DSST Principles of Statistics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Skewness sign

Positive skewness means the right tail is longer (mean typically pulled right of the median); negative skewness means the left tail is longer. A symmetric distribution has skewness zero.

Excess kurtosis

Excess kurtosis equals fourth standardized moment minus 3, so the normal distribution scores 0. Positive values indicate heavier tails than normal (leptokurtic); negative values indicate lighter tails (platykurtic).

Standard error of the mean

The standard deviation of the sampling distribution of the sample mean, equal to σ/√n or estimated by s/√n. It quantifies how much x̄ varies from sample to sample.

Central limit theorem

For independent observations with finite variance, the sampling distribution of the sample mean approaches normality as sample size grows, regardless of the population shape, enabling z and t inference.

Type I versus Type II error

Type I error is rejecting a true null hypothesis, with probability α. Type II error is failing to reject a false null, with probability β; power equals 1 minus β.

Meaning of a p-value

The probability, computed assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the observed one. It is not the probability the null is true.

Power determinants

Statistical power rises when the true effect is larger, sample size is larger, population variability is smaller, or the chosen significance level α is larger. Power equals one minus Type II error rate.

Law of large numbers

As sample size increases, the sample mean converges toward the population mean. This describes accuracy of estimation, distinct from the central limit theorem, which describes the shape of the sampling distribution.

Binomial mean and variance

For a binomial count with n independent trials and success probability p, the mean is np and the variance is np(1−p). The standard deviation is the square root of np(1−p).

Poisson distribution properties

For a Poisson count with rate λ, the probability of k events is e^(−λ)λ^k/k!. Mean and variance both equal λ, and events occur independently in a fixed interval.

Geometric distribution

Models the trial number of the first success in repeated independent trials with probability p. Probability of success on trial k is (1−p)^(k−1)p; mean is 1/p; it is memoryless.

Hypergeometric distribution

Models the number of successes when sampling n items without replacement from a finite population of size N containing K successes. Trials are dependent, unlike the binomial case.

Finite population correction

The factor (N−n)/(N−1) multiplies the variance of a sample mean or proportion when sampling without replacement from a finite population of size N. It approaches 1 for large N.

Continuous uniform distribution

Has constant density 1/(b−a) over the interval from a to b. Its mean is (a+b)/2 and its variance is (b−a)²/12. Probability equals interval length divided by (b−a).

Empirical rule

For approximately normal data, about 68% of observations lie within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.

Standard normal transformation

Any normal variable converts to standard normal via z = (x − μ)/σ. The resulting z has mean 0 and standard deviation 1, allowing use of standard normal tables.

Exponential memorylessness

For an exponential waiting time with rate λ, the conditional probability of waiting t more units given survival to time s equals the unconditional probability of waiting t units.

Sum of independent normals

If independent variables are normally distributed, their sum is normal with means added and variances added. Standard deviations do not add; combine variances first, then take the square root.

Simple regression slope formula

In least squares regression, the slope equals r times sy divided by sx, where r is the correlation and sy, sx are the sample standard deviations of response and predictor.

Regression intercept meaning

The predicted value of y when the predictor x equals zero. It is often outside the observed data range, so it may lack a meaningful physical interpretation.

Coefficient of determination

R² equals the proportion of total variation in the response explained by the regression, computed as SSR/SST or 1 − SSE/SST. In simple regression it equals r².

Residual standard error

An estimate of the standard deviation of the regression errors, computed as the square root of SSE divided by n minus 2 in simple linear regression.

Slope t test

Tests whether the population slope equals zero using t = b1/SE(b1) with n minus 2 degrees of freedom. In simple regression its square equals the overall F statistic.

Regression assumptions

Errors should be independent, normally distributed, have constant variance, and relate linearly to predictors. In multiple regression, predictors should not be severely multicollinear.

Extrapolation risk

Predicting the response for predictor values outside the observed data range. The fitted linear relationship may not hold there, so such predictions are unreliable.

Adjusted R-squared

A version of R² that penalizes adding predictors, adjusting for degrees of freedom. It rises only when a new predictor improves fit more than expected by chance.

Confidence interval structure

An interval of the form estimate plus or minus a critical value times a standard error. The critical value comes from z or t depending on whether σ is known.

t critical values

Used for means when σ is unknown, with n minus 1 degrees of freedom. They exceed corresponding z values and approach them as sample size increases.

Confidence interval interpretation

If sampling were repeated many times, about 95% of intervals built by this procedure would contain the true parameter. A single realized interval either contains it or does not.

Sample size for margin of error

To achieve margin E for a mean, required n equals (zσ/E)², rounded up. Quadrupling sample size halves the margin of error.

Interval-test duality

A two-sided hypothesis test at significance α rejects the null value exactly when that value lies outside the corresponding 100(1−α)% confidence interval for the parameter.

Prediction versus confidence interval

A prediction interval for a single new observation is wider than a confidence interval for the mean response, because it adds the variability of the individual observation around the fitted value.