Find the mean and standard deviation using short-cut method. | $x_i$ | $f_i$ | |---|---| | 60 | 2 | | 61 | 1 | | 62 | 12 | | 63 | 29 | | 64 | 25 | | 65 | 12 | | 66 | 10 | | 67 | 4 | | 68 | 5 |

Find the mean and standard deviation using short-c – 1st PUC Mathematics

Exercise 13.2 · Q6

Find the mean and standard deviation using short-cut method.

$x_i$	$f_i$
60	2
61	1
62	12
63	29
64	25
65	12
66	10
67	4
68	5

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Concept: Short-cut method for mean and standard deviation (step-deviation).

We choose an assumed mean $A = 64$ (midpoint of the data) and class size $h = 1$ . Define $u_i = \frac{x_i - A}{h} = x_i - 64$ .

Step 1 – Compute $f_i u_i$ and $f_i u_i^2$

$x_i$	$f_i$	$u_i$	$f_i u_i$	$f_i u_i^2$
60	2	-4	-8	32
61	1	-3	-3	9
62	12	-2	-24	48
63	29	-1	-29	29
64	25	0	0	0
65	12	1	12	12
66	10	2	20	40
67	4	3	12	36
68	5	4	20	80
Total	100		0	286

Step 2 – Mean …

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Mean $= 64$ and standard deviation $\approx 1.69$ .

Short-cut (step-deviation) method

Take assumed mean $A = 64$ and $h = 1$ , so $d_i = x_i - 64$ . Then $\bar{x} = A + \dfrac{\sum f_i d_i}{\sum f_i}$ and $\sigma = h\sqrt{\dfrac{\sum f_i d_i^2}{\sum f_i} - \left(\dfrac{\sum f_i d_i}{\sum f_i}\right)^2}$ .

Step-by-step solution

$x_i$	$f_i$	$d_i = x_i-64$	$f_i d_i$	$f_i d_i^2$
60	2	$-4$	$-8$	32
61	1	$-3$	$-3$	9
62	12	$-2$	$-24$	48
63	29	$-1$	$-29$	29
64	25	0	0	0
65	12	1	12	12
66	10	2	20	40

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

COMEDK 2024Set 2024-M1 markMCQ
$a, b, 8,5,10$
›Reveal solutionSolution
We use the definitions of mean and variance to set up two equations in $a$ and $b$ , then solve the system to find which option satisfies both.

We are given five numbers: $a, b, 8, 5, 10$ . Their mean is 6 and their variance is 6.80. We need to find which pair $(a, b)$ from the options fits both conditions.

Concept & Intuition

The mean gives us a linear equation: the sum of all numbers equals $5 \times 6 = 30$ .

The variance gives us a quadratic condition: the average of the squared deviations from the mean equals 6.80.

Together, these two equations will restrict $a$ and $b$ to a specific pair — we just check which option satisfies both.

Step-by-step solution

Use the mean to get a relation between $a$ and $b$ .

$\frac{a + b + 8 + 5 + 10}{5} = 6$
Multiply both sides by 5:
$a + b + 23 = 30 \quad \Rightarrow \quad a + b = 7$

Use the variance formula. Variance $\sigma^2$ is the average of $(x_i - \text{mean})^2$ . Here mean = 6, so:

$\sigma^2 = \frac{(a-6)^2 + (b-6)^2 + (8-6)^2 + (5-6)^2 + (10-6)^2}{5} = 6.80$
Compute the known terms:
$(8-6)^2 = 4,\quad (5-6)^2 = 1,\quad (10-6)^2 = 16$
Sum of known squares = $4 + 1 + 16 = 21$ .

So:
$\frac{(a-6)^2 + (b-6)^2 + 21}{5} = 6.80$
Multiply by 5:
$(a-6)^2 + (b-6)^2 + 21 = 34$ $(a-6)^2 + (b-6)^2 = 13$

Solve the system. From step 1: $b = 7 - a$ . Substitute into the squared equation:

$(a-6)^2 + (7 - a - 6)^2 = 13$
Simplify the second term: $7 - a - 6 = 1 - a$ , so:
$(a-6)^2 + (1 - a)^2 = 13$
Expand:
$(a^2 - 12a + 36) + (1 - 2a + a^2) = 13$
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Concept understanding — Variance and Standard Deviation of Data

Variance and Standard Deviation of Data

Short-cut (step-deviation) method

Step-by-step solution

Short-cut (step-deviation) method

Step-by-step solution

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