Fitting Histogram Data to a Normal Distribution - How to incorporate chi-sq?

Hi

I fitted a gaussian to a set of histogram data I had but my advisor told me that chi-sq is not minimized and the profile was computed using mu and stdev only and it isn't working with the three parameter mean stdev and chi-sq minimization. Can someone explain to me how normfit and normpdf do chi-sq minimization? I will admit that using built-in functions in matlab sometimes prevents me from being able to know what exactly is going on. Here is a jist of what my code is doing:

y=S(:,7)-Avg;

% Fit
N = sum(~isnan(y));
[muhat,sigmahat] = normfit(y);

% Bin
xmin = -5; % lower bound
xmax = 5; % upper bound
h = 0.1; % bin width
edges = xmin:h:xmax;
% Histogram
n = histc(y,edges);
bar(edges,n,'histc');
hbar = findobj(gca,'Type','patch');
set(hbar,'FaceColor','y','EdgeColor','k')
hold;
z=(-3*sigmahat+muhat:0.1*sigmaha:3*sigmahat+muhat);
f = N*h*normpdf(z,muhat,sigmahat);
plot(z,f,'r-','Linewidth',1.5);
xlim([-3.5 3.5]); set(gca,'XTick',[-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5]) 
ylim([0 80]);grid on;
xlabel('Position Variations (um)');
ylabel('Frequency')
legend('Histogram','Normal Distribution Fit')
title('Average Subtracted Measurements - New Scanner (Data Set 1)');
text(-2.5,200,sprintf('mu = %1.2g\nstd = %1.4g',muhat,sigmahat));
hold off;

Answer 1

[Tom Lane]

The normfit function estimates mu and sigma from the raw data.

It's possible to estimate mu and sigma from the histogram bin counts instead. The histogram gives observed values O for each bin. Using mu and sigma you can compute expected values E for each bin. You could compute a chi-square measure such as sum(((O-E).^2 ./ E), and you could select mu and sigma to minimize that. But that's not what normfit does. You might be able to carry this out yourself using fminsearch.

[Marmi Afrin]

I was thinking of getting error bars on the histogram and fitting from there. For E computation are you referring to mu and sigma from the raw data or mu and sigma from the data generated above to plot the histograms?

[Tom Lane]

I am thinking of E being a function of mu and sigma, and using something like fminsearch to select the values of mu and sigma that minimize the chi-square criterion.