Uncertainty in Young's modulus

Checking of the obtained results

FIG.  6: Variation of the experimental points to the general linear regression line.

A visual monitoring of result (4) and of it's $V\! RF\protect$ is provided by FIG. 6, which plots $y-y_{prev}$ according to $x$. One can see that the points are distributed in a horizontal band whose height measures approximately $100 N$. This result can be checked as follows. From FIG. 2, one has a $\Delta F\approx 13000 N$. Since $V\! RF\approx 16900$, the standard deviation is reduce by a factor $\approx 130$ and thus $\Delta F_{res}\approx 100 N$.

Comparison with the Code of Practice #7

The Code of Practice #7 [3] recommends to choose the interval $\left[ \alpha , \beta \right]$ which minimizes the quantity $\Delta a/a$, where $a$ is the slope of the regression line in $\left( \Delta L_{0}, F\right)$ and $\Delta a$ is defined by :

$$\Delta a\doteq \sqrt{\frac{\left( 1-r^{2}\right) {\mathrm{var}}\left( y \right) }{\left( n-2\right) {\mathrm{var}}\left( x \right) }}$$ (5)

This preconization is equivalent to maximize the quantity :

$$\left( \frac{a}{\Delta a}\right) ^{2} = \left( n-2\right) V\! RF\frac{cov^{2}}{{\mathrm{var}}\left( x \right) {\mathrm{var}}\left( y \right) }$$

$$= \left( n-2\right) \left( V\! RF-1\right)$$ (6)

Considering the values of $N$ and $V\! RF\protect$ in the vicinity of the extremum, the two procedures are equivalent for the choice of $\left[ \alpha , \beta \right]$ and therefore provide the same estimate of the Young's modulus. But they differ in the resulting confidence interval.

Uncertainty on $a$

Formula (5) has been presented in [3] as the uncertainty carried by $a$, and is introduced as a consequence of the formula

$$y_{i}=a x_{i}+b+t\times s \sqrt{\frac{n+1}{n}+\frac{1}{n}\frac{\left( x-\overline{x}\right) ^{2}}{{\mathrm{var}}\left( x \right) }}$$ (7)

in which $s$ is the estimator of the standard deviation $\sigma _{res}$ of the residual variable $z_{i}\doteq y_{i}-y_{prev}=y_{i}-a x_{i}-b$, and $t$ a Student-Fischer+'s variable. Applied to a small-sized sample, one obtains FIG. 7 where the covering factor $t=1$ has been chosen (in this example, $V\! RF\approx 2$).

FIG.  7: Regression line founded on a small sample.

A majority of authors obtains EQ. 7 by supposing that the residual variables are independent and identically distributed normal variables, but the hypothesis of normality can be released due to the number of points (about a hundred) involved in the regression. Nevertheless, we have tested the distribution of the variables $\left( y_{j}-\widehat{a}x_{j}-\widehat{b}\right) /s$ by gathering them in ten classes and used the $\chi ^{2}$ test to compare the obtained histogram with the normal law. We have obtained $\chi ^{2}=5.52$. Since $\nu =7$, it comes $\chi _{red}^{2}=-0.4$ and thus there is no rejection of the normality hypothesis.

FIG.  8: Residual variable (thin) and moving average (thick).

But, in any case, a strong prerequisite of EQ. (7) is the independence of the residual variables $z_{j}=y_{j}-\widehat{a}x_{j}-\widehat{b}$. But, for the set of test pieces involved in our study, the autocorrelation of the $z_{j}$, i.e. $cov\left( z_{i}, z_{i+1}\right)$, was approximately $0.5$. Therefore, the independence hypothesis cannot be defended.

Even the hypothesis of a long range independence is dubious, as shown FIG. 8. In this figure, the thin line joins the points $\left( x_{j}, z_{j}\right)$, while the thick one joins the points obtained by a moving average. This lack of independence is correlated to the fact that the effective variability of $a$ is rather about $\sqrt{1/V\! RF}\approx 1/130$ than about $\sqrt{1/\left( n\times V\! RF\right) }\approx 1/1000$.