How to normalize a histogram in MATLAB

Question

How to normalize a histogram such that the area under the probability density function is equal to 1

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For some Distributions  Cauchy I think  I have found that trapz will overestimate the area  and so the pdf will change depending on the number of bins you select  In which case I do   N h  hist q f  theta 30000     there Is a large range but most of the bins will be empty plot h N  sum N  mean diff h      r

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Since 2014b  Matlab has these normalization routines embedded natively in the histogram function  see the help file for the 6 routines this function offers   Here is an example using the PDF normalization  the sum of all the bins is 1    data   2 randn 5000 1    5                generate normal random  m 5  std 2  h   histogram data  Normalization   pdf       PDF normalization   The corresponding PDF is  Nbins   h NumBins  edges   h BinEdges   x   zeros 1 Nbins   for counter 1 Nbins     midPointShift   abs edges counter -edges counter 1   2      x counter    edges counter  midPointShift  end  mu   mean data   sigma   std data    f   exp - x-mu   2   2 sigma 2     sigma sqrt 2 pi      The two together gives  hold on  plot x f  LineWidth  1 5      An improvement that might very well be due to the success of the actual question and accepted answer      EDIT - The use of hist and histc is not recommended now  and histogram should be used instead  Beware that none of the 6 ways of creating bins with this new function will produce the bins  hist and histc produce  There is a Matlab script to update former code to fit the way  histogram is called  bin edges instead of bin centers - link   By doing so  one can compare the pdf normalization methods of  abcd  trapz and sum  and Matlab  pdf    The 3 pdf normalization method give nearly identical results  within the range of eps    TEST   A   randn 10000 1   centers   -6 0 5 6  d   diff centers  2  edges    centers 1 -d 1   centers 1 end-1  d  centers end  d end    edges 2 end    edges 2 end  eps edges 2 end     figure  subplot 2 2 1   hist A centers   title  HIST not normalized     subplot 2 2 2   h   histogram A edges   title  HISTOGRAM not normalized     subplot 2 2 3   counts  centers    hist A centers    get the count with hist bar centers counts trapz centers counts   title  HIST with PDF normalization      subplot 2 2 4  h   histogram A edges  Normalization   pdf   title  HISTOGRAM with PDF normalization     dx   diff centers 1 2   normalization difference trapz   abs counts trapz centers counts  - h Values   normalization difference sum   abs counts sum counts dx  - h Values    max normalization difference trapz  max normalization difference sum      The maximum difference between the new PDF normalization and the former one is 5 5511e-17

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The area of abcd s PDF is not one  which is impossible like pointed out in many comments   Assumptions done in many answers here   Assume constant distance between consecutive edges   Probability under pdf should be 1  The normalization should be done as Normalization with probability  not as Normalization with pdf  in histogram   and hist       Fig  1 Output of hist   approach  Fig  2 Output of histogram   approach     The max amplitude differs between two approaches which proposes that there are some mistake in hist   s approach because histogram   s approach uses the standard normalization   I assume the mistake with hist   s approach here is about the normalization as partially pdf  not completely as probability    Code with hist    deprecated   Some remarks    First check  sum f  N gives 1 if Nbins manually set   pdf requires the width of the bin  dx  in the graph g   Code   http   stackoverflow com a 5321546 54964 N 10000  Nbins 50   f x  hist randn N 1  Nbins     create histogram from ND   METHOD 4  Count Densities  not Sums  figure 3  dx diff x 1 2      width of bin g 1 sqrt 2 pi  exp -0 5 x  2     dx    pdf of ND with dx   1 0000 bar x  f sum f   hold on plot x g  r   hold off   Output is in Fig  1    Code with histogram    Some remarks    First check  a  sum f  is 1 if Nbins adjusted with histogram   s Normalization as probability  b  sum f  N is 1 if Nbins is manually set without normalization    pdf requires the width of the bin  dx  in the graph g   Code     METHOD 5  with histogram     http   stackoverflow com a 38809232 54964 N 10000   figure 4   h   histogram randn N 1    Normalization    probability     hist   deprecated  Nbins h NumBins  edges h BinEdges   x zeros 1 Nbins   f h Values  for counter 1 Nbins     midPointShift abs edges counter -edges counter 1   2    same constant for all     x counter  edges counter  midPointShift  end dx diff x 1 2      constast for all g 1 sqrt 2 pi  exp -0 5 x  2     dx    pdf of ND   Use if Nbins manually set  new area sum f  N   diff of consecutive edges constant   Use if histogarm   Normalization probability new area sum f    1 0000   No bar   needed here with histogram   Normalization probability hold on  plot x g  r   hold off   Output in Fig  2 and expected output is met  area 1 0000    Matlab  2016a System  Linux Ubuntu 16 04 64 bit Linux kernel 4 6

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My answer to this is the same as in an answer to your earlier question  For a probability density function  the integral over the entire space is 1  Dividing by the sum will not give you the correct density  To get the right density  you must divide by the area  To illustrate my point  try the following example    f  x    hist randn 10000  1   50     Create histogram from a normal distribution  g   1   sqrt 2   pi    exp -0 5   x    2     pdf of the normal distribution    METHOD 1  DIVIDE BY SUM figure 1  bar x  f   sum f    hold on plot x  g   r    hold off    METHOD 2  DIVIDE BY AREA figure 2  bar x  f   trapz x  f    hold on plot x  g   r    hold off   You can see for yourself which method agrees with the correct answer  red curve      Another method  more straightforward than method 2  to normalize the histogram is to divide by sum f   dx  which expresses the integral of the probability density function  i e     METHOD 3  DIVIDE BY AREA USING sum   figure 3  dx   diff x 1 2   bar x  f   sum f   dx    hold on plot x  g   r    hold off

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There is an excellent three part guide for Histogram Adjustments in MATLAB  broken original link  archive org link    the first part is on Histogram Stretching

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hist can not only plot an histogram but also return you the count of elements in each bin  so you can get that count  normalize it by dividing each bin by the total and plotting the result using bar  Example   Y   rand 10 1   C   hist Y   C   C    sum C   bar C    or if you want a one-liner   bar hist Y     sum hist Y      Documentation    hist bar   Edit  This solution answers the question How to have the sum of all bins equal to 1  This approximation is valid only if your bin size is small relative to the variance of your data  The sum used here correspond to a simple quadrature formula  more complex ones can be used like trapz as proposed by R  M

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f x  hist data    The area for each individual bar is height width  Since MATLAB will choose equidistant points for the bars  so the width is   delta x   x 2  - x 1    Now if we sum up all the individual bars the total area will come out as  A sum f  delta x   So the correctly scaled plot is obtained by  bar x  f sum f   x 2 -x 1

[matlab] How to normalize a histogram in MATLAB?

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