Right-hand side of Henry

The right of Henry is a method to visualize the chances that Gaussian has a distribution to be . It makes it possible to quickly read the Moyenne and the standard deviation of such a distribution.

Principle

If X is a Gaussian variable of average

\ overline {X}

and of variance

\ sigma^2

and if NR is a variable of normal Loi centered reduced, there are the following equalities:

$P (X < X) = P \ left (\ frac {X \ overline {X}} {\ sigma} < \ frac {X \ overline
{X}} {\ sigma} \ right) = P (NR < T) = \ Phi (T)$ , with $t = \ frac {X \ overline {X}} {\ sigma}$

(one notes $\ Phi$ the function of distribution of the reduced centered normal law).

For each value X _i of the variable X , one can (using a function table $\ Phi$ ):

to calculate $P (X < x_i)$
to deduce from it $t_i$ such as $\ Phi (t_i) = P (X < x_i)$

If the variable is Gaussian, the points of coordinates ( X _i; T _i) is aligned on the line of equation

t = \ frac {X \ overline {X}} {\ sigma}

Numerical example

During an examination noted on 20, one obtains the following results:

10% of the candidates obtained less than 4
30% of the candidates obtained less than 8
60% of the candidates obtained less than 12
80% of the candidates obtained less than 16

One seeks to determine if the distribution of the notes is Gaussian, and, if so, which is worth its hope and its standard deviation.

One thus knows 4 values X _i, and, for these 4 values, one knows P ( X < X _i).

By using the table wikisource: Function table of distribution of the reduced centered normal law, one determines the T _i corresponding:

It is then enough to trace the points of coordinates ( X _i; T _i).

The points appear aligned; the line cuts the x-axis at the point of X-coordinate 11 and the directing coefficient is (0,84 +1,28) approximately /12, which would give a standard deviation of 12/2,12 = 5,7. That lets think that the distribution is Gaussian parameters $m$ , $\ sigma^2$ , where $m = 11$ and $\ sigma = 5,7$ .

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