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STATISTICS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSTATISTICS3 Marks
Question
Calculate the mean, variance and standard deviation for the following distribution:
Class 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Frequency 3 7 12 15 8 3 2
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Answer

Here, we construct the following table:
Class Frequency $(f_i)$ Mid-point$(x_i)$ $f_ix_i$ $(x_i - \overline x)^2$ $f_i(x_i - \overline x)^2$
30-40 3 35 105 729 2187
40-50 7 45 315 289 2023
50-60 12 55 660 49 588
60-70 15 65 975 9 135
70-80 8 75 600 169 1352
80-90 3 85 255 529 1587
90-100 2 95 190 1089 2178
  N = 50   $\sum f_i x_i$= 3100   $ \sum f_{i}\left(x_{i}-\overline{x}\right)^{2}$= 10050
Thus, N = 50, $\sum f_i x_i$ = 3100
$\therefore$ Mean $\overline x$ = $\frac{1}{N} \sum f_{i} x_{i}$ = $\frac{3100}{50}$ = 62
Variance, $\sigma^2$ = $\frac{1}{N} \sum f_{i}\left(x_{i}-\overline{x}\right)^{2}$ = $\frac{1}{50}$ $\times$ 10050 = 201 and standard deviation, $\sigma$ = $\sqrt{201}$ = 14.18

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