Statistics for Tourism Research: Central Tendency, Dispersion, Skewness and Kurtosis, Correlation and Regression (Scatter Plots, Lines of Best Fit, Pearson and Spearman, Bivariate and Multivariate Regression), Discrete and Continuous Distributions, the Normal and Sampling Distributions, and Hypothesis Testing with t, ANOVA, Chi-Square, Run, Sign, Wald–Wolfowitz, Kruskal–Wallis and Kolmogorov–Smirnov Tests

The mean is the sum of values divided by their number; the median is the middle value of the ordered data; the mode is the most frequent value. For symmetrical data they coincide; for the spending of tourists, where a few big spenders pull the distribution to the right, the mean exceeds the median, and the median is the better "typical" figure. Dispersion measures spread: the range (largest minus smallest), the quartile deviation, the mean deviation, the variance, which is the average squared deviation from the mean (for a sample, Σ(x − x̄)² ÷ (n − 1)), and the standard deviation, its square root, in the original units. The coefficient of variation, standard deviation ÷ mean × 100, compares spread between series with different units or levels.

The sample variance is 40 ÷ (5 − 1) = 10, the sample standard deviation √10 ≈ 3.16 thousand rupees, and the coefficient of variation 3.16 ÷ 8 × 100 ≈ 39.5 per cent. The range is 12 − 4 = 8. If the five were the whole population, the variance would be 40 ÷ 5 = 8; dividing by n − 1 corrects the tendency of a sample to understate the population's spread.

Skewness measures asymmetry. A distribution with a long tail to the right is positively skewed (mean > median > mode), as tourist spending and length of stay usually are; a long tail to the left is negative skew. Karl Pearson's coefficient is (mean − mode) ÷ standard deviation, or approximately 3(mean − median) ÷ standard deviation. Kurtosis measures the peakedness and weight of the tails: a normal curve is mesokurtic (moment coefficient β₂ = 3), a sharper, heavier-tailed curve leptokurtic (β₂ > 3) and a flatter one platykurtic (β₂ < 3). A discrete distribution takes separate values, such as the number of trips a household makes (binomial and Poisson distributions); a continuous one takes any value in a range, such as spending or time. The normal distribution is continuous, symmetrical and bell-shaped, fixed by its mean and standard deviation: about 68 per cent of values lie within one standard deviation of the mean, 95 per cent within 1.96 (about two) and 99.7 per cent within three. Any normal value can be standardised as z = (x − mean) ÷ standard deviation.

A sampling distribution is the distribution of a statistic, such as the sample mean, over all possible samples of a given size from a population. By the central limit theorem the sampling distribution of the mean is approximately normal for large samples (commonly n ≥ 30), whatever the shape of the population, with a mean equal to the population mean and a standard deviation, the standard error, equal to σ ÷ √n. This is what allows a survey of a few hundred visitors to estimate the average spending of all visitors, with a confidence interval of the sample mean ± 1.96 standard errors at the 95 per cent level.

A scatter plot of two variables, for example advertising spend and arrivals, shows the form and strength of their relationship before any coefficient is computed. Pearson's product-moment correlation coefficient r measures the strength of a linear relationship between two interval or ratio variables, from −1 (perfect negative) through 0 (no linear relationship) to +1 (perfect positive). Spearman's rank correlation coefficient ρ (rho) works on ranks, so it suits ordinal data such as the ranking of hotels by two judges or relationships that are monotonic but not linear: ρ = 1 − 6Σd² ÷ [n(n² − 1)], where d is the difference between the two ranks of each item. Correlation does not show causation: arrivals and ice-cream sales both rise in summer without either causing the other.

Here ρ = 1 − (6 × 4) ÷ [5 × (25 − 1)] = 1 − 24 ÷ 120 = 1 − 0.2 = 0.8, a strong agreement between the judges. Regression goes further than correlation: it estimates how one variable changes with another, fitting the line of best fit by least squares, which minimises the sum of squared vertical deviations of the points from the line. In bivariate (simple) regression, y = a + bx, where b is the slope (the change in y for a one-unit change in x) and a the intercept; if the fitted line of daily visitors on the number of attractions open is y = 50 + 2.5x, a site with 10 attractions is predicted to draw 50 + 25 = 75 visitors. Multiple (multivariate) regression, y = a + b₁x₁ + b₂x₂ + ..., explains y by several predictors at once, such as demand by income, price and exchange rate; R² gives the share of the variation in y explained, and highly correlated predictors (multicollinearity) make the separate coefficients unreliable.

A hypothesis test decides whether sample evidence is strong enough to reject a null hypothesis (H₀, usually "no difference" or "no relationship") in favour of an alternative (H₁). The significance level α, commonly 0.05, is the probability of a Type I error, rejecting a true H₀; a Type II error is failing to reject a false H₀, and the power of a test is 1 minus its probability. The p-value is the probability, if H₀ were true, of a result at least as extreme as the one observed; H₀ is rejected when p < α. A two-tailed test looks for a difference in either direction, a one-tailed test in one stated direction. Parametric tests (t, ANOVA, Pearson's r) assume interval or ratio data, a normal distribution (or large samples) and, for comparisons, similar variances; non-parametric tests make fewer assumptions, work on ranks, signs or counts, and suit ordinal data, small samples and skewed distributions, at some cost in power when the parametric assumptions do hold.

The choice follows from three questions: what kind of data (categorical counts, ordinal ranks, or interval and ratio measurements), how many groups or samples, and whether the samples are independent or related. For two independent groups on interval data use the independent t-test, and on ordinal data the Mann–Whitney U test or the Wald–Wolfowitz test; for two related measurements on interval data the paired t-test, and on ordinal data the Wilcoxon signed-rank test or, using only direction, the sign test; for three or more independent groups ANOVA on interval data and Kruskal–Wallis on ordinal data; for association between two categorical variables the chi-square test; for the relationship between two measured variables Pearson's r, and between two ranked variables Spearman's ρ; for the fit of a sample to a distribution the Kolmogorov–Smirnov test (or chi-square goodness of fit); and for the randomness of a sequence the run test.