One hundred test specimens made of grey cast iron FG 300 are tested on a universal testing machine to determine the ultimate tensile strength \left(S_{u t}\right) of the material. The results are tabulated as follows:
| Frequency | Class interval (N/mm²) |
| 2 | 261–280 |
| 12 | 281–300 |
| 50 | 301–320 |
| 32 | 321–340 |
| 4 | 341–360 |
Calculate: (i) the mean; (ii) the variance; and (iii) the standard deviation for this sample.
Step I Mean
| f_{i} X_{i}^{2} | f_{i} X_{i} | Frequency \left(f_{i}\right) |
Class mark \left(X_{i}\right) |
| 145800 | 540 | 2 | 270 |
| 1009200 | 3480 | 12 | 290 |
| 4805000 | 15500 | 50 | 310 |
| 3484800 | 10560 | 32 | 330 |
| 490000 | 1400 | 4 | 350 |
| 9934800 | 31480 | 100 | Total |
From Eq. 24.2,
\mu=\frac{\sum f_{i} X_{i}}{\sum f_{i}} (24.2)
\mu=\frac{\sum f_{i} X_{i}}{\sum f_{i}}=\frac{31480}{100}=314.8 N / mm ^{2} . (i).
Step II Variance
From Eq. 24.6,
S^{2}=\frac{\sum f_{i} X_{i}^{2}-\frac{\left(\sum f_{i} X_{i}\right)^{2}}{N}}{(N-1)} (24.6).
S^{2}=\frac{\sum f_{i} X_{i}^{2}-\frac{\left(\sum f_{i} X_{i}\right)^{2}}{N}}{(N-1)} .
=\frac{9934800-\frac{(31480)^{2}}{100}}{(100-1)} .
=251.47\left( N / mm ^{2}\right)^{2} (ii).
Step III Standard deviation
S=\sqrt{251.47}=15.86 N / mm ^{2} . (iii).