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Question
The diameter of circles (in mm) drawn in the design are given below.
| Diameters | 33 − 36 | 37 − 40 | 41 − 44 | 45 − 48 | 49 − 52 |
| Number of circles | 15 | 17 | 21 | 22 | 25 |
Calculate the standard deviation.
Solution
Assumed mean = 42.5
| Diameters | Number of Circles (fi) |
Mid value xi |
d = xi − A = xi − 42.5 |
fidi | fidi2 |
| 33 − 36 | 15 | 34.5 | − 8 | − 120 | 960 |
| 37 − 40 | 17 | 38.5 | − 4 | − 68 | 272 |
| 41 − 44 | 21 | 42.5 | 0 | 0 | 0 |
| 45 − 48 | 22 | 46.5 | 4 | 88 | 352 |
| 49 − 52 | 25 | 50.5 | 8 | 200 | 1600 |
| `sumf_"i"` = 100 | `sumf_"i""d"_"i"` = 100 | `sumf_"i""d"_"i"^2` = 3184 |
Here `sumf_"i"` = 100, `sumf_"i""d"_"i"` = 100, `sumf_"i""d"_"i"^2` = 3184
Standard deviation (σ) = `sqrt((sumf_"i""d"_"i"^2)/(sumf_"i") - ((sumf_"i""d"_"i")/(sumf_"i"))^2`
= `sqrt(3184/100 - (100/100)^2`
= `sqrt(31.84 - 1)`
σ = `sqrt(30.84)`
Standard deviation (σ) = 5.55
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