$$
>
> **Step 3 — $Q_3$ (size of $3N/4=50.25$th item).** Falls in class 50-60 ($L=50,\ f=9,\ 	ext{c.f.}=46,\ h=10$).
> $$Q_3=L+\dfrac{\frac{3N}{4}-	ext{c.f.}}{f}	imes h=50+\dfrac{50.25-46}{9}	imes10=50+\dfrac{42.5}{9}=50+4.72=54.72.$$
>
> **Step 4 — Quartile deviation.**
> $$QD=\dfrac{Q_3-Q_1}{2}=\dfrac{54.72-22.5}{2}=\dfrac{32.22}{2}\approx16.11.$$
>
> ---
>
> **Part (b) — Karl Pearson's coefficient of correlation.**
>
> **Step 1 — Sums ($n=8$).**
>
> | $X$ | $Y$ | $X^2$ | $Y^2$ | $XY$ |
> | --- | --- | --- | --- | --- |
> | 65 | 67 | 4225 | 4489 | 4355 |
> | 66 | 68 | 4356 | 4624 | 4488 |
> | 67 | 64 | 4489 | 4096 | 4288 |
> | 67 | 68 | 4489 | 4624 | 4556 |
> | 68 | 72 | 4624 | 5184 | 4896 |
> | 69 | 70 | 4761 | 4900 | 4830 |
> | 71 | 69 | 5041 | 4761 | 4899 |
> | 73 | 70 | 5329 | 4900 | 5110 |
> | $\Sigma X=546$ | $\Sigma Y=548$ | $\Sigma X^2=37314$ | $\Sigma Y^2=37578$ | $\Sigma XY=37422$ |
>
> **Step 2 — Formula.** …

Concept understanding — Measures Of Dispersion

Measures of Dispersion

The Core Idea

Why It Matters in Commerce and Humanities

What the NCERT Textbook Emphasises

Key Points to Remember

Step 1 — List genuine measures of dispersion

Step 2 — Identify the intruder

Step 1 — First blank: based on all observations

Step 2 — Second blank: central 50% of observations

Step 1 — Recall the measures of dispersion

Step 2 — Identify the odd one out

$X$	$Y$	$X^2$	$Y^2$	$XY$
65	67	4225	4489	4355
66	68	4356	4624	4488
67	64	4489	4096	4288
67	68	4489	4624	4556
68	72	4624	5184	4896
69	70	4761	4900	4830
71	69	5041	4761	4899
73	70	5329	4900	5110
$\Sigma X=546$	$\Sigma Y=548$	$\Sigma X^2=37314$	$\Sigma Y^2=37578$	$\Sigma XY=37422$

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