Statistics

rng(7)
data = 2 + 0.7 * randn(1, 4000);

histogram(data, 45)
hold on
mu = mean(data); sd = std(data);
x = linspace(min(data), max(data), 300);
binw = (max(data) - min(data)) / 45;
plot(x, numel(data) * binw * exp(-(x - mu).^2 / (2*sd^2)) / (sd*sqrt(2*pi)), 'LineWidth', 2)
hold off
title(sprintf('μ = %.3f,  σ = %.3f', mu, sd))
legend('data', 'normal fit')

rng(3)
x = linspace(0, 5, 25);
y = 0.6*x.^2 - 1.5*x + 2 + 0.6*randn(size(x));

p = polyfit(x, y, 2);
xf = linspace(0, 5, 200);

subplot(2, 1, 1)
errorbar(x, y, 0.6*ones(size(x)), 'o')
hold on, plot(xf, polyval(p, xf), 'LineWidth', 2), hold off
title(sprintf('fit: %.2f x² %+.2f x %+.2f', p(1), p(2), p(3)))
legend('data ± σ', 'quadratic fit')

subplot(2, 1, 2)
stem(x, y - polyval(p, x))
yline(0, 'k:')
title('residuals')

rng(11)
n = 3000;
x = rand(1, n); y = rand(1, n);
inside = x.^2 + y.^2 <= 1;
scatter(x, y, 8, inside, 'filled')
colormap coolwarm
hold on
s = linspace(0, pi/2, 200);
plot(cos(s), sin(s), 'k', 'LineWidth', 1.5)
hold off
axis equal
title(sprintf('π ≈ 4 · %d / %d = %.4f', sum(inside), n, 4*mean(inside)))

% Means of uniform random numbers look more normal as n grows
rng(5)
ns = [1 2 5 30];
for k = 1:4
  subplot(2, 2, k)
  m = mean(rand(ns(k), 3000), 1);
  histogram(m, 30)
  title(sprintf('mean of %d uniform', ns(k)))
end

rng(2)
n = 300;
a = randn(n, 1); b = a + 0.5*randn(n, 1); c = -a + randn(n, 1);
d = randn(n, 1); e = b + d;
data = [a b c d e];
C = zeros(5);
for i = 1:5
  for j = 1:5
    r = corrcoef(data(:, i), data(:, j)); C(i, j) = r(1, 2);
  end
end
imagesc(C)
colormap coolwarm
clim([-1 1])
colorbar
title('Correlation between five variables')