10.5 Continuous Distributions


• Histograms are useful for grouped data, but in the cases where the data is continuous, we use distributions of probability.


• In general

- the area under the graph = 1.00000.......

- the graphs often stretch (asymptotically) to infinity


• In specific, some of the distribution properties are,



• In addition the centre of the distribution can vary (i.e. the average or mean)


• More on distribution later




10.5.1 Describing Distribution Centers With Numbers


• The best known method is the average. This gives the centre of a distribution







• Another good measure is the median


- If odd number of samples it is the middle number

- if an even number of samples, it is the average of the left and right bounding numbers



• If the numbers are grouped the median becomes



• Mode can be useful for identifying repeated patterns

- a mode is a repeated value that occurs the most. Multiple modes are possible.





10.5.2 Dispersion As A Measure of Distribution


• The range of values covered by a distribution are important


- Range is the difference between the highest and lowest numbers.



• Standard deviation is a classical measure of grouping (for normal distributions??????????)


- the equation is,




• When we use a standard deviation, we can estimate the distribution of the samples.



• By adding standard deviations to increase the range size, the percentage of samples included are,



• Other formulas for standard deviation are,






10.5.3 The Shape of the Distribution


• Skewed functions



• this lack of symmetry tends to indicate a bias in the data (and hence in the real world)


• a skew factor can be calculated



10.5.4 Kurtosis


• This is a peaking in the data



• This is best used for comparison to other values. i.e. you can watch the trends in the values of a4.



10.5.5 Generalizing From a Few to Many




10.5.6 The Normal Curve


• this is a good curve that tends to represent distributions of things in nature (also called Gaussian)


• This distribution can be fitted for populations (μ, σ), or for samples (X, s)



• The area under the curve is 1, and therefore will enclose 100% of the population.


• the parameters vary the shape of the distribution



• The area under the curve indicates the cumulative probability of some event



• When applied to quality ±3σ are used to define a typical “process variability” for the product. This is also known as the upper and lower natural limits (UNL & LNL)


******************* LOOK INTO USE OF SYMBOLS, and UNL, LNL, UCL, LCL, etc.



10.5.7 Probability plots


• Procedure