Distributions and histograms

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Distributions and histograms
Grades
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Distributions and histograms
Grades
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Distributions and histograms
Grades
Grades
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Distributions and histograms
Grades
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Distributions and histograms
Grades
As you collect an increasing amount of data (give
the same exam to an increasing number of
students), the histogram is expected to converge
to some well-defined shape the limiting
distribution f(x)
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Limiting distribution
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f(x)
Given the functional shape of distribution, f(x),
how do we know, what fraction of grades falls
into a particular interval?
B
???
A
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is the fraction of grades falling into the
interval from A to B. What is the meaning of
f(x) itself, though?
f(x)
probability to get an answer between x and
xdx is
f(x)dx
1
dx
Probability to get an answer in an interval
between x and xdx is proportional to both the
size of the interval, dx, and the value of f(x).
Therefore f(x) is called probability density.
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The probability to get an answer between x and
xdx is f(x)dx
f(x)
Given f(x), how do we calculate
of times xk appears
score
dx
is the fraction of students scoring xk or
probability of the score xk
f(xk)dxk
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Limiting bell-shaped distribution
Physics 2Cb
Grades
Most of the random processes in physics,
including measurements subject to random noise or
errors, result in the bell-shaped curves of the
normal (Gaussian) distribution. You also get a
normal distribution if the measured parameter is
a sum of many random factors.
Thats why we usually assume that the measured
values are normally distributed. If we get a
clearly different distribution, it is usually a
sign that something special or fishy is going on.
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Gauss distribution and random errors
Gauss (aka Gaussian and normal) distribution GX,s
is a function of variable x with 2 parameters
X and s. The parameters are fixed for each
particular distribution and define its shape.
The variable changes from -8 to 8
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Lets see how the shape of GX,s depends on X and
s.
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Gauss distribution changing X
Gauss distribution has its maximum at x
X. Therefore, changing X, changes position of
the maximum, but does not change the shape of the
distribution.
X45
X93
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Gauss distribution changing s
s appears in the equation twice and sets both
height and width of the distribution. Large s
corresponds to a wider distribution with a lower
peak. The area under the distribution is always
preserved, because of the normalization
s 7
s 15
X45
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Gauss distribution how do we calculate the mean
and standard deviation of the distribution?
The point of maximum of GX,s is called the most
probable x. For Gauss distribution it is
It turns out, they are already there!
probability
f(x)dx
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Gauss distribution the meaning of s
The area under a segment from X -s to Xs
accounts for 68 of the total area under the
bell-shaped curve.
That is, 68 of the measured points fall within s
from the best estimate
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What about the probabilities to find a point
within 0.5s from X, 1.7s from X or in general ts
from X ?
To find those probabilities we need to calculate
Unfortunately, we cannot do it analytically and
have to look it up in a table
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t 1
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t 1.47