Write done the formula of normal cumulative function to comp

Write done the formula of normal cumulative function to compute the defect ppm of a six sigma process. Then compute the ppm number.

Solution

Here the meaning of term ppm is parts per million.

Usually defects are measured in ppm but statistical processes are understood in terms of standard deviations.i.e sigma.

ppm is a value that represents the part of a whole number in units of 1/1000000.

1ppm = 1/1000000 = 0.000001 = 10^-6

Normal cumulative function means normal cumulative distribution function.

It can be abbrevated as c.d.f.= P(X < = x) , for all x

c.d.f = P((X - mu)/sigma < = (x - mu)/sigma )

= P( Z <= (x - mu)/sigma )

This is a formula of normal cumulative function to compute the defect ppm of a six sigma process.

ppm number = P( Z <= (x - mu)/sigma ) * 1000000

For example , suppose our x = -3.2 , mu = 0 and sigma = 1

( Note - If in normal distribution we don\'t given the values of mu and sigma then we take mu is 0 and sigma is 1 that is standard nomal distribution.)

then ppm number = P ( Z <= ( -3.2 - 0) / 1) * 1000000

=P ( Z <= -3.2) * 1000000

= 0.000687 * 1000000

ppm number = 687.