Kerala Plus Two Statistics (DHSE) flashcards

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

Primary vs secondary data

Primary data are collected first-hand by the investigator for the present purpose. Secondary data were collected earlier by someone else and are reused, so their reliability and original scope must be checked.

Inclusive class correction factor

When classes are written inclusively, such as 10–19, 20–29, the gap between the upper limit of one class and the lower limit of the next is halved. That half-gap is subtracted from upper limits and added to lower limits to get continuous boundaries.

Ogive

A cumulative frequency curve plotted against class boundaries. The less-than ogive rises to the total frequency, the more-than ogive falls from it, and their intersection gives the median graphically.

Coefficient of variation

The ratio of standard deviation to mean, expressed as a percentage. Because units cancel, it compares consistency of two series measured in different units; a smaller value indicates greater consistency.

Median of grouped data

The value L + [(N/2 − cf)/f]h, where L is the lower boundary of the median class, cf the cumulative frequency before it, f its frequency, h the class width, and N the total frequency.

Empirical mode formula

For moderately skewed frequency distributions, mode = 3 median − 2 mean. The relation fails for highly skewed or bimodal data and is only an approximation.

Change of origin and scale

Adding a constant to every observation shifts the mean by that constant but leaves variance and standard deviation unchanged. Multiplying by k multiplies the mean and standard deviation by k and the variance by k squared.

Quartile deviation

Half the interquartile range, (Q₃ − Q₁)/2. It measures spread of the middle fifty per cent and ignores extreme values, so it is resistant to outliers.

Product moment correlation

Karl Pearson's r equals the covariance of x and y divided by the product of their standard deviations. It is unit-free, lies between minus one and plus one, and measures only linear association.

Regression coefficient bxy

The slope of the line predicting x from y, equal to r times the ratio of the standard deviation of x to that of y. Its sign always matches the sign of the correlation coefficient.

Spearman rank correlation

A correlation computed on ranks rather than values, using 1 − 6Σd²/[n(n² − 1)], where d is the difference between paired ranks. Tied observations are given the average of the ranks they span.

Regression lines intersection

The two regression lines, y on x and x on y, always intersect at the point of means (x̄, ȳ). They coincide only in the perfect linear case where the correlation is plus or minus one.

Mutually exclusive events

Events that cannot occur in the same trial, so their intersection is empty and P(A ∪ B) = P(A) + P(B). They are not the same as independent events unless one has probability zero.

Classical probability

When outcomes are equally likely and the sample space is finite, the probability of an event equals the number of outcomes favourable to it divided by the total number of outcomes.

Addition theorem

For any two events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtracted term corrects for outcomes counted twice because the events overlap.

Independent events

Events whose occurrence does not alter each other's probability, characterised by P(A ∩ B) = P(A)P(B). Independence is a multiplicative condition, not a statement about being disjoint.

Conditional probability

The probability of A given that B has occurred, written P(A|B) and equal to P(A ∩ B)/P(B) when P(B) is positive. It restricts the sample space to B.

Total probability theorem

If B₁ to Bₙ partition the sample space, then P(A) equals the sum of P(Bᵢ)P(A|Bᵢ). It expresses the overall probability of A as a weighted average over the causes.

Bayes' theorem

A rule reversing conditional probability: P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ) divided by ΣP(Bⱼ)P(A|Bⱼ). It converts prior probabilities into posterior probabilities after observing A.

Prior and posterior probability

The prior is the probability of a cause before evidence is observed. The posterior is its revised probability after the evidence, obtained through Bayes' theorem.

Random variable

A real-valued function defined on the sample space that assigns a number to each outcome. It is discrete when its possible values are countable and continuous when they form an interval.

Probability mass function

For a discrete random variable, the function giving p(x) = P(X = x). It must be non-negative everywhere and its values over the support must sum to one.

Distribution function

F(x) = P(X ≤ x). It is non-decreasing and right-continuous, tends to zero as x tends to minus infinity and to one as x tends to plus infinity.

Variance of a random variable

The expectation of the squared deviation from the mean, equal to E(X²) − [E(X)]². For a linear transformation aX + b, the variance becomes a² times the original variance.

Expectation of a function

For Y = g(X), the expectation is the sum of g(x)p(x) over the support. Unless g is linear, this differs from g evaluated at the mean of X.

Variance of a sum

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y). The covariance term disappears only when X and Y are independent, giving the familiar additive rule.

Covariance

Cov(X, Y) = E(XY) − E(X)E(Y), measuring joint linear variation. It depends on units, so it is standardised by the two standard deviations to give correlation.

Binomial distribution

The distribution of the number of successes in n independent trials with constant success probability p. Its mean is np and its variance is npq, where q = 1 − p.

Poisson distribution

A distribution for counts of rare events, with P(X = r) = e^(−λ)λ^r/r!. Its mean and variance both equal λ, and it approximates the binomial when n is large and p is small.

Finite population correction

When sampling without replacement from a finite population, the standard error of the mean is multiplied by the square root of (N − n)/(N − 1). It tends to one for large populations.

Central limit theorem

For large sample size, the sampling distribution of the sample mean is approximately normal with mean μ and variance σ²/n, whatever the shape of the population distribution.

Unbiased estimator

An estimator whose expected value equals the true parameter for every sample size. The sample mean is unbiased for μ and the divisor n − 1 makes s² unbiased for σ².

Type I error

Rejecting a null hypothesis that is actually true. Its probability is the significance level α, fixed in advance, and it is reduced by choosing a smaller critical region.

One-sample t statistic

For a normal population with unknown standard deviation, t = (x̄ − μ₀)/(s/√n) follows a t distribution with n − 1 degrees of freedom. It is used when the sample is small.

Power of a test

The probability of rejecting the null hypothesis when it is false, equal to one minus the Type II error probability. Power increases with larger samples and larger true departures from the null value.