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13.Statistics
hard
The data is obtained in tabular form as follows.
| ${x_i}$ | $60$ | $61$ | $62$ | $63$ | $64$ | $65$ | $66$ | $67$ | $68$ |
| ${f_i}$ | $2$ | $1$ | $12$ | $29$ | $25$ | $12$ | $10$ | $4$ | $5$ |
A
$1.69$
B
$1.69$
C
$1.69$
D
$1.69$
Solution
The data is obtained in tabular form as follows.
| ${x_i}$ | ${f_i}$ | ${f_i} = \frac{{{x_i} – 64}}{1}$ | ${y_i}^2$ | ${f_i}{y_i}$ | ${f_i}{y_i}^2$ |
| $60$ | $2$ | $-4$ | $16$ | $-8$ | $32$ |
| $61$ | $1$ | $-3$ | $9$ | $-3$ | $9$ |
| $62$ | $12$ | $-2$ | $4$ | $-24$ | $48$ |
| $63$ | $29$ | $-1$ | $1$ | $-29$ | $29$ |
| $64$ | $25$ | $0$ | $0$ | $0$ | $0$ |
| $65$ | $12$ | $1$ | $1$ | $12$ | $12$ |
| $66$ | $10$ | $2$ | $4$ | $20$ | $40$ |
| $67$ | $4$ | $3$ | $9$ | $12$ | $36$ |
| $68$ | $5$ | $4$ | $16$ | $20$ | $80$ |
| $100$ | $220$ | $0$ | $286$ |
Mean, $\bar x = A\frac{{\sum\limits_{i = 1}^9 {{f_i}{y_i}} }}{N} \times h = 64 + \frac{0}{{100}} \times 1 = 64 + 0 = 64$
Variance, ${\sigma ^2} = \frac{{{h^2}}}{{{N^2}}}\left[ {N\sum\limits_{i = 1}^9 {{f_i}{y_i}^2 – \left( {\sum\limits_{i = 1}^9 {{f_i}{y_i}^2} } \right)} } \right]$
$=\frac{1}{100^{2}}[100 \times 286-0]$
$=2.86$
$\therefore$ Standard deviation $(\sigma)=\sqrt{2.86}=1.69$
Standard 11
Mathematics
