Statistics

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Descriptive statistics summarize and describe the main features of a dataset, using measures like mean, median, mode, and standard deviation. Inferential statistics go further by using sample data to make generalizations, predictions, or inferences about a larger population, often involving hypothesis testing and confidence intervals.

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Descriptive statistics tell you what the data in front of you looks like. Inferential statistics use that data to make educated guesses about a bigger group you haven't fully measured.

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The mean is the arithmetic average, calculated by summing all values and dividing by the count. The median is the middle value when data is sorted, robust to outliers. The mode is the most frequently occurring value. For skewed distributions, the median is often preferred over the mean as a measure of central tendency.

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The mean is the 'fair share' average. The median is the exact middle number. The mode is the most popular number. They each tell a different story about what's 'typical.'

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Standard deviation measures how spread out data values are from the mean, calculated as the square root of variance. A low standard deviation means data points cluster near the mean, while a high one indicates wide dispersion. It's fundamental in statistics for understanding variability, detecting outliers, and in the context of the normal distribution, roughly 68% of data falls within one standard deviation.

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Standard deviation tells you how 'scattered' the numbers are — a small value means they're all huddled close to the average, and a big value means they're spread all over the place.

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Correlation measures the strength and direction of a linear relationship between two variables, commonly using Pearson's correlation coefficient (r), which ranges from -1 to +1. Correlation does not imply causation because a third (confounding) variable may explain the relationship, or the association may be coincidental. Establishing causation requires controlled experiments or rigorous causal inference methods.

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Correlation means two things move together — when one goes up, the other tends to go up too. But that doesn't mean one causes the other; maybe a third factor is behind both.

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A population includes every member of the group you're interested in, while a sample is a subset selected for study. Population values are called parameters (like mu for the population mean), while sample values are called statistics (like x-bar for the sample mean). Since measuring an entire population is usually impractical, we use samples and statistical inference to draw conclusions about the population.

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The population is everyone you care about. The sample is the smaller group you actually survey. You use the sample to make your best guess about the whole population.

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Probability deals with predicting the likelihood of future events given known parameters (the forward problem). Statistics works in reverse — using observed data to infer the unknown parameters or processes that generated it (the inverse problem). Probability provides the mathematical foundation on which statistical methods are built.

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Probability is like predicting what will happen if you roll dice. Statistics is like looking at the dice rolls that already happened and figuring out whether the dice are fair.

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The normal (Gaussian) distribution is a symmetric, bell-shaped probability distribution defined by its mean and standard deviation. It's important because many natural phenomena follow it, the Central Limit Theorem shows that sample means approach normality regardless of the underlying distribution, and it forms the basis for many statistical tests like t-tests and z-tests.

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The normal distribution is a bell-shaped curve where most values cluster around the middle and fewer appear toward the edges — it shows up everywhere in nature, like heights or test scores.

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The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the original population's distribution shape, provided the sample size is sufficiently large (typically n >= 30). This is the theoretical foundation for many inferential statistics methods, including hypothesis tests and confidence intervals for means.

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The Central Limit Theorem says that if you repeatedly take groups of data and calculate each group's average, those averages will form a bell curve, even if the original data wasn't bell-shaped at all.

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A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter. The confidence level (commonly 95%) indicates the long-run proportion of such intervals that would contain the true value. A narrower interval implies more precision, while a wider interval reflects greater uncertainty.

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A confidence interval is like saying 'I'm pretty sure the real answer falls somewhere between these two numbers,' with the confidence level telling you how sure you are.

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A hypothesis test evaluates two competing claims about a population: the null hypothesis (H0), representing no effect or no difference, and the alternative hypothesis (Ha), representing the effect or difference you're testing for. You calculate a test statistic from the sample and its corresponding p-value, then compare the p-value to a significance level (like 0.05). If the p-value is below the threshold, you reject H0.

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A hypothesis test is like a courtroom trial for data — you assume the null hypothesis (nothing unusual) is true unless the evidence (p-value) is strong enough to prove otherwise.

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The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. Common misinterpretations include believing it's the probability that the null hypothesis is true, or the probability the result occurred by chance — neither is correct. A small p-value indicates the data is inconsistent with the null, not that the effect is practically significant.

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The p-value is the probability of seeing data at least as extreme as yours if nothing real is going on (null hypothesis is true). Common mistakes include thinking it's the probability the null is true, or the probability your result is due to chance.

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A Type I error (false positive) occurs when you reject a true null hypothesis, while a Type II error (false negative) occurs when you fail to reject a false null hypothesis. The probability of a Type I error is controlled by the significance level (alpha), while the probability of a Type II error (beta) is reduced by increasing sample size or effect size. Statistical power is 1 - beta.

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A Type I error is like convicting an innocent person. A Type II error is like letting a guilty person go free. You have to balance how many of each type of mistake you're willing to tolerate.

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A t-test compares the means of one or two groups, used when the population standard deviation is unknown and/or the sample size is small. A one-sample t-test compares a sample mean to a known value, an independent two-sample t-test compares means of two separate groups, and a paired t-test compares means from the same group at two times. The t-distribution adjusts for the extra uncertainty of estimating the standard deviation from a small sample.

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A t-test is like asking 'are these two groups really different, or did they just happen to look different this time?' It works especially well when you don't have a lot of data.

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Parametric tests assume the data follows a known distribution (usually normal) and work with the raw values, offering greater statistical power when assumptions are met. Non-parametric tests make fewer or no distributional assumptions, working with ranks or orderings instead of raw values, making them more robust when data is skewed, ordinal, or violates normality assumptions.

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Parametric tests are like using a precise ruler — great when everything fits the standard assumptions. Non-parametric tests are more flexible, working even when the data doesn't follow the usual rules.

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Regression analysis models the relationship between a dependent variable and one or more independent variables. Linear regression fits a straight line to minimize the sum of squared residuals. R-squared measures the proportion of variance in the dependent variable explained by the model. Multiple regression includes more than one predictor, and the method can be extended to non-linear relationships.

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Regression draws the best-fitting line through a scatter of data points to describe how one thing changes in response to another, and tells you how closely the line matches the data.

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Sampling bias occurs when the sample is not representative of the population because certain groups are systematically over- or under-represented. Common types include selection bias, undercoverage, and voluntary response bias. It leads to misleading conclusions because the results cannot be generalized to the population. Random sampling is the primary method for reducing bias.

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Sampling bias is like only asking your friends what they think and assuming everyone in the country agrees — your group wasn't chosen fairly, so the results are skewed.

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A two-tailed test checks for any difference in either direction (greater than or less than), splitting the significance level between both tails of the distribution. A one-tailed test checks for a difference in only one direction (greater than or less than), concentrating the entire significance level in one tail. One-tailed tests are used when you have a directional hypothesis based on theory or prior evidence.

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A two-tailed test asks 'is this different in any way?' A one-tailed test asks 'is this bigger (or smaller) specifically?' One-tailed tests are more powerful but require you to know which direction you're looking for.

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The chi-square test is used for categorical data. A chi-square goodness-of-fit test checks whether observed frequencies match expected frequencies for a single variable. A chi-square test of independence checks whether two categorical variables are related in a contingency table. Both compare observed versus expected counts, with large differences producing high chi-square statistics and small p-values.

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The chi-square test checks whether the numbers you observed match what you'd expect, like comparing actual candy counts from a bag to what the label claims.

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ANOVA (Analysis of Variance) tests whether the means of three or more groups are significantly different, using the F-statistic which compares between-group variance to within-group variance. It's used instead of multiple t-tests to control the overall Type I error rate. If ANOVA finds a significant difference, post-hoc tests like Tukey's HSD identify which specific groups differ.

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ANOVA is like asking 'are all these groups really the same, or is at least one different?' It's the tool you use when you have more than two groups to compare.

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Frequentist statistics treats probability as the long-run frequency of events and evaluates hypotheses using p-values and confidence intervals without incorporating prior beliefs. Bayesian statistics treats probability as a degree of belief, combining prior information with observed data via Bayes' theorem to produce a posterior distribution that updates as new data arrives. Bayesian methods are preferred when incorporating domain knowledge or when data is limited.

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Frequentists only trust what the current data shows. Bayesians combine what they already believed with what the new data shows, updating their beliefs as evidence accumulates.