Statistics tutoring in English for business students
A statistics tutor for business and economics students in Spain, working entirely in English if that suits you. Descriptive statistics, probability, distributions, confidence intervals, hypothesis testing and regression, one to one and online.
Statistics is the module where people can do every calculation and still lose marks, because the exam asks what the number means. A p-value of 0.03 is not an answer on its own. We spend as much time on the sentence that follows the calculation as on the calculation itself, since that is where the marks actually sit.
Descriptive statistics and what the summary hides
Measures of location, mean, median and mode, and measures of dispersion, range, variance, standard deviation and the coefficient of variation. Skewness and kurtosis, and reading a boxplot or a histogram. The point is knowing which summary to use: the mean is the wrong measure for skewed income data, and the coefficient of variation is what lets you compare dispersion across variables measured in different units.
Probability and the rules people get backwards
Sample space and events, the addition and multiplication rules, conditional probability and independence, the law of total probability and Bayes' theorem. Bayes is where most people struggle, and it becomes much easier with a tree diagram or a contingency table than with the formula alone, so that is how we work it. We also cover combinatorics, enough for the counting problems your paper will contain.
Distributions: binomial, Poisson and normal
Discrete and continuous random variables, expected value and variance. The binomial for a fixed number of independent trials, the Poisson for counts in an interval of time or space, and the normal for continuous data, with standardisation and reading the tables. We cover the normal approximation to the binomial and, importantly, the central limit theorem, which is what licenses most of the inference that follows.
Sampling, estimation and confidence intervals
Sampling distributions and the standard error, point estimators and what makes one unbiased or efficient. Confidence intervals for a mean with known and unknown variance, hence when to use the normal and when the t distribution, for a proportion, and for a difference between two groups. We also work on sample size determination, which is a standard exam question and takes two minutes once you can rearrange the formula.
Hypothesis testing, and how to write the conclusion
Setting up the null and alternative hypotheses, the significance level, type I and type II errors, and the difference between a one-tailed and a two-tailed test. Tests on a mean, a proportion and the difference between two groups, plus chi-square for independence and goodness of fit. The part we spend most time on is the conclusion, written in the language of the original problem rather than as a bare statement about rejecting a hypothesis.
Correlation and regression
Covariance and the correlation coefficient, then simple linear regression by least squares, interpreting the slope and the intercept in the units of the problem, and the coefficient of determination. From there, prediction and the danger of extrapolating outside the observed range, and an introduction to multiple regression with the interpretation of partial coefficients and the model significance tests.
Where almost everyone gets stuck
Choosing the distribution by habit rather than by the wording
Binomial when there is a fixed number of independent trials with a constant probability. Poisson when you are counting occurrences within an interval of time or space. Normal when the problem gives you the mean and standard deviation of a continuous, symmetric variable described as normal, or when the central limit theorem applies. Being continuous does not make a variable normal, and the question wording is where the answer is.
Reading a p-value as the probability that the hypothesis is true
It is not. The p-value is the probability of observing data at least as extreme as yours if the null hypothesis were true. That distinction is examined directly in written questions, and it also stops you from writing that you have proved the null hypothesis, which you never can.
Treating correlation as causation, or extrapolating beyond the data
A high coefficient of determination tells you the line fits the observations you have. It says nothing about cause, and nothing at all about what happens outside the range of the data you fitted. Exam questions frequently ask you to predict at a value outside that range precisely to see whether you flag it.
Worked example: simple linear regression, R squared and prediction
A company records five months of advertising spend X (in thousands of euros) and sales Y (in tens of thousands of euros). The data are: X = 1, 2, 3, 4, 5 and Y = 3, 5, 4, 8, 10. Find the least squares regression line of Y on X, the coefficient of determination R squared, and the predicted sales for an advertising spend of 6 thousand euros.
- Compute the means: mean of X = (1+2+3+4+5)/5 = 15/5 = 3; mean of Y = (3+5+4+8+10)/5 = 30/5 = 6.
- Work with deviations from the mean. In X: -2, -1, 0, 1, 2. In Y: -3, -1, -2, 2, 4. So Sxx = 4+1+0+1+4 = 10; Syy = 9+1+4+4+16 = 34; Sxy = (-2)(-3)+(-1)(-1)+(0)(-2)+(1)(2)+(2)(4) = 6+1+0+2+8 = 17.
- Find the slope and intercept: b = Sxy/Sxx = 17/10 = 1.7; a = mean of Y - b times mean of X = 6 - 1.7 x 3 = 6 - 5.1 = 0.9. The fitted line is Y = 0.9 + 1.7X.
- Assess the fit: R squared = Sxy^2/(Sxx x Syy) = 17^2/(10 x 34) = 289/340 = 0.85. The model explains 85% of the variability in sales, and the correlation coefficient is r = +0.922 (the square root of 0.85, positive because Sxy is positive).
- Substitute X = 6 into the line: Y = 0.9 + 1.7 x 6 = 0.9 + 10.2 = 11.1. Interpret the slope in units: for every extra thousand euros of advertising, estimated sales rise by 1.7 tens of thousands of euros, that is 17,000 euros.
SolutionThe regression line is Y = 0.9 + 1.7X, with R squared = 0.85 (r = 0.922). For an advertising spend of 6 thousand euros the predicted sales are 11.1 tens of thousands of euros, that is 111,000 euros. One detail that earns marks: X = 6 lies outside the observed range (1 to 5), so this is an extrapolation and it is worth saying so in the exam.
About these lessons in particular
Do you cover the software as well, such as Excel, R or SPSS?
Yes, if your course is assessed on it. We start with the calculation by hand so you know what the output means, and then reproduce it in whichever tool your module uses.
My module is called Econometrics. Is that the same?
It is the natural continuation. If your course is mainly regression, hypothesis testing and model diagnostics, we can work on it. Send me the syllabus and I will tell you honestly whether I am the right fit.
Can the lessons be in English?
Yes, completely. I live and study in the United States, so statistical terminology in English is what I use daily, and we can switch to Spanish whenever a concept lands better that way.
I also teach
Shall we work on it together?
Tell me where you are, which university you are at and when the exam is. I will get back to you as soon as I can.