Download The inflexion points of a normal curve - Ing-Stat

Some thoughts about sigma and the normal distribution
Sometimes you find figures like figure 1 below where sigma () is indicated as a straight line from
the mean to a point on the curve (usually called the inflexion point. See also figure 2 and 3).
Questions that arise are e.g. ’what does it mean’ or ’is it of any importance’ or ’do I need this information’ etc. Maybe it is an unimportant piece of information for the practical analysist and only
a mathematical entity?
If you ask the person that designed the figure you will get answers like the ’inflexion point is
where the curve changes directions’ or ’where the second derivative changes signs’. Most likely
will the practical analysist shake his head meaning that he has not become any wiser.
If we start with the formula that is the mother of all 2 (variance) for all the different distributions
that we can think of, we will see this:
 2  E( X   ) 2
This is read as the ’expected squared deviation from the mean’. If we use mathematical tools and
develop this formula we will end with expressions for  for all the distribution that we need when
studying problems in our environment (see other Ing-Stat documents for details). The practical
analysist is usually happy with these end results.
A remark (I). It seems unnecessary to decorate the text with words or expressions that lack
meaning for the practical analysist. Also, figure 1 incorrectly ties  hard to the normal distribution
while the truth is that we find  in all the distributions. (There is at least one pathological exemption. See the literature for information of the Cauchy distribution.)
Figure 1. In a normal distribution N(0, 1)
the standard deviation is sometimes indicated as the distance from the mean to
the point of inflexion.
This is true but of doubtful value for anyone that is not an mathematical expert.
Sometimes this is described as where
the curve changes directions. It is easy to
see that on a correctly drawn normal
distribution the point is not easy to find. It
is also defined as where the second
derivative changes signs.
Figure 2 and 3 show the first and second
derivatives respectively.
The standard normal distribution
The height of the normal curve
0.4
0.3
The inflexion point
0.2
0.1
0.0
-3
-2
-1
0
Values of the varible X
1
2
3
A remark (II). When we talk about the areas under the normal curve e.g. the area (probability) to
the left of - we usually use the expression the normal distribution as to indicate how the area is
distributed over the x-axis. When we talk about the very curve we often use the expression the
normal density function.
A remark (III). It is rather surprising that people seems to accept the normal distribution without
too much questioning. If you walk in to a workshop and say that there is a positive probability that
the deviations are infinitely large, people would laugh at you. However, this is what the normal
distribution states. But on the other hand this reminds us that the normal distribution is only a
model and not the reality. The thought of the inifinite tails was hard to swallow when developing
the statistical theory for an error distribution more than 200 years ago. Then some of the scientists
used a certain limit for the deviations. However, this lead to very complicated distributions.
© Ing-Stat – statistics for the industry www.ing-stat.nu
Rev D . 2009-12-05 . 1(3)
Figure 2. This curve is the so-called first
derivative of the normal distribution or
rather the normal density function in
figure 1.
To the practical analysist this is a completely useless curve.
1st derivative of the standard normal distribution
0.2
1st derivative
0.1
(The first derivative of a function is zero
where the original function, here the
normal density function in figure 1, has a
maximum (or a minimum). This fact
seems to be supported by diagram to the
left as the curve is 0 on the y-axis at the
mean on the x-axis.)
0.0
-0.1
-0.2
-3
-2
-1
0
Values of the varible X
1
2
3
Figure 3. This curve is the so-called first
derivative of the first derivative of the
normal density function. This is usually
called the second derivative of the
normal distribution.
It is obvious, as previously mentioned,
that this curve changes signs (i.e. passes
0) at exactly ±1 sigma.
Again, to the practical analysist this is a
completely useless information.
2nd derivative of the standard normal distribution
0.2
0.1
2nd derivative
0.0
-0.1
-0.2
-0.3
-0.4
-3
-2
-1
0
Values of the varible X
1
2
3
# These lines creates the diagrams
set c1
-3:3/0.005
end
let k1 = -1
let k2 = 1
let k3 = 0.3989*E()**((-(-k1)**2)/2)
let c2 = 0.3989*E()**((-c1**2)/2)
plot c2*c1;
wtitle 'The normal curve';
marker k1 k3;
marker k2 k3;
text k2 k3 ' The inflexion point';
tsize 0.9;
line 0 k3 1 k3;
lestyle 2 2;
title 'The standard normal distribution';
tsize 1.1;
grid 1;
grid 2;
data 0.10 0.97 0.10 0.90;
etype 0;
axlabel 1 'Values of the varible X';
tsize 0.9;
axlabel 2 'The height of the normal curve';
tsize 0.9;
connect;
size 2;
Scale 1;
© Ing-Stat – statistics for the industry www.ing-stat.nu
Rev D . 2009-12-05 . 2(3)
tsize 0.6;
HDisplay 0 0 0 0;
Scale 2;
min 0;
tsize 0.6;
HDisplay 0 0 0 0;
nodt.
# --------------------------------------------------- First derivative
let c3 = -0.3989*E()**((-c1**2)/2)*c1
plot c3*c1;
wtitle '1st derivative ';
title '1st derivative of the standard normal distribution';
tsize 1.1;
grid 1;
grid 2;
data 0.10 0.97 0.10 0.90;
etype 0;
axlabel 1 'Values of the varible X';
tsize 0.9;
axlabel 2 '1st derivative';
tsize 0.9;
connect;
size 2;
Scale 1;
tsize 0.6;
HDisplay 0 0 0 0;
Scale 2;
tsize 0.6;
HDisplay 0 0 0 0;
nodt.
# --------------------------------------------------- Second derivative
let c4 = -0.3989*E()**((-c1**2)/2) + 0.3989*E()**((-c1**2)/2)*c1**2
plot c4*c1;
wtitle '2nd derivative ';
title '2nd derivative of the standard normal distribution';
tsize 1.1;
marker -1 0;
marker 1 0;
grid 1;
grid 2;
data 0.10 0.97 0.10 0.90;
etype 0;
axlabel 1 'Values of the varible X';
tsize 0.9;
axlabel 2 '2nd derivative';
tsize 0.9;
connect;
size 2;
Scale 1;
tsize 0.6;
HDisplay 0 0 0 0;
Scale 2;
tsize 0.6;
HDisplay 0 0 0 0;
nodt.
© Ing-Stat – statistics for the industry www.ing-stat.nu
Rev D . 2009-12-05 . 3(3)