Histogram

The histogram a tool for the Management of quality

The histogram is simple means and rapid to represent the distribution of a parameter obtained during a manufacture.

Example:

* hardness of a series of parts after a Heat treatment,

* concentration of an element in the composition of Alloy S produced by a foundry,

* mass of food preparation in a can

* etc

The histogram is a “visual” tool which makes it possible to detect certain anomalies or to make a diagnosis before engaging a step of improvement. Used within this framework, the histogram is a “qualitative” tool. To be able well to undertake the study of the dispersion of one parameter using one or several histograms, it is necessary to have a good knowledge of the studied parameter. In the same way, it is necessary to know the conditions of data-gathering: frequency of measurement, measuring instrument used, possibility of mixture of batches, possibility of tri etc

Construction of a histogram

Data-gathering

The first phase is the data-gathering in the course of manufacture. This collection can be realized either in an exceptional way at the time of the study of the parameter or by using an automatic statement or handbook made during a control carried out within the framework of the monitoring of the manufactoring process.

Without it being really possible to give a minimum number, it is necessary that the number of recorded values is sufficient. The more one has a high number of values, the more interpretation will be easy.

Many classes

The first operation is to determine the number of classes of the histogram. Generally, within the framework of an analysis of this type, one uses classes of identical width.

The number of classes depends on the number of values NR one has.

The number of class K can be given by the following formula:

$K= 1 + \ frac {10 \ log (NR)}{3}$

or more simply

$K = \ sqrt {NR} \,$

However, the histogram being a visual tool, it is possible to vary the number of classes. This makes it possible to see the histogram with a number different of classes and thus to find the best compromise which will facilitate interpretation. The use of a dedicated software or more simply of a spreadsheet facilitates this operation.

Class-intervals

The amplitude W of the histogram is

$w = \ hbox {maximum value} - \ hbox {minimal value} \,$

The theoretical amplitude H of each class is then:

$H = \ frac {W} {K}$

It is necessary to round this value with a multiple of resolution of the Measuring instrument (round-off with excess).

Example: That is to say the Mass of a culinary preparation before conditioning. The calculation of amplitude of class gives h_th = 0,014 kg. The resolution of the balance used is of 0,001 kg. One rounds the value H to 0,015 kg.

The classes can be of the _inférieure type; limit _supérieure [or limit _inférieure; limit _supérieure].

The minimal value of the first class is given by the minimal value of the series minus a half-resolution.

Example: the smallest value recorded during manufacture of the culinary preparation is of 0,498 G. The limit inferior will be: 0,498 - (0,001/2) = 0,4975 kg.

For more facility, it is preferable to take values “round” for example 0,495 kg

Example

That is to say the manufacture of food intakes, the weighing of the rations before packing gives the series of measure following in kg:

The characteristics of the statement are the following ones:

*Le many samples: Wide N=64

*L': w=0,098 kg

*Valor minimal: 0,498 kg

* maximum Value: 0,596 kg

One from of deduced the following parameters for the histogram:

*Le many classes is of 7 (by using the formula with the logarithm)

*L' amplitude of class is 0,098/7 = 0,014 kg which one rounds to 0,015 kg (resolution of the balance: 0,001 kg)

*La minimal value of the first class are of 0,498 - (0,001/2) = 0,4975. By preoccupation with a facility for interpretation, one can round this value to 0,495 kg.

The following histogram is obtained

Interpretation of a histogram

The distribution of much of industrial parameters often corresponds to a normal Loi. One often compares the histogram obtained with the profile “out of bell” of the normal law. This comparison is visual and even if it can be a first approach, it does not constitute a test of “normality”. For that, it is necessary to carry out a test of which one of most traditional is the Droite of Henry.

Foot-note: the distribution according to the normal law, if it is extremely frequent, is not systematic. It will be checked that the distribution does not correspond to a distribution of defect of form (example: measure offsetting in a tube, position of objects launched in the direction of a wall of which some rebound on this wall).

Interpretation can, for example, to give the following results:

Foot-note: in the case of histogram showing a mixture of two batches having a different average, there exist cases where dispersion present this aspect without to accuse a mixture. It is for example the case of the measurement of a cylindrical part but which presents a defect of the ovalization type. The two averages then represent the large Diamètre and the small diameter. It is the knowledge of the process and/or the product which makes it possible to carry out this type of interpretation.

See too

Internal bonds

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