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Further Statistics 2 formulae

The 24 results the Further Statistics 2 notes display, each with the conditions it needs to be true and a link to the lesson that states it. Of the 24, the formulae booklet prints 13 and 11 are yours to memorise.

Further Maths · 10 lessons with formulae.

A result the Edexcel formulae booklet prints is marked as such; one it does not print is marked to learn; and a result badged derived in the lesson is a working line the notes build in front of you rather than a standard result to memorise.

10 lessons

Least squares regression and residuals

b = \frac{S_{xy}}{S_{xx}}, \qquad a = \text{mean of y} - b \times \text{mean of x}

IN THE FORMULAE BOOKLETConditionsThe least squares line of y on x, with Sxx not zero. It is not the line to use for predicting x from y.

\text{RSS} = S_{yy} - \frac{(S_{xy})^{2}}{S_{xx}}

IN THE FORMULAE BOOKLETConditionsThe least squares line of y on x, with Sxx not zero.

Continuous random variables: density and distribution functions

\text{P}(a < X \le b) = \text{∫}_{a}^{b} \text{f}(x) \, dx

NOT IN THE BOOKLET: LEARN ITConditionsX continuous with density f, where f ≥ 0 and integrates to 1 across the whole range.

\text{f}(x) = \frac{d\text{F}(x)}{dx}

NOT IN THE BOOKLET: LEARN ITConditionsWherever F is differentiable.

Σ \, \text{P}(X = c) + \text{∫} \text{f}(x) \, dx = 1

NOT IN THE BOOKLET: LEARN ITConditionsA variable that is part discrete and part continuous: the sum runs over the values carrying a lump of probability, the integral over the range where the density is non-zero, and f ≥ 0 there.

\text{P}(X > x) = e^{-λx}, \qquad \text{F}(x) = 1 - e^{-λx}

NOT IN THE BOOKLET: LEARN ITConditionsX the wait to the next event, where events occur singly, independently and at a constant mean rate λ > 0, and x ≥ 0, measured in the same time unit as λ. Below zero F is 0.

\text{f}(x) = λe^{-λx} \text{ for } x \ge 0

NOT IN THE BOOKLET: LEARN ITConditionsThe exponential distribution with rate λ > 0. The density is 0 for x < 0, and λ is a rate, so a mean wait of m corresponds to λ = 1/m.

\text{P}(X > s + t \, | \, X > s) = \text{P}(X > t)

NOT IN THE BOOKLET: LEARN ITConditionsX exponential with rate λ > 0, and s, t ≥ 0. This memoryless property belongs to the exponential distribution alone among continuous models.

Mean, variance and skewness of continuous variables

\text{E}(X) = \text{∫} x \, \text{f}(x) \, dx, \qquad \text{Var}(X) = \text{E}(X^{2}) - [\text{E}(X)]^{2}

IN THE FORMULAE BOOKLETConditionsIntegrated across the whole range on which f is non-zero, and both integrals must converge.

\text{E}(X) = \frac{1}{λ}, \qquad \text{Var}(X) = \frac{1}{λ^{2}}

NOT IN THE BOOKLET: LEARN ITConditionsX exponential with rate λ > 0. The standard deviation is 1/λ as well, matching the mean, and the median (ln 2)/λ falls below both.

The continuous uniform distribution

\text{F}(x) = 0 \text{ for } x < a, \qquad \frac{x - a}{b - a} \text{ for } a \le x \le b, \qquad 1 \text{ for } x > b

NOT IN THE BOOKLET: LEARN ITConditionsa < b. Defined for every real x: the ramp holds on [a, b], and the two flat branches carry F to 0 below a and to 1 above b.

\text{E}(X) = \frac{a + b}{2}, \qquad \text{Var}(X) = \frac{(b - a)^{2}}{12}

IN THE FORMULAE BOOKLETConditionsX uniform on [a, b] with a < b.

Correlation coefficients: product moment and Spearman

r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}

IN THE FORMULAE BOOKLETConditionsSxx and Syy both non-zero. It measures linear association only, so a strong curved relationship can still give r near zero.

r_{s} = 1 - \frac{6 Σ d^{2}}{n(n^{2} - 1)}

IN THE FORMULAE BOOKLETConditionsNo tied ranks. With ties, rank the tied values by their mean rank and compute the product moment coefficient of those ranks instead.

Combinations of normal random variables

aX ± bY \text{ is normal, with mean } aμ_{x} ± bμ_{y}

NOT IN THE BOOKLET: LEARN ITConditionsX and Y independent, and each normally distributed.

\text{Var}(aX ± bY) = a^{2}σ_{x}^{2} + b^{2}σ_{y}^{2}

IN THE FORMULAE BOOKLETConditionsX and Y independent. Without independence a covariance term appears, whichever sign the combination carries.

Estimators, standard error and confidence intervals

s^{2} = \frac{1}{n - 1} Σ (x_{i} - \text{sample mean})^{2}

IN THE FORMULAE BOOKLETConditionsA random sample of n ≥ 2 observations; this is the unbiased estimator of the population variance.

s^{2} = \frac{1}{n - 1}(Σ x^{2} - \frac{(Σ x)^{2}}{n})

NOT IN THE BOOKLET: LEARN ITConditionsA random sample of n ≥ 2 observations; algebraically the same estimator as the one above.

\text{sample mean} ± z \times \frac{σ}{\sqrt{n}}, \qquad z = 1.96 \text{ at 95 per cent}

NOT IN THE BOOKLET: LEARN ITConditionsA random sample with σ known, and either a normal population or a sample large enough for the central limit theorem.

Comparing two normal means

\frac{(\text{difference of means}) - (μ_{x} - μ_{y})}{\sqrt{\frac{σ_{x}^{2}}{n_{x}} + \frac{σ_{y}^{2}}{n_{y}}}} \text{ is N}(0, 1)

IN THE FORMULAE BOOKLETConditionsIndependent random samples from normal populations with known variances; for non-normal populations both samples must be large.

Testing variances: chi-squared and the F-distribution

\frac{(n - 1)S^{2}}{σ^{2}} \text{ is } χ^{2} \text{ on } n - 1 \text{ degrees of freedom}

IN THE FORMULAE BOOKLETConditionsA random sample from a normal population. The result is sensitive to departures from normality in a way the t test is not.

\frac{S_{1}^{2}}{S_{2}^{2}} \text{ is } F \text{ on } n_{1} - 1 \text{ and } n_{2} - 1 \text{ degrees of freedom}

IN THE FORMULAE BOOKLETConditionsIndependent random samples from two normal populations, equal variances under the null hypothesis, and the larger estimate placed on top.

Confidence intervals and tests with the t-distribution

\frac{\text{sample mean} - μ}{\frac{S}{\sqrt{n}}} \text{ is } t \text{ on } n - 1 \text{ degrees of freedom}

IN THE FORMULAE BOOKLETConditionsA random sample from a normal population with σ unknown, S being the sample standard deviation with divisor n − 1.

s^{2} = \frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2}

IN THE FORMULAE BOOKLETConditionsTwo independent random samples from normal populations assumed to share a common variance.

Every formula in the library is indexed by unit on the equation index, and the Further Statistics 2 notes are where these results are derived and used. The 11 the booklet does not print are on the list of every formula to memorise, alongside the rest of the course.

Also: Further Pure 2 formulae · Further Mechanics 1 formulae