Find the mean and standard deviation using short-cut method. x_i6… - Vidyadip

Question

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

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

✓

Answer

Let $y_i=\frac{x_i-a}{i}=x_i-64[\because i=1$ and ' $a$ ' is assumed to be 64$]$

$x_i$	$f_i$	$y_i=x_i-64$	$f_i y_i$	$y_i^2$	$f_i y_i^2$
60	2	-4	-8	16	32
61	1	-3	-3	9	9
62	12	-2	-24	4	48
63	29	-1	-29	1	29
64	25	0	0	0	0
65	12	1	12	1	12
66	10	2	20	4	40
67	4	3	12	9	36
68	5	4	20	16	80
Total	100		0	60	286

Here,
$\quad \begin{aligned} \text { Mean, } \bar{x} & =64+\frac{\sum f_i y_i}{\sum f_i} \\ & =64+0 \\ & =64\end{aligned}$
Variance,
$
\begin{aligned}
\sigma^2 & =\frac{1}{\sum f_i}\left[\sum f_i y_i^2-n \bar{y}^2\right] \\
& =\frac{1}{100}[286]=2.86 \\
\sigma & =\sqrt{2.86}=1.69 . \text { (Approx.) }
\end{aligned}
$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks Questions" from Data Interpretation→Data Interpretation — all question types→Applied Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the first four moments about zero for the set of the number $1,3,5,7$.

Whatever be the value of $t$, prove that the locus of the point of intersection of the lines $x \cos t+y \sin t$ $=a$ and $x \sin t-y \cos t=b$ is a circle.

A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag, if (i) they can be of any colour. (ii) two must be white and two red. (iii) they must all be of the same colour.

Find the equation of circle which passes through the points $(0,2),(3,0)$ and $(3,2)$. Find also the centre and radius of this circle.

A machine with useful life of seven years costs ₹ 10,000 while another machine with useful life of five years costs ₹ 8000. The first machine saves labour expenses of ₹ 1900 annually and the second one saves labour expenses of ₹ 2200 annually. Determine preferred course of action. Assume cost of borrowing as 10% compounded per annum.
Given, P(7,0.1) = 4.86842 P(5,0.1) = 3.79079

A and B can do a piece of work in 45 days and 40 days respectively. They began to do the work together but A leaves after some days and then B complete the remaining work in 23 days. Find the number of days after which A left the work.

Two cyclists start from the same place to ride in the same direction. $A$ starts at noon at8 km / hr and $B$ at 1.30 p . m. at 10 km / h. How far will $A$ have ridden before he is overtaken by $B$ ? Find also at what times $A$ and $B$ will be 5 km apart.

The profits in lakhs of rupees of an industrial house for 2002, 2003, 2004, 2005, 2006, 2007 and 2008 are 5, 8, 9, 6, 12, 15 and 24 respectively. Represent these data using both line diagram and bar diagram.

$A \cap(B-C)=(A \cap B)-(A \cap C)$

If $x=3^{\frac{1}{3}}+3^{-\frac{1}{3}}$, then find the value of $3 x^3-9 x$.