Higher National Certificate in Engineering Unit 36 – LO 1.4

Learning Outcome LO1.3 LO 1.4: relate the characteristics of the normal curve to the distribution of the means of small samples

The Normal Distribution x _ x ± σ - 2x ± σ - 3x ± σ - Range

Sampling and Averages If the length of a product say a single straw is measured it is clear that occasionally a length will be found which is towards one end of the ‘tails’ of the process’s normal distribution. This occurrence if taken on its own may lead to the wrong conclusion that the cutting process requires adjustment. If on the other hand a sample of say 5 is taken it is extremely unlikely that all 5 lengths will lie towards one extreme of the distribution.

Sampling and Averages If therefore we take the average or mean length of the sample of 5 (i.e. x-bar) we shall have a much more reliable indicator of the state of the process. Sample means will vary with each sample taken but the variation will not be as great as that for single pieces.

Sampling and Averages

Sampling and Averages

Sampling and Averages Population of individual values i.e. single pieces x ; Mean = μ; s.d. = σ Distribution of sample means x-bar ; Mean = x-bar-bar; SE (standard error of means) = σ/√n where n is the sample size.

Sampling and Averages Comparison of the two frequency diagrams on the previous slide shows that the scatter of the sample averages is much less that the scatter of the individual pieces. In the distribution of the mean lengths from samples of the straws, the standard deviation of the means called the standard error of the means, and denoted by the symbol SE is σ/√n where n is the sample size

Sampling and Averages SE has the same characteristics as any standard deviation and normal tables may be used to evaluate probabilities related to the distribution of sample averages. We call it by a different name to avoid confusion with the population standard deviation.

Sampling and Averages The smaller spread of the distribution of sample averages provides the basis for a useful means of detecting changes in processes. Any change in the process mean, unless it is extremely large, will be difficult to detect from individual results alone

Sampling and Averages Spread of individual pieces 1000 1012 Volume (ml) Using the spread of mean values enables a shift the process mean to be more readily identified as there is less of a cross-over of measurements Spread of sample means Volume (ml)

Sampling and Averages The previous slide shows the parent distributions for two periods in a paint filing process between which the average has risen from 1000ml to 1012ml. The area between the ends of the distributions is common to both distributions and if a volume estimate occurs in the shaded portion, say at 1010ml, it could suggest either a volume from above the average from the distribution centered on 1000ml or below the average from the distribution centered on 1012ml. A large number of readings would need to be taken in order to establish that a change was confirmed.

Sampling and Averages The distribution of sample means however reveals the change much more readily, the overlap of distributions for such a change being much smaller. A sample of 1010ml would almost certainly not come form the distribution centered on 1000ml. Therefore on a chart for sample means, plotted against time, the change in level would be revealed almost immediately. For this reason sample means rather than individual values are used where possible and appropriate to control the centring of a process!

The Central Limit Theorem This states that when we draw samples of size n from a population with a mean μ, and a standard deviation σ, then as n increases in size, the distribution of sample means approaches a normal distribution with a mean μ and a standard error of the means σ/√n. This tells us that even if the individual values are not normally distributed, the distribution of the means will tend to have a normal distribution and the larger the sample size the greater will be this tendency.

The Central Limit Theorem It also tells us that the Grand or Process Mean x-bar-bar will be a very good approximation of the true mean of the population μ. Even if n is small say 5 for instance, and the population is not normally distributed, the distribution of sample means will be very close to normal. This provided the basis for the Mean Control Chart (x-bar chart)!