1–4
Motorola Sensor Device Data
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frequencies of an event to established confidence intervals.
The relationship between failure rate and the chi-square
distribution is as follows:
L1
2
, d.f.
2t
Where:
λ
=
=
=
=
=
=
=
failure rate
lower one side confidence limit
chi–square function
risk, (1–confidence level)
degrees of freedom = 2 (r + 1)
number of failures
device hours
L1
χ
2
α
d.f.
r
t
Chi-square values for 60% and 90% confidence intervals
for up to 12 failures is shown in Table 1.
As indicated by the table, when no failures occur, an
estimate for the chi-square distribution interval is obtainable.
This interval estimate can then be used to solve for the
failure rate, as shown in the equation above. If no failures
occur, the failure rate estimate is solely a function of the
accumulated device hours. This estimate can vary dramati-
cally as additional device hours are accumulated.
As a means of showing the influence of device hours with
no failures on the failure rate value, a graphical representa-
tion of cumulative device hours versus the failure rate
measured in FITs is shown in Figure 1.
A descriptive example between two potential vendors best
serves to demonstrate the point. If vendor A is introducing a
new product and they have put a total of 1,000 parts on a
high temperature storage test for 500 hours each, their
corresponding cumulative device hours would be 500,000
device hours. Vendor B has been in the business for several
years on the same product and has tested a total of 500,000
parts for 10 hours each to the same conditions as part of an
in-line burn-in test for a total of 5,000,000 device hours. The
corresponding failure rate for a 60% confidence level for
vendor A would be 1,833 FITs, vendor B would have a FIT
rate of 183 FITs.
Table 1. Chi-Square Table
Chi-Square Distribution Function
60% Confidence Level
90% Confidence Level
No. Fails
χ
2 Quantity
No. Fails
χ
2
Quantity
0
1.833
0
4.605
1
4.045
1
7.779
2
6.211
2
10.645
3
8.351
3
13.362
4
10.473
4
15.987
5
12.584
5
18.549
6
14.685
6
21.064
7
16.780
7
23.542
8
18.868
8
25.989
9
20.951
9
28.412
10
23.031
10
30.813
11
25.106
11
33.196
12
27.179
12
35.563
CUMULATIVE DEVICE HOURS, [t]
105
108
107
109
104
100
10
1
0.1
1,000
106
1
10
100
1,000
104
105
106
107
108
109
F
Figure 1. Depiction of the influence on the cumulative device hours with no failures
and the Failure Rate as measured in FITs.
F
Freescale Semiconductor, Inc.
n
.