Basic Statistics

## Learning Objectives - Calculate descriptive statistics - Perform statistical analysis - Use built-in statistical functions - Analyze data distributions ## Descriptive Statistics ### Central Tendency ```matlab data = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]; % Mean (average) mean(data) % 55 % Median (middle value) median(data) % 55 % Mode (most frequent) mode(data) % 10 (first occurrence if multimodal) ``` ### Variation/Spread ```matlab data = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]; % Standard deviation std(data) % Sample std (N-1) std(data, 1) % Population std (N) % Variance var(data) % Sample variance (N-1) var(data, 1) % Population variance (N) % Range range(data) % 90 (max - min) ``` ### Quartiles and Percentiles ```matlab data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; % Quartiles q1 = quantile(data, 0.25) % 3.25 q2 = quantile(data, 0.50) % 5.5 (same as median) q3 = quantile(data, 0.75) % 7.75 % Using prctile p25 = prctile(data, 25) % 3.25 p50 = prctile(data, 50) % 5.5 p75 = prctile(data, 75) % 7.75 % IQR (Interquartile Range) iqr(data) % 4.5 ``` ## Statistical Functions ### Basic Functions ```matlab data = [5, 10, 15, 20, 25]; min(data) % 5 max(data) % 25 sum(data) % 75 prod(data) % 312500 (5*10*15*20*25) cumsum(data) % [5, 15, 30, 50, 75] cumprod(data) % [5, 50, 750, 15000, 375000] ``` ### Summary Statistics ```matlab data = randn(1000, 1) + 5; % Normal distribution % All basic stats summary = [mean(data), median(data), mode(data)', std(data), var(data)]; summary = [min(data), max(data), range(data)]; % Using histfit for visualization histfit(data) ``` ## Random Numbers ### Random Number Generation ```matlab % Uniform random (0 to 1) r = rand(1, 5); % [0.8147, 0.9058, ...] % Uniform random in range [a, b] r = a + (b-a) * rand(1, 5); % Normal (Gaussian) random r = randn(1, 5); % Mean 0, std 1 % Normal with specific mean and std r = mean + std * randn(1, 5); % Random integers r = randi([1, 10], 1, 5); % Random integers 1-10 ``` ### Random Sampling ```matlab data = 1:100; % Random permutation perm = randperm(100); % Random ordering 1-100 sample = data(perm(1:10)); % 10 random samples % With replacement (bootstrap) idx = randi(length(data), 1, 10); sample = data(idx); % Shuffle array shuffled = data(randperm(length(data))); ``` ## Distributions ### Normal Distribution ```matlab % PDF (Probability Density Function) x = -3:0.1:3; y = normpdf(x, 0, 1); % Mean 0, Std 1 % CDF (Cumulative Distribution Function) p = normcdf(x, 0, 1); % Inverse CDF (Quantile function) x = norminv(0.975, 0, 1); % 97.5th percentile % Random numbers from normal r = normrnd(0, 1, 100, 1); ``` ### Distribution Functions | Function | Distribution | |----------|--------------| | normpdf, normcdf, norminv, normrnd | Normal | | exppdf, expcdf, expinv, exprnd | Exponential | | poisspdf, poisscdf, poissinv, poissrnd | Poisson | | binopdf, binocdf, binoinv, binornd | Binomial | | unifpdf, unifcdf, unifinv, unifrnd | Uniform | ### Distribution Fitting ```matlab % Fit normal distribution to data data = normrnd(5, 2, 1000, 1); pd = fitdist(data, 'Normal'); % Get parameters mu = pd.mu % Estimated mean sigma = pd.sigma % Estimated std % Goodness of fit [h, p] = kstest(data); % Kolmogorov-Smirnov test ``` ## Hypothesis Testing ### t-Test ```matlab % One-sample t-test (mean = mu) data = [10.2, 9.8, 10.1, 10.3, 9.9]; [h, p, ci] = ttest(data, 10); % Two-sample t-test group1 = randn(100, 1) + 5; group2 = randn(100, 1) + 6; [h, p, ci] = ttest2(group1, group2); ``` ### Other Tests ```matlab % Chi-square test observed = [10, 20, 30]; expected = [15, 15, 30]; [h, p] = chisquare(observed, expected); % Kolmogorov-Smirnov test data = normrnd(0, 1, 100, 1); [h, p] = kstest(data); % Test if normal % ANOVA group1 = randn(30, 1) + 5; group2 = randn(30, 1) + 6; group3 = randn(30, 1) + 7; [p, tbl] = anova1([group1, group2, group3]); ``` ## Correlation and Covariance ```matlab x = 1:10; y = 2 * x + randn(1, 10); % Linear relationship % Correlation coefficient r = corrcoef(x, y); % 2x2 matrix r = r(1, 2); % Correlation: ~0.99 % Covariance C = cov(x, y); % 2x2 covariance matrix % Pearson correlation (linear) [R, P] = corr(x', y'); % R=corr, P=p-value ``` ## Regression ### Simple Linear Regression ```matlab x = 1:10; y = 2 * x + 5 + randn(1, 10); % Fit linear model mdl = fitlm(x', y'); % Predictions y_pred = predict(mdl, x'); % Statistics coef = mdl.Coefficients.Estimate; % [intercept, slope] rsq = mdl.Rsquared.Ordinary; % R-squared ``` ### Polynomial Regression ```matlab x = 0:0.1:5; y = x.^2 - 2*x + 1 + randn(1, 51) * 0.5; % Fit polynomial (degree 2) p = polyfit(x, y, 2); % [1, -2, ~1] % Evaluate y_fit = polyval(p, x); % Plot plot(x, y, 'o', x, y_fit, '-') ``` ## Moving Statistics ```matlab data = randn(100, 1); % Moving average (window = 5) window = 5; ma = conv(data, ones(window, 1)/window, 'same'); % Moving sum ms = conv(data, ones(window, 1), 'same'); % Cumulative statistics cumsum(data) cummax(data) cummin(data) ``` ## Data Binning ```matlab data = randn(1000, 1) * 10 + 50; % Histogram counts [n, edges] = hist(data, 20); % Bin centers bin_centers = (edges(1:end-1) + edges(2:end)) / 2; % Discretize data bins = discretize(data, 0:10:100); % Bin into 0-10, 10-20, etc. ``` ## Summary Statistics by Group ```matlab % Using groupsummary (R2019b+) data = table(); data.Value = randn(100, 1); data.Group = categorical(randi(3, 100, 1)); % Summary by group summary = groupsummary(data, 'Group', {'mean', 'std', 'median'}); % Using grpstats (older) stats = grpstats(data.Value, data.Group, 'mean'); ``` ## Normality Tests ```matlab data = randn(1000, 1); % Normal data % Jarque-Bera test [h, p] = jbtest(data); % Kolmogorov-Smirnov test [h, p] = kstest(data); % Lilliefors test [h, p] = lillietest(data); ``` ## Summary - `mean()`, `median()`, `mode()` for central tendency - `std()`, `var()`, `range()` for spread - `quantile()`, `prctile()` for percentiles - `rand()`, `randn()`, `randi()` for random numbers - `fitdist()` to fit distributions to data - `ttest()`, `ttest2()` for hypothesis testing - `corrcoef()` for correlation - `fitlm()` for linear regression - Use vectorized operations for efficiency