Larson nomogram

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Larson.jpg

The Larson nomogram is a two-dimensional diagram of the binomial distribution . In this nomogram , the values ​​of the distribution function (this is the probability sum) can be approximately determined graphically.

Surname

The Larson nomogram is named after Harry R. Larson , who published this nomogram under the title "A nomograph of the cumulative binomial distribution" in December 1966 in Industrial Quality Control.

Practical meaning

The most important applications of the Larson nomogram are not in mathematics or science, but in quality assurance . In industrial series production in particular, the Larson nomogram is an important aid for assessing the quality situation, for sampling and for control card technology. Less well known is that the Larson nomogram can also be used to derive information about administrative and service processes.

Advantages and disadvantages

The advantage of the Larson nomogram is that you do not have to be a mathematician to use it, nor do you have to have any special theoretical knowledge of statistics. Results can only be achieved very easily in a few seconds with the nomogram printed on paper (or a photocopy) plus pencil and ruler. A small disadvantage is that the results only have an approximate accuracy - the small deviations from the mathematically exact values, however, are mostly of no relevance in practice.

Alternatives

  • For small numbers, the binomial distribution can be calculated using a formula.
  • For larger numbers, you can use pocket calculators or, even better, programmable pocket calculators.
  • The values ​​can be read from certain statistical tables.
  • There are special computer programs for calculating the binomial distribution.

Formula symbols in the Larson nomogram

The formula symbols used in quality assurance differ in part from the formula symbols in specialist mathematical books. The following notation is common in quality assurance:

  • The distribution function has the symbols G . The more understandable name of the distribution function is probability sum - this is the probability that there will be up to x defective units in a sample. The term “up to x defective units” becomes clearer if one imagines that e.g. B. up to 2 defective units in the sample would still be permissible. The QA specialist is then not interested in the probability that there are exactly 2 defective units in the sample - he is interested in the probability that up to 2 defective units are in the sample and this is equal to the sum of the probabilities for 0 defective units, for 1 incorrect unit and for 2 incorrect units - up to 2. The numerical values ​​of the distribution function G can be read off on the scale on the right edge of the LARSON nomogram. Attention: The scale is not shown in%. A probability sum of 1% results in a value of 0.01 on the scale.
  • The proportion of defective units in the population has the symbol p . In quality assurance, the population means the amount of parts from which the sample was taken. The population must be large compared to the sample, because otherwise the mathematical laws of binomial distribution do not apply. The scale of the proportion of defective units p is on the left edge of the Larson nomogram. Attention: This scale is also not shown in%. A proportion of defective units of 10% therefore corresponds to a value of 0.10 on the scale.
  • The number of defective units in the sample has the symbol x . In connection with the binomial distribution, a unit can only be faulty or not faulty. Intermediate values ​​such as B. "second choice" for porcelain are not permitted here.
  • The sample size has the symbol n . No geometric size is meant here - the sample size in quality assurance is the number of parts that make up the sample. A random sample can be taken from any part in series production. But you could also take a sample of bills or letters and evaluate them.
  • In the middle of the Larson nomogram is a network of lines for the number of defective units in sample x and lines for sample size n . The sample size and the number of defective units are always natural numbers , but the lines for n = 0 and n = 1 are missing in the diagram, because in quality assurance samples only make sense from n = 2. However, x = 0 can often occur - then there is no faulty unit in the sample.

How it works in theory

The numbers x and n form an intersection in the line network of the Larson nomogram. This point of intersection lies on a straight line between the p value on the left scale and the G value on the right scale. So only three of the four values ​​(numbers) G , x , n and p need to be known, then you can determine the missing value (number) by drawing a line.

How it works in practice

You take

  • Printed or copied Larson nomogram
  • Transparent ruler, length approx. 25 cm
  • Pointed pencil HB
  • Highlighter
  • Eraser for corrections
  • When p and / or G are known, mark the value (s) on the scale (s)
  • When x and n are known, circle or mark the intersection in the network of lines. From n> 10 and from x> 10 all lines are no longer available - then take “invisible” intermediate lines into account
  • If only x or only n are known, it is better to mark the respective line with a fluorescent highlighter instead of a pencil. "Invisible" intermediate lines can easily be drawn in with a pencil.
  • G - p - x - n: Of these, three values ​​or numbers must be known. If fewer than three values ​​(numbers) are known, the problem cannot be solved. If all four values ​​(numbers) are already known, there is nothing left that needs to be determined (except to check for correctness)
  • Mark the three known values ​​or numbers in the line network and on the scales - draw a straight, thin pencil line - then you can read the missing value (or the missing number) on a scale or in the line network.

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