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Question
Find mean, variance and S.D. of the following data:
| Classes | 10 – 20 | 20 – 30 | 30 – 40 | 40 – 50 | 50 – 60 | 60 – 70 | 70 – 80 | 80 – 90 | 90 – 100 |
| Frequency | 7 | 14 | 6 | 13 | 9 | 15 | 11 | 10 | 15 |
Solution
We prepare the following table to compute mean, variance, and S.D.:
| C.I. | Mid value (xi) |
fi |
ui = `(x_"i" - "A")/"h"` A = 55 |
fiui | fiui2 = fiui x ui |
| 10 – 20 | 15 | 7 | – 4 | – 28 | 112 |
| 20 – 30 | 25 | 14 | – 3 | – 42 | 126 |
| 30 – 40 | 35 | 6 | – 2 | – 12 | 24 |
| 40 – 50 | 45 | 13 | – 1 | – 13 | 13 |
| 50 – 60 | 55 | 9 | 0 | 0 | 0 |
| 60 – 70 | 65 | 15 | 1 | 15 | 15 |
| 70 – 80 | 75 | 11 | 2 | 22 | 44 |
| 80 – 90 | 85 | 10 | 3 | 30 | 90 |
| 90 – 100 | 95 | 15 | 4 | 60 | 240 |
| `sumf_"i"` = 100 | `sumf_"i"u_"i"` = 32 | `sumf_"i"u_"i"^2` = 664 |
From the table, `sumf_"i"u_"i" = 32, sumf_"i" = 100, sumf_"i"u_"i"^2 = 644`
∴ `bar(u) = (sumf_"i"u_"i")/(sumf_"i")`
`bar(u) = 32/100`
`bar(u)` = 0.32
∴ `bar(x) = "A" + "h"bar(u)`
= 55 + 10(0.32)
= 58.2
`sigma_u^2 = (sumf_"i"u_"i"^2)/(sumf_"i") - (baru)^2`
`= 644/100 - (0.32)^2`
= 6.64 – 0.1024
= 6.5376
Var(x) = `sigma_x^2`
`= "h"^2sigma_u^2`
= 100 × 6.5376
= 653.76
S.D. = `sigma_x`
`= sqrt(653.76)`
= 25.56
Hence, mean = 58.2, variance 653.76, S.D. = 25.56
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