Find the mean- and standard deviation for the following data -i begin array -c- c-c- c-c- c-c- hline Year render -10-20-30-40-50-60Whline No of person cumulative -15-32-51-78-97-1091Whline lend array Marks- 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Frequency - 1 6 6 8 8 2 2 3 0 2 1 0 0 0 1 ii

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Standard XII

Mathematics

Standard Deviation

Find the mean...

Question

Find the mean , and standard deviation for the following data :

(i) $\begin{matrix} Y e a r r e n d e r : & 10 & 20 & 30 & 40 & 50 & 60 N o o f p e r s o n (c u m u l a t i v e) : & 15 & 32 & 51 & 78 & 97 & 109 \end{matrix}$

(ii) $\begin{matrix} M a r k s : & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 F r e q u e n c y : & 1 & 6 & 6 & 8 & 8 & 2 & 2 & 3 & 0 & 2 & 1 & 0 & 0 & 0 & 1 \end{matrix}$

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Solution

$\begin{matrix} x_{i} & C u m - F r e q & f_{i} & f_{i} x_{i} & f_{i} x_{i}^{2} 10 & 15 & 15 & 150 & 1500 20 & 32 & 17 & 340 & 6800 30 & 51 & 19 & 570 & 17100 40 & 78 & 27 & 1080 & 43200 50 & 97 & 19 & 950 & 47500 60 & 109 & 12 & 720 & 43200 N = 109 & T o t a l = 3810 & T o t a l = 159300 \end{matrix}$

Mean $¯ ¯¯¯ ¯ X = \frac{3810}{109} = 34.95$

Var $= \frac{159300}{109} - (34.95)^{2} = 239.96$

SD = $\sqrt{239.96} = 15.49$

$\begin{matrix} x_{i} & f_{i} & f_{i} x_{i} & f_{i} x_{i}^{2} 2 & 1 & 2 & 4 3 & 6 & 18 & 54 4 & 6 & 24 & 96 5 & 8 & 40 & 200 6 & 8 & 48 & 288 7 & 2 & 14 & 98 8 & 2 & 16 & 128 9 & 3 & 27 & 243 10 & 0 & 0 & 0 11 & 2 & 22 & 242 12 & 1 & 12 & 144 13 & 0 & 0 & 0 14 & 0 & 0 & 0 15 & 0 & 0 & 0 16 & 1 & 16 & 256 N = 40 & T o t a l = 239 & T o t a l = 1753 \end{matrix}$

Mean $= \frac{239}{40} = 5.975$

Var $= \frac{1753}{40} - (5.975)^{2} = 8.12$

SD = $\sqrt{8.12} = 2.85$

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Find the mean , and standard deviation for the following data : (i) Year render:102030405060No of person (cumulative):1532517897109 (ii) Marks:2345678910111213141516Frequency:166882230210001