Junior Statistician Qualification Examination flashcards

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

Sample variance formula

s squared equals the sum of squared deviations from the mean divided by n minus one. The n minus one denominator corrects the downward bias of the population variance estimate.

Coefficient of variation

The standard deviation divided by the mean, usually multiplied by 100 and reported as a percentage. It compares relative dispersion between data sets measured in different units or with very different means.

Interquartile range rule

IQR equals the third quartile minus the first quartile. Values below Q1 minus 1.5 IQR or above Q3 plus 1.5 IQR are commonly flagged as potential outliers for boxplot display.

Grouped data median

For class data, the median equals L plus the fraction n over 2 minus cumulative frequency below, divided by class frequency, times class width. L is the lower boundary of the median class.

Bayes theorem

The posterior probability of event A given B equals the likelihood of B given A times the prior of A, divided by the total probability of B summed over a partition of the sample space.

Binomial distribution conditions

Fixed number of independent trials, two mutually exclusive outcomes per trial, and constant success probability. Its mean is np and its variance is np times one minus p.

Poisson mean and variance

A count distribution with probability of k events equal to e to the minus lambda times lambda to the k over k factorial. Both mean and variance equal lambda.

Standard normal standardization

Subtract the mean and divide by the standard deviation to obtain a z score. About 68 percent of a normal distribution lies within one standard deviation, 95 percent within two, 99.7 percent within three.

Variance of a sum

For constants a and b, the variance of aX plus bY equals a squared times variance of X plus b squared times variance of Y plus twice ab times their covariance. The covariance term vanishes under independence.

Standard error of the mean

The standard deviation of the sampling distribution of the sample mean, equal to sigma divided by the square root of n. It is estimated by s over root n when sigma is unknown.

Confidence interval for a mean

Sample mean plus or minus a critical value times the standard error. Use the normal critical value when sigma is known and the t critical value with n minus one degrees of freedom otherwise.

Central limit theorem

For sufficiently large n, the sampling distribution of the sample mean is approximately normal with mean mu and variance sigma squared over n, regardless of the population shape.

Estimator mean squared error

Mean squared error equals the variance of the estimator plus the square of its bias. An unbiased estimator has zero bias, so its mean squared error reduces to its variance.

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 that the null hypothesis is true.

Type I and Type II errors

Type I error rejects a true null hypothesis and has probability alpha. Type II error fails to reject a false null hypothesis and has probability beta. Power equals one minus beta.

Paired t test

Compute differences within each matched pair, then run a one-sample t test on the mean difference. Degrees of freedom equal the number of pairs minus one.

Chi-square goodness of fit

Sum over categories of observed minus expected, squared, divided by expected. Degrees of freedom equal number of categories minus one minus the number of parameters estimated from the data.

Factors affecting test power

Power rises with larger sample size, larger true effect, larger alpha, and smaller variability. A one-sided test in the correct direction is more powerful than a two-sided test at the same alpha.

Least squares slope

The regression slope equals the covariance of x and y divided by the variance of x, equivalently the sum of cross products over the sum of squared x deviations. The line passes through both means.

Coefficient of determination

R squared equals the regression sum of squares divided by the total sum of squares, giving the proportion of variation in the response explained by the model. It ranges from zero to one.

Pearson correlation coefficient

A measure of linear association between minus one and plus one, computed as the covariance divided by the product of the two standard deviations. It is unchanged by linear rescaling of either variable.

Prediction versus confidence interval

A prediction interval covers a single future observation and is wider. A confidence interval covers the mean response at a given x and is narrower.

Stratified versus cluster sampling

Stratified sampling draws from every stratum and usually reduces variance. Cluster sampling selects entire groups and usually increases variance but lowers cost per sampled unit.

Randomization and blocking

Random assignment of treatments to units removes systematic bias from unknown confounders. Blocking groups similar units first so that treatment comparisons are made within homogeneous blocks.

Interaction in factorial design

An interaction exists when the effect of one factor on the response changes depending on the level of another factor. Main effects alone then give an incomplete description.

Sample size for a proportion

Required n equals the square of the critical z times p times one minus p, divided by the squared margin of error. Use p equal to 0.5 when no prior estimate exists.

Mann-Whitney U test

A rank-based test comparing two independent samples without assuming normality. It tests whether one distribution tends to produce larger values than the other.

Wilcoxon signed-rank test

A paired nonparametric test that ranks absolute differences and compares the sum of positive ranks. It assumes the differences are symmetrically distributed about their median.

McNemar test

A test for paired binary data that uses only the discordant pairs. The statistic is the squared difference of the two discordant counts divided by their sum.

Odds ratio interpretation

The odds of the outcome in the exposed group divided by the odds in the unexposed group, computed as ad over bc in a two-by-two table. It approximates relative risk when the outcome is rare.

Laspeyres and Paasche indices

Laspeyres uses base-period quantities and tends to overstate price change. Paasche uses current-period quantities and tends to understate it. Fisher's index is their geometric mean.

Exponential smoothing recurrence

The new smoothed value equals alpha times the current observation plus one minus alpha times the previous smoothed value. A larger alpha places more weight on recent observations.

Seasonal index by ratio method

Divide each observation by a centered moving average, average those ratios by season, then rescale so the seasonal indices sum to the number of seasons, or to 400 percent for quarters.

Floating-point comparison rule

Binary floating-point cannot represent most decimals exactly, so equality tests between computed values often fail. Compare absolute differences against a small tolerance such as one times ten to the minus eight.

Dummy variable coding

For a categorical predictor with k levels, create k minus one indicator variables and treat the omitted level as the reference. This avoids perfect collinearity with the intercept.