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STATISTICS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSTATISTICS4 Marks
Question
Find the mean, variance and standard deviation using short cut method.
Height in cm 70-75 75-80 80-85 85-90 90-95 95-100 100-105 105-110 110-115
No. of children 3 4 7 7 15 9 6 6 3
✓

Answer

Height in cms. Mid values $x_i$ $f_i$ $u = \frac{{x - 92.5}}{5}$ fu $fu^2$
70-75 72.5 3 - 4 - 12 48
75-80 77.5 4 - 3 - 12 36
80-85 82.5 7 - 2 - 14 28
85-90 87.5 7 - 1 - 7 7
90-95 92.5 15 0 0 0
95-100 97.5 9 1 9 9
100-105 102.5 6 2 12 24
105-110 107.5 6 3 18 54
110-115 112.5 3 4 12 48
    60   6 254
Mean $(\bar x) = A + \frac{{\Sigma fu}}{N} \times h = 92.5 + \frac{6}{{60}} \times 5 = 92.5 + 0.5 = 93$
Variance $({\sigma ^2}) = \frac{{{h^2}}}{{{N^2}}}[N\Sigma f{u^2} - {(\Sigma fu)^2}]$
$= \frac{{{{(5)}^2}}}{{{{(60)}^2}}}[60 \times 254 - {(6)^2}]$
$ = \frac{{25}}{{3600}}[15240 - 36] = \frac{{25}}{{3600}} \times 15204 = 105.58$
Standard deviation $(\sigma ) = \sqrt {105.58} = 10.27$

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