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STATISTICS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSTATISTICS3 Marks
Question
Find the mean and variance for: First $10$ multiples of $3$
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Answer

The First $10$ multiples of $3$ are given by,
$3, 6, 9, 12, 15, 18, 21, 24, 27, 30$
We know that Mean, $\overline{\mathrm{x}}=\frac{\sum_{i=1}^{\mathrm{a}} \mathrm{x}_{\mathrm{i}}}{\mathrm{n}}$
$\therefore$ $\overline{\mathrm{x}}=\frac{3+6+9+12+15+18+21+24+27+30}{10}$ = $\frac{165}{10}$ = 16.5
From the given data, we can form the table:
$x_i$ Deviation from mean (xi - $\overline{\mathbf{X}}$) (xi - $\overline{\mathbf{X}}$)$^2$
3 3 - 16.5 = 13.5 182.25
6 6 - 16.5 = 10.5 110.25
9 9 - 16.5 = 7.5 56.25
12 12 - 16.5 = -4.5 20.25
15 15 - 16.5 = -1.5 2.25
18 18 - 16.5 = 1.5 2.25
21 21 - 16.5 = 4.5 20.25
24 24 - 16.5 = 7.5 56.25
27 27 - 16.5 = 10.5 110.25
30 30 - 16.5 = 13.5 182.25
    $\sum_{i=1}^{10}\left(x_{i}-\bar{x}\right)^{2}$ = 742.5
We know that Variance, $\sigma^{2}=\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{a}}\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}$
$\therefore$ $\sigma^{2}$ = (1/10) $\times$ 742.5 = 74.25
Mean = 16.5 and Variance = 74.25

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