Plot of a fitted normal distribution

Hello!

I have following problem:

My data consists of a vector x, which contains the possible results of a property (e.g. x = particle size), and a vector y, which contains the relative frequency of x.

For example:

x = 0,1 ; 0,5 ; 1 ; 5 ; 10 ; 50 ; 100; 500

y = 7% ; 9% ; 12% ; 15% ; 30% ; 13% ; 8% ; 6%

I don´t have any absolute frequencies.

I want to plot a fitted normal distribution of this data.

Thank You!

Best regards

Leo

[Adam Danz]

Could you replace the x and y lines with something we can copy-paste into matlab without errors?

[Leo]

Sorry, my mistake. FYI: I write the relative frequencies without the percent sign.

x

0.1000 0.5000 1.0000 5.0000 10.0000 50.0000 100.0000 500.0000

y

0.0700 0.0900 0.1200 0.1500 0.3000 0.1300 0.0800 0.0600

Answer 1

[Adam Danz]

x  = [0.011 0.013 0.015 0.017 0.02 0.023 0.026 0.03 0.034 0.039 0.044 0.051 0.058 0.067 0.076 0.087 0.1 0.115, ...
    0.131 0.15 0.172 0.197 0.226 0.259 0.296 0.339 0.389 0.445 0.51 0.584 0.669 0.766 0.877 1.005 1.151 1.318, ...
    1.51 1.729 1.981 2.269 2.599 2.976 3.409 3.905 4.472 5.122 5.867 6.72 7.697 8.816 10.097 11.565 13.246, ...
    15.172 17.377 19.904 22.797 26.111 29.907 34.255 39.234 44.938 51.471 58.953 67.523 77.34 88.583 101.46, ... 
    116.21 133.103 152.453 174.616 200 229.075 262.376 300.518 344.206 394.244 451.556 517.2 592.387 678.504, ... 
    777.141 890.116 1019.515 1167.725 1337.481 1531.914 1754.613 2009.687 2301.841 2636.467 3000];
y = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00036 0.0012533, ...
    0.00147 0.00151 0.0013867 0.0011333 0 0 0 0 0 0 0 0 0.00084 0.0025433 0.0058567 0.012833 0.025703 0.048463, ...
    0.079867 0.10807 0.11754 0.12896 0.13169 0.11523 0.086417 0.056513 0.03343 0.01883 0.010793 0.0059967, ...
    0.0033333 0 0 0 0 0 0 0 0 0 0 0 0 0 0];

f = fit(log(x(:)),y(:),'gauss1');
x0 = linspace(min(log(x)), max(log(x)), 100); 
y0 = feval(f,x0); 

clf()
hold on
plot(log(x),y,'b-','LineWidth', 3, 'DisplayName', 'Data')
plot(x0,y0, 'r--', 'LineWidth', 3, 'DisplayName', 'fit')
xlabel('log(x)')
ylabel('y')
title([formula(f), newline(), strjoin(compose('%s=%.3f  ',string(coeffnames(f)), coeffvalues(f)'))], ...
    'FontName','FixedWidth')
legend()
grid on 
box on

[Leo]

Thank you very much!

The coefficients are perfect me, but I have one more problem: The x-axis should be shown as x and not as log(x). Only the scale should be logarithmic. This means, that I need a scale with 1,10,100,1000.

I know that your solution is the same, but I have to present this graph in an axis-system with x and y and not with log(x) and y.

Best regards and thank you again!

[Adam Danz]

plot(exp(x0),y0, 'r--', 'LineWidth', 3, 'DisplayName', 'fit')

and

set(gca,'xscale','log')

Ignore the log(x) xlabel....

[Leo]

Thank you! It looks great. Thank you for your support.

My last question: Is it possible to solve integrals with matlab?

I want to find the value1 for which 10% of the area under the fitted curve is "left" from this value1.

And i want to find the value2 for which 10% of the area under the fitted curve is "right" from the value2.

And i want to find the value3 for which the area under the fitted curve is splitted in 50% und 50%.

I want to find the median and the other two values.

Best regards!

[Adam Danz]

Glad I could help.

Regards to your new question, here are all of the post-processing functions that accepts the model (f) as an input.

https://www.mathworks.com/help/curvefit/fit-postprocessing.html

See the integrate function. Again, the fun input will be f, your model.

[Leo]

Thank you for your support!!! You helped me a lot!

Unfortunately i don´t get the values, which I need, because I need the coefficients not for log(x). I need the median and co. of the particle size x and don´t of the log(particle size) and it is not correct for the distribution to say median = 10^log(median[in log]).

"I want to find the value1 for which 10% of the area under the fitted curve is "left" from this value1.

And i want to find the value2 for which 10% of the area under the fitted curve is "right" from the value2.

And i want to find the value3 for which the area under the fitted curve is splitted in 50% und 50%.

I want to find the median and the other two values."

[Adam Danz]

The distribution is not gaussian without the log transform.

You can compute the mean, median, etc from the fit and then transform the results back to their original scale using the same method I showed with the previous plot, using exp().

[Leo]

Ok, thank you!

Best regards! :)

[Adam Danz]

Glad I could help.

Answer 2

[David Hill]

x=[ 0.1000    0.5000    1.0000    5.0000   10.0000   50.0000  100.0000  500.0000];
y=[0.0700    0.0900    0.1200    0.1500    0.3000    0.1300    0.0800    0.0600];
z=[];
y=floor(100*y);
x=log10(x);%looks like your data is logarithmic (if you don't take log(x), normal distribution will not be great) 
for k=1:length(x)
  z=[z,repmat(x(k),1,y(k))];%replicate the data based on the frequencies
end
pd=fitdist(z','Normal');
X=-3:.1:5;
Z = normpdf(X,pd.mu,pd.sigma);
plot(X,Z);

[Leo]

First of all thank you!

But my problem is not yet solved.

I will show you other data, which should clarify my problem.

I will plot this distribution on a logartihmic x-axis. In the axis system with linear y-axis and log x-axis this data should "look" like a normal distribution. By using the term semilogx(x,y), i get a graph, which is too angular, because the connection between points is linear. Furthermore i would get the mean value and D90 and D10 of the distribution.

I send new data:

x =

1.0e+03 *

Columns 1 through 10

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Columns 11 through 20

0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

Columns 21 through 30

0.0002 0.0002 0.0002 0.0003 0.0003 0.0003 0.0004 0.0004 0.0005 0.0006

Columns 31 through 40

0.0007 0.0008 0.0009 0.0010 0.0012 0.0013 0.0015 0.0017 0.0020 0.0023

Columns 41 through 50

0.0026 0.0030 0.0034 0.0039 0.0045 0.0051 0.0059 0.0067 0.0077 0.0088

Columns 51 through 60

0.0101 0.0116 0.0132 0.0152 0.0174 0.0199 0.0228 0.0261 0.0299 0.0343

Columns 61 through 70

0.0392 0.0449 0.0515 0.0590 0.0675 0.0773 0.0886 0.1015 0.1162 0.1331

Columns 71 through 80

0.1525 0.1746 0.2000 0.2291 0.2624 0.3005 0.3442 0.3942 0.4516 0.5172

Columns 81 through 90

0.5924 0.6785 0.7771 0.8901 1.0195 1.1677 1.3375 1.5319 1.7546 2.0097

Columns 91 through 93

2.3018 2.6365 3.0000

y =

Columns 1 through 10

0 0 0 0 0 0 0 0 0 0

Columns 11 through 20

0 0 0 0 0 0 0 0 0 0

Columns 21 through 30

0 0 0 0 0 0 0 0 0 0

Columns 31 through 40

0 0 0 0 0 0 0 0 0 0

Columns 41 through 50

0 0 0 0 0 0 0.0004 0.0013 0.0015 0.0015

Columns 51 through 60

0.0014 0.0011 0 0 0 0 0 0 0 0

Columns 61 through 70

0.0008 0.0025 0.0059 0.0128 0.0257 0.0485 0.0799 0.1081 0.1175 0.1290

Columns 71 through 80

0.1317 0.1152 0.0864 0.0565 0.0334 0.0188 0.0108 0.0060 0.0033 0

Columns 81 through 90

0 0 0 0 0 0 0 0 0 0

Columns 91 through 93

0 0 0

Hopefully you can solve my problem. Thank you!!!

[Adam Danz]

We can't access data the way you've shared it.

One option is to use formatted code to supply data in a copy-paste format such as

x = [ 1 2 3
    4 5 6
    7 8 9];

Another option is to save the data to a mat file and attach the file.

[Leo]

x = 1000*[0.000011000000000 0.000013000000000 0.000015000000000 0.000017000000000 0.000020000000000 0.000023000000000 0.000026000000000 0.000030000000000 0.000034000000000 0.000039000000000 0.000044000000000 0.000051000000000 0.000058000000000 0.000067000000000 0.000076000000000 0.000087000000000 0.000100000000000 0.000115000000000 0.000131000000000 0.000150000000000 0.000172000000000 0.000197000000000 0.000226000000000 0.000259000000000 0.000296000000000 0.000339000000000 0.000389000000000 0.000445000000000 0.000510000000000 0.000584000000000 0.000669000000000 0.000766000000000 0.000877000000000 0.001005000000000 0.001151000000000 0.001318000000000 0.001510000000000 0.001729000000000 0.001981000000000 0.002269000000000 0.002599000000000 0.002976000000000 0.003409000000000 0.003905000000000 0.004472000000000 0.005122000000000 0.005867000000000 0.006720000000000 0.007697000000000 0.008816000000000 0.010097000000000 0.011565000000000 0.013246000000000 0.015172000000000 0.017377000000000 0.019904000000000 0.022797000000000 0.026111000000000 0.029907000000000 0.034255000000000 0.039234000000000 0.044938000000000 0.051471000000000 0.058953000000000 0.067523000000000 0.077340000000000 0.088583000000000 0.101460000000000 0.116210000000000 0.133103000000000 0.152453000000000 0.174616000000000 0.200000000000000 0.229075000000000 0.262376000000000 0.300518000000000 0.344206000000000 0.394244000000000 0.451556000000000 0.517200000000000 0.592387000000000 0.678504000000000 0.777141000000000 0.890116000000000 1.019515000000000 1.167725000000000 1.337481000000000 1.531914000000000 1.754613000000000 2.009687000000000 2.301841000000000 2.636467000000000 3.000000000000000]

y = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000360000000000 0.001253333333333 0.001470000000000 0.001510000000000 0.001386666666667 0.001133333333333 0 0 0 0 0 0 0 0 0.000840000000000 0.002543333333333 0.005856666666667 0.012833333333333 0.025703333333333 0.048463333333333 0.079866666666667 0.108070000000000 0.117543333333333 0.128960000000000 0.131686666666667 0.115226666666667 0.086416666666667 0.056513333333333 0.033430000000000 0.018830000000000 0.010793333333333 0.005996666666667 0.003333333333333 0 0 0 0 0 0 0 0 0 0 0 0 0 0]