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Preventing fatigue crack propagation in aircraft struc-
tures is an important element of aircraft safety. An engineering
study to investigate fatigue crack in $$n=9$$ cyclically loaded
wing boxes reported the following crack lengths (in $$\mathrm{mm}$$ )
$$2.13,2.96,3.02,1.82,1.15,1.37,2.04,2.47,2.60 .$$

(a) Calculate the sample mean.
(b) Calculate the sample variance and sample standard
deviation.

$$n=9$$

crack lengths:

$$2.13,2.96,3.02,1.82$$
$$1.15,1.37,2.04,2.47$$
2.6

(a) Calculate sample mean.

$$\overline{x}=\frac{\sum_{i=1}^{n} x_{i}}{n}=\frac{19.56}{9}=2.173 \mathrm{mm}$$

(b) sample variance

$$S^{2}=\frac {\sum_{i=1}^{n} {x_{i}^{2}-\frac {\left(\sum x_{i}\right)^{2}}{n}}}{n-1}$$

$$\sum x_{i}=19.56 \mathrm {m m}$$

$$\sum x_{i}^{2}=45.953 \mathrm{mm}^{2}$$

$$s^{2}=\frac{45.953-\frac{19.56^{2}}{9}}{9-1}$$

$$=\frac{3.443}{8}=0.4303 \mathrm{mm}^{2}$$

$$S=\sqrt{s^{2}}=0.6560 \mathrm{mm}$$

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