Question

1. Evaluate the mean and standard deviation for the following.

(a) The young modulus^(GN/m^2) for stainless steel is given by:

Young modulus 193 194 195 196 197 198 199 200

Frequency 7 7 10 19 13 9 8 3

B) The tensile strength of the aluminum alloy is given by:

Tensile strength T / frequency

320^ ≤ T < 350 4

350^ ≤ T < 380 12

380^ ≤ T < 410 18

410^ ≤ T < 440 16

440^ ≤ T < 470 15

470^ ≤ T < 500 10

500^ ≤ T < 530 3

530^ ≤ T < 560 7

Answer 1

a)

Sample size $N=76.$

$\overline{x}=\frac{1}{N}\sum x_in_i\\ x_i\text{ --- value (Young's modulus)}\\ n_i\text{ --- corresponding frequency}\\ \overline{s}=\sqrt{\frac{1}{N}\sum (x_i-\overline{x})^2n_i}$

b)

Sample size $N=85.$

$\overline{p_n}(x)= \begin{cases} \frac{l_j}{N\cdot h}, x\in [a+jh,a+(j+1)h)\\ 0,\ x\notin (a,b) \end{cases}\\ l_j\text{ --- number of values in the interval } [a+jh,a+(j+1)h)$

$h=30$

$x_i^*=\frac{x_i+x_{i-1}}{2}\\ \overline{s}=\sqrt{\frac{1}{N}\sum(x_i^*)^2n_i-(\frac{1}{N}\sum x_i^*n_i)^2}\\ \overline{x}=\frac{1}{N}\sum x_i^*n_i$