Carillon Technologies Limited

Normal Probability Paper and the Relationship to Specifications

Background

  • Normal distributions have been depicted as bell shaped frequency curves, with the area under the curve corresponding to percentages of the total population.
  • These bell-shaped curves are useful in visualizing a distribution, but they are not well suited to graphical predict outcomes from limited data.
  • This predictive analysis function is served by the use of Normal Probability Paper.

    • The adjacent figure shows the relationship between the bell-shaded frequency curve or a Normal distribution and the equivalent cumulative distribution plotted on Normal Probability Paper.

    Both graphs have data values as their horizontal scale, and the dashed lines for the specification limits appear in the same place on both.

    The vertical scale for the bell-shaped curve is linear with frequency (or percentage rate); as compared to the the vertical scale of the Normal Probability Paper, which shows cumulative percentages, increases by equally-spaced standard deviation units on either side of the mean.

    Because of this arrangement of scales, the cumulative percentages of any Normal distribution plot as a straight line on Normal Probability Paper.

    The characteristic of straight-line plotting is very desirable in graphic forecasting because it is much easier to estimate a straight line than to try to draw a bell-shaped curve.

    NNP Analysis Steps

    Background Steps - Collecting the Data
    Link to Background Page

    Loading the Data:

    Load the points into the Excel spreadsheet to determine the "median rank" for each point.

    The rank is determined by first ordering the data from low to high, assigning an order number and then calculating the median rank.

    For n=5 points the order numbers (i ) will be 1,2,3,4,5 and the median.

    The median rank (j) is calculated by j = (i - 0.3)/(n+0.4).

    Plotting Steps

    Record the median rank (%) on the graph portion of the NNP. Using the left-hand vertical scale of ascending percentages, locate each Plot Point along the vertical line representing its Value.

    After all the Plot Points have been entered on the graph,, draw a "best fit" straight line through them.

    This graph haw been drawn on Normal Probability Paper, which means that plot points which come from a normal distribution will lie approximately on a straight line. Since the fit will seldom be exact (all points on a line), location of the "best fit" line will require some judgment. In general, the "best fit" straight line should be located so that any points not directly on the line will be as close as possible to it and appear to lie randomly on either side of it. Be especially wary of any tendency to bias the line in either an optimistic or pessimistic direction.

    If. in trying to locate a "best fit" straight line, the points off the line are not random but appear to have a pattern, e.g., they start to curve to the right toward the top of the graph, this signifies two very important things: (a) the distribution is not normal meaning that the forecasting techniques described here must be altered, and (b) this evidence of non-normality may be the first clue to assignable causes in the process that may jeopardize its capability of producing consistently to specification.

    Extend the "best fit" line beyond the actual data values until it crosses the bold horizontal lines at plus and minus 3 sigma.

    Draw the lower and upper specification limits on the graph as dashed vertical lines at the appropriate values

    Determine the 99.73% or +/- 3 sigma limits and 99.994% or +/- 4 sigma limits of the distribution.

    Software generated plot - Not as sensitive, will always find a straight line, if you use the trendline option.

    The normaldist.xls program provides a plot of measurement versus Z value.

    The z value of 0 represents the mean value estimate - in the adjacent graph - approximately 165-170 hours.

    Minus 1 z represents the average - 1 std. deviation - approximately 75.

    Plus 1 z represents the average + 1 std. deviation - approximately 255-260 hours.

    Click here to download the normaldist.xls Excel program

    Link to http://www.weibull.com to download a pdf file of the normal probability paper - Excellent Site

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