Title: InkMaths / Statistics essentials
On every note of this deck, unless a note says otherwise:
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Front: CHECK YOURSELF
One fair die is rolled; X is the score. Name the distribution of X. Does the total of two dice have the same one?
Back: X is discrete uniform: six values, each with probability 1/6, because every score is equally likely. The total of two dice is not — 7 has six ways and 2 has one — so equal likelihood is the test, and naming it is all the specification asks.

Front: SPOT THE ERROR
Say what is wrong with this claim:
In a histogram, the tallest bar holds the most values.
Back: Height is frequency density, so a tall narrow bar can hold fewer values than a low wide one. Area, height times width, is what counts values.

Front: CHECK YOURSELF
A firm has 900 full-time and 300 part-time staff. How many of each go into a stratified sample of 60?
Back: 60 × 900/1200 = 45 full-time and 60 × 300/1200 = 15 part-time. Each stratum keeps its share of the population.

Front: STANDARD RESULT — LEARN IT
Addition rule:
P(A ∪ B) = ?\(P(A ∪ B) = ?\)P(A ∪ B) = ?
Back: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)\(P(A ∪ B) = P(A) + P(B) − P(A ∩ B)\)P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Front: STANDARD RESULT — LEARN IT
Outlier fences:
outliers lie beyond ?\(\text{outliers lie beyond } ?\)outliers lie beyond ?
Back: Q_1 - 1.5 × IQR and Q_3 + 1.5 × IQR\(Q_{1} - 1.5 \times \text{IQR} \, \text{ and } \, Q_{3} + 1.5 \times \text{IQR}\)Q_1 - 1.5 × IQR and Q_3 + 1.5 × IQR

Front: RECALL
State the effect of coding on the mean and the standard deviation.
Back: If y = (x − a)/b then the mean of x is b times the mean of y plus a. The standard deviation of x is |b| times the standard deviation of y, since no spread comes out negative and exam codes take b above zero anyway. Subtracting a shifts everything and changes no spread at all, so the a never appears in the second line.

Front: DEFINE
Define: actual significance level.
Back: The probability of landing in the critical region when the null hypothesis is true. Because the binomial is discrete you rarely hit 5 per cent exactly, so the real false-alarm rate usually sits a little under the nominal one.

Front: STANDARD RESULT — LEARN IT
Conditional probability:
P(A|B) = ?\(P(A|B) = ?\)P(A|B) = ?
Back: P(A|B) = P(A ∩ B)P(B)\(P(A|B) = \frac{P(A ∩ B)}{P(B)}\)P(A|B) = (P(A ∩ B))/P(B)

Front: SPOT THE ERROR
Say what is wrong with this claim:
Once you have the regression line you can predict y for any x you like.
Back: Inside the range of the data the line is supported by evidence. Outside it you are guessing that a pattern you never observed keeps going. Examiners award the mark for the word extrapolation and for saying the prediction is unreliable, so name the range.

Front: SPOT THE ERROR
Say what is wrong with this claim:
A strong correlation shows that one variable causes the other.
Back: Correlation measures how two variables move together, nothing more. A third factor can drive both, the way summer heat lifts ice cream sales and drownings at once. Causation needs an experiment, not an r value.

Front: STANDARD RESULT — LEARN IT
Variance from sums:
σ^2 = ? from Σ x and Σ x^2\(σ^{2} = ? \, \text{from } Σ x \text{ and } Σ x^{2}\)σ^2 = ? from Σ x and Σ x^2
Back: σ^2 = Σ x^2n - (Σ xn)^2\(σ^{2} = \frac{Σ x^{2}}{n} - (\frac{Σ x}{n})^{2}\)σ^2 = (Σ x^2)/n - ((Σ x)/n)^2

Front: DEFINE
Define: independent events.
Back: Events A and B are independent when P(A ∩ B) = P(A) × P(B); equivalently, when P(A|B) = P(A), so B carries no news about A.

Front: STANDARD RESULT — LEARN IT
Standardising:
X is N(μ, σ^2): Z = ?\(X \text{ is N}(μ, σ^{2}): Z = ?\)X is N(μ, σ^2): Z = ?
Back: Z = X - μσ, which is N(0, 1)\(Z = \frac{X - μ}{σ}, \, \text{which is N}(0, 1)\)Z = ((X - μ)/σ), which is N(0, 1)

Front: STANDARD RESULT — LEARN IT
Histograms:
frequency density = ?\(\text{frequency density} = ?\)frequency density = ?
Back: frequency density = frequencyclass width\(\text{frequency density} = \frac{\text{frequency}}{\text{class width}}\)frequency density = frequency/class width

Front: CHECK YOURSELF
X ~ B(20, 0.15). Write P(X ≥ 5) in calculator-ready cumulative form.
Back: P(X ≥ 5) = 1 − P(X ≤ 4). At least five means not four or fewer, so the cut is at 4 and not at 5.

Front: STANDARD RESULT — LEARN IT
The sample mean's distribution:
mean of n obs of N(μ, σ^2): ?\(\text{mean of } n \text{ obs of } N(μ, σ^{2}): ?\)mean of n obs of N(μ, σ^2): ?
Back: N(μ, σ^2/n)\(N(μ, \, σ^{2}/n)\)N(μ, σ^2/n)