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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

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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.

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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,

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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

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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.

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10.5.5 Generalizing From a Few to Many

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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.

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10.5.7 Probability plots

• Procedure