Monday, June 2, 2014

Normal curve, Area under the normal curve and control limits

Normal Curve

A very famous and common type of population distribution is known as "Normal distribution or Gaussian distribution". The data is distributed under the normal curve symmetrically. The normal curve is a symmetrical, uni modal, bell shaped with mean median and mode having the same value. Much of the variation in nature and in industry follows the frequency distribution of normal curves. The normal curve is such a good description of the variations that occur to most quality characteristics in industry that it is the basis for many quality control techniques. There is definite relationship among mean , the standard deviation and the normal curve. Normal curves with different mean but same standard deviation may cause of shift of normal curve and hence change their location as shown in fig.....
Normal curves which have same mean but different standard deviation may cause of the normal curve at the same place Normalas shown in fig.....
The fig. explained that larger the S.D. , flatter the curve (data are widely dispersed) and smaller the S.D., the more peaked the curve(data are narrowly dispersed). If the standard deviation is zero then all  values are identical to the mean and there is no curve.

The normal distribution is fully defined by the population mean and population S.D.. Also the above fig's show that these two parameters are independent. In other words a change in one parameter has no effect on the other.

Area under the normal curve

The area under the normal curve is given by the relationship of mean and standard deviation and mention below
  • 68.27%  observations are included with in the limits of "mean+/- 1Std. Dev."
  • 95.45% observations are included with in the limits of "mean+/- 2Std. Dev."
  • 99.73% observations are included with in the limits of "mean+/- 3Std. Dev."
  • 100% observations are included in between +/- infinity. this is because the normal curve never touches the X-Axis
These percentages hold true regardless of the shape of the normal curve.

The area under the normal curve is more elaborate in below fig. in which 10 over bowling spell of a bowler in one day match is shown. The Yellow balls delivered in between "mean+/- 1S.D. which is 68.27%. The Blue balls delivered between +/- 1S.D. and +/- 2 S.D. but the total of Yellow and Blue balls results 95.45% deliveries. The Red balls are delivered between +/- 2 S.D. and +/- 3 S.D. and the total delivered balls percentage will come out as 99.73%.

Normal curve and control limits of control charts

An important fact is that 99.73 %  observations fall between mean+/- 3S.D. and it becomes the base of control limits (Upper Control Limit and Lower Control Limit)  of the control charts as shown in below fig.

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