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A statistical analysis of microhardness data in boron softened Ni3Al
Scripta
METALLURGICA
V o l . 22, pp. Printed in
725-727, 1988 the U.S.A.
A STATISTICAL ANALYSIS OF MICROHARDNESS
Pergamon Press plc All rights reserved
DATA IN BORON SOFTENED Ni3AI
X. R. Qian and Y. T. Chou Department of Materials Science and Engineering Lehigh University, Bethlehem, PA 18015, U.S.A. (Received
February
25,
1988)
Introduction In a recent paper [i], the present authors reported that the microhardness of stoichiometric Ni^AI alloy could be lowered by the addition of boron. The lowering in J hardness was observed both in grain interiors and on grain boundaries, indicating that under localized deformation boron could soften the material. Although the data are convinclng, they only represent a random sample drawn from the population. The calculated means, x, and standard deviations, s, are referred to the sample set. For a more general conclusion, it is appropriate to make estimates about the population mean, ~, and population standard deviation, o, by using statistical methods. Such an analysis is presented in this note. In particular, the 95% confidence intervals for ~ and o are evaluated; the difference between the ~'s of the boron-free and boron-doped alloys is estimated. Statistical Analysis In the previous study, two polycrystalline stoichiometric alloys were examined, one doped with 0.2 at.% boron and the other free of boron. Vickers hardness was measured separately in the grain interiors and along the grain boundaries. Each sample set has a size of 30 and each sample point, x., represents the averaged hardness value for an individual grain (or boundary), The samplel mean, x, - and standard deviation, s, were then calculated using the expressions.
-
1
x
= n
2 s
=
~
i xi .
1 n-i
i
.
=
i
.
2,
.
..
n
(I)
Z (xi-x) 2 i
(2)
where n is the sample size and E represents the s u n a t i o n taken over the n terms. The data for x and s reported in the previous study [I] are listed in Table I. To evaluate the confidence interval for ~, we assume that the population is represented by a normal distribution. Then the random variable t = (x - ~)/s/~n has a Student's t distribution with n-I degrees of freedom [2]. A 95% confidence interval can be obtained by assigning a value of 0.95 for the probability of finding t, i.e., t0.05 P (-t0.05
< t < t0.05) = I
f(t)dt = 0.95
J -t0.05
725 0036-9748/88 $3.00 + Copyright (c) 1 9 8 8 P e r g a m o n
.00 Press
plc
(3)
726
MICROHARDNESS
IN N i 3 A I
Vol.
22,
No.
where f(t) is the distribution function, and the number t o 0~ is called the 5% level of t and locates points which cut off 2.5% of area under the distribution curve. In the present case, n = 30 and from Table IV in Ref. 2, 2 tO = 2.045. The calculated values for the 95% confidence intervals are listed in Table . ~% results show that with a 95% confidence interval, ther mean hardness of the boron-doped alloy is lower than that of the boron-free alloy, regardless of whether the measurements are taken in the grains or on the grain boundaries. TABLE I Calculated Values of n, x and s for Four Sample Sets Grain Interior
Grain Boundary
Variable Boron free
n x
Boron doped
30 225.2 6.9
s
30 201.7 6.3
Boron free
Boron doped
30 329.2 10.3
30 283.7 I0.0
TABLE 2 Estimation of 95% Confidence
Interval for
Alloy
Boron free Boron doped
Grain Interior
Grain Boundary
(222.6, 227.8) (199.3, 204.1)
(325.4, 333.0) (280.0, 287.4)
Similarly, by using the X 2 distribution, obtained from the inequality (n-l)s 2 < 2 X2 where
XI 2
is the probability
a 95% confidence
interval for o 2 can be
02 < (n-l)s 2 2 XI
of 0.975 for X 2 > XI 2 and X22 is the probability
(4) of 0.025
for X 2 > X 2. In the present case X 2 = 15.93 and X22 = 46.00. The confidence intervals for o were t~en obtained, as given in T~ble 3. The results imply that the true o's for the doped and undoped alloys may be very close or even equal. To t e s t the equality of the o's, we first assume that o F = o n and prove that the hypothesis is acceptable. Here, subscript F refers to the boron-free alloy and subscript D refers to the boron-doped alloy. To check on this assumption, we consider the F distribution with F defined by 2 sF F =
2
(5)
sD with n~-I and nn-i degrees of freedom [2]. for hardness In-the grain interiors and F , However, the 95% confidence interval for g ~ Ref. 2) which is greater than both Fin and for both cases.
Using the data in Table I, we obtain Fi_ = 1.20 = 1.06 for hardness on the grain bounda~L~es. with 58 degrees of freedom is 2.21 (Table V in Fg b. Thus the hypothesis, o F = o D, is acceptable
5
Vol.
22,
No.
S
MICROHARDNESS
IN N i 3 A I
727
TABLE 3 Estimation of o 95% Confidence Interval for o Alloy Grain Interior
Boron free Boron doped
Grain Boundary
(5.5, 9.3) (5.0, 8.5)
(8.2, 13.9) (7.9, 13.5)
Once the equality a F = ~D is established, one can proceed to estimate the difference between the population means, ~F and ~D" For this case, it is convenient to choose a new random variable
1/2 (xF-xD) - (BF-~D) t =
I
nFnD(nF+nD-2)
[(nF_l)SF2 + (nD-l)SD 2 31/2
nF + nD
1
(6)
which has the Student's t distribution with (nF + n,- 2) degrees of freedom. The 95% confidence interval is then given by Itl < t o o~ wit~ t o O5 = 2.002. Thus the 95% confidence interval for the difference between the t w o V ~ n s is "-(7)
20.1 < ~F - ~D < 26.9 for the hardness values in grain interiors, and 40.3 < ~F - ~D < 50.7
(8)
for the values on grain boundaries.
Conclusion The statistical analysis presented above confirms the findings that in microhardness measurement the addition of boron softens the Ni3AI alloy. Acknowledgement The work was sponsored by the Division of Materials Sciences, U.S. Department of Energy under Grant DE-FG02-86ER45256. References I. 2.
X. R. Qian and Y. T. Chou, Materials Letters, in press (1988). P. G. Hoel, Introduction to Mathematical Statistics, 5th edition, John Wiley, New York (1984).
METALLURGICA
V o l . 22, pp. Printed in
725-727, 1988 the U.S.A.
A STATISTICAL ANALYSIS OF MICROHARDNESS
Pergamon Press plc All rights reserved
DATA IN BORON SOFTENED Ni3AI
X. R. Qian and Y. T. Chou Department of Materials Science and Engineering Lehigh University, Bethlehem, PA 18015, U.S.A. (Received
February
25,
1988)
Introduction In a recent paper [i], the present authors reported that the microhardness of stoichiometric Ni^AI alloy could be lowered by the addition of boron. The lowering in J hardness was observed both in grain interiors and on grain boundaries, indicating that under localized deformation boron could soften the material. Although the data are convinclng, they only represent a random sample drawn from the population. The calculated means, x, and standard deviations, s, are referred to the sample set. For a more general conclusion, it is appropriate to make estimates about the population mean, ~, and population standard deviation, o, by using statistical methods. Such an analysis is presented in this note. In particular, the 95% confidence intervals for ~ and o are evaluated; the difference between the ~'s of the boron-free and boron-doped alloys is estimated. Statistical Analysis In the previous study, two polycrystalline stoichiometric alloys were examined, one doped with 0.2 at.% boron and the other free of boron. Vickers hardness was measured separately in the grain interiors and along the grain boundaries. Each sample set has a size of 30 and each sample point, x., represents the averaged hardness value for an individual grain (or boundary), The samplel mean, x, - and standard deviation, s, were then calculated using the expressions.
-
1
x
= n
2 s
=
~
i xi .
1 n-i
i
.
=
i
.
2,
.
..
n
(I)
Z (xi-x) 2 i
(2)
where n is the sample size and E represents the s u n a t i o n taken over the n terms. The data for x and s reported in the previous study [I] are listed in Table I. To evaluate the confidence interval for ~, we assume that the population is represented by a normal distribution. Then the random variable t = (x - ~)/s/~n has a Student's t distribution with n-I degrees of freedom [2]. A 95% confidence interval can be obtained by assigning a value of 0.95 for the probability of finding t, i.e., t0.05 P (-t0.05
< t < t0.05) = I
f(t)dt = 0.95
J -t0.05
725 0036-9748/88 $3.00 + Copyright (c) 1 9 8 8 P e r g a m o n
.00 Press
plc
(3)
726
MICROHARDNESS
IN N i 3 A I
Vol.
22,
No.
where f(t) is the distribution function, and the number t o 0~ is called the 5% level of t and locates points which cut off 2.5% of area under the distribution curve. In the present case, n = 30 and from Table IV in Ref. 2, 2 tO = 2.045. The calculated values for the 95% confidence intervals are listed in Table . ~% results show that with a 95% confidence interval, ther mean hardness of the boron-doped alloy is lower than that of the boron-free alloy, regardless of whether the measurements are taken in the grains or on the grain boundaries. TABLE I Calculated Values of n, x and s for Four Sample Sets Grain Interior
Grain Boundary
Variable Boron free
n x
Boron doped
30 225.2 6.9
s
30 201.7 6.3
Boron free
Boron doped
30 329.2 10.3
30 283.7 I0.0
TABLE 2 Estimation of 95% Confidence
Interval for
Alloy
Boron free Boron doped
Grain Interior
Grain Boundary
(222.6, 227.8) (199.3, 204.1)
(325.4, 333.0) (280.0, 287.4)
Similarly, by using the X 2 distribution, obtained from the inequality (n-l)s 2 < 2 X2 where
XI 2
is the probability
a 95% confidence
interval for o 2 can be
02 < (n-l)s 2 2 XI
of 0.975 for X 2 > XI 2 and X22 is the probability
(4) of 0.025
for X 2 > X 2. In the present case X 2 = 15.93 and X22 = 46.00. The confidence intervals for o were t~en obtained, as given in T~ble 3. The results imply that the true o's for the doped and undoped alloys may be very close or even equal. To t e s t the equality of the o's, we first assume that o F = o n and prove that the hypothesis is acceptable. Here, subscript F refers to the boron-free alloy and subscript D refers to the boron-doped alloy. To check on this assumption, we consider the F distribution with F defined by 2 sF F =
2
(5)
sD with n~-I and nn-i degrees of freedom [2]. for hardness In-the grain interiors and F , However, the 95% confidence interval for g ~ Ref. 2) which is greater than both Fin and for both cases.
Using the data in Table I, we obtain Fi_ = 1.20 = 1.06 for hardness on the grain bounda~L~es. with 58 degrees of freedom is 2.21 (Table V in Fg b. Thus the hypothesis, o F = o D, is acceptable
5
Vol.
22,
No.
S
MICROHARDNESS
IN N i 3 A I
727
TABLE 3 Estimation of o 95% Confidence Interval for o Alloy Grain Interior
Boron free Boron doped
Grain Boundary
(5.5, 9.3) (5.0, 8.5)
(8.2, 13.9) (7.9, 13.5)
Once the equality a F = ~D is established, one can proceed to estimate the difference between the population means, ~F and ~D" For this case, it is convenient to choose a new random variable
1/2 (xF-xD) - (BF-~D) t =
I
nFnD(nF+nD-2)
[(nF_l)SF2 + (nD-l)SD 2 31/2
nF + nD
1
(6)
which has the Student's t distribution with (nF + n,- 2) degrees of freedom. The 95% confidence interval is then given by Itl < t o o~ wit~ t o O5 = 2.002. Thus the 95% confidence interval for the difference between the t w o V ~ n s is "-(7)
20.1 < ~F - ~D < 26.9 for the hardness values in grain interiors, and 40.3 < ~F - ~D < 50.7
(8)
for the values on grain boundaries.
Conclusion The statistical analysis presented above confirms the findings that in microhardness measurement the addition of boron softens the Ni3AI alloy. Acknowledgement The work was sponsored by the Division of Materials Sciences, U.S. Department of Energy under Grant DE-FG02-86ER45256. References I. 2.
X. R. Qian and Y. T. Chou, Materials Letters, in press (1988). P. G. Hoel, Introduction to Mathematical Statistics, 5th edition, John Wiley, New York (1984).