IB Maths AI · Topic 4 · 32 cards

Statistics and probability flashcards

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What are the types of data, with an example of each?
Categorical (eye colour), discrete numerical (number of siblings) and continuous numerical (height). Continuous data can take any value in an interval; discrete data takes separate values.
How do random sampling methods and non-random methods differ, with examples?
Random methods give every member a known chance: simple random, systematic, stratified (proportional groups). Non-random methods such as convenience and quota sampling are quicker but more likely to be biased.
What rule identifies outliers using quartiles? Q1 = 20, Q3 = 30: what are the limits?
Values below \(Q_1-1.5\times IQR\) or above \(Q_3+1.5\times IQR\). With \(IQR=10\), the limits are 5 and 45.
How do you estimate the mean from a grouped frequency table?
Use the midpoint of each class: \(\bar x\approx\dfrac{\sum fx}{n}\). For classes 0–10 (f=4) and 10–20 (f=6): \(\dfrac{4(5)+6(15)}{10}=11\).
What do the standard deviation and variance measure, and how do you get them quickly?
The spread of data about the mean (variance = SD²). Enter the data in the GDC list and use 1-variable statistics; use the population SD for a whole population (σ) and the sample value for a sample.
How do you read the median and quartiles from a cumulative frequency graph?
Read across from 25%, 50% and 75% of the total frequency to the curve, then down to the horizontal axis: these give \(Q_1\), the median and \(Q_3\). IQR \(=Q_3-Q_1\).
What does Pearson's correlation coefficient \(r\) measure, and what are its limits?
The strength and direction of a linear relationship, with \(-1\le r\le1\). Values near ±1 are strong. It does not detect curved relationships, and correlation does not imply causation.
How do you use a regression line \(y=ax+b\) responsibly?
Use it to predict \(y\) from \(x\) within the data range (interpolation). Predictions outside the range (extrapolation) are unreliable, and the line should not be used to predict \(x\) from \(y\).
What is Spearman's rank correlation, and when is it better than Pearson's?
Rank each variable, then find the correlation of the ranks, \(r_s\). It measures any monotonic (consistently increasing or decreasing) relationship and is less affected by outliers.
What is the formula for \(P(A\cup B)\)? Find it if \(P(A)=0.5\), \(P(B)=0.4\), \(P(A\cap B)=0.2\).
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.5+0.4-0.2=0.7\). Also \(P(A')=1-P(A)\).
What is conditional probability? Find \(P(A|B)\) if \(P(A\cap B)=0.2\) and \(P(B)=0.4\).
\(P(A|B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{0.2}{0.4}=0.5\): the probability of \(A\) given that \(B\) has occurred.
How do independent events differ from mutually exclusive events?
Independent: \(P(A\cap B)=P(A)P(B)\), so one does not affect the other. Mutually exclusive: \(P(A\cap B)=0\), they cannot both happen. Events with non-zero probabilities cannot be both.
How do you use a tree diagram? A bag has 3 red and 2 blue; two are drawn without replacement. Find P(both red).
Multiply along branches, add the probabilities of different routes to the same outcome. Without replacement the second probabilities change: \(\tfrac35\times\tfrac24=\tfrac3{10}\).
When is a binomial model \(X\sim B(n,p)\) appropriate, and what are its mean and variance?
Fixed number of trials \(n\), two outcomes, constant probability \(p\), independent trials. Mean \(np\) and variance \(np(1-p)\). For \(B(10,0.3)\), the mean is 3.
How do you find binomial probabilities on the GDC? Find \(P(X=2)\) for \(B(5,0.5)\) and write \(P(X\ge3)\) using cdf.
Use pdf for \(P(X=k)\): here \(\tfrac{10}{32}=0.3125\). Use cdf for \(P(X\le k)\). \(P(X\ge3)=1-P(X\le2)\).
What are the key features of a normal distribution \(N(\mu,\sigma^2)\)?
Bell-shaped and symmetric about the mean \(\mu\); total area 1. About 68% of values lie within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.
How do you standardise a normal value, and how do you find a value from a given probability?
\(z=\dfrac{x-\mu}\sigma\); for \(x=65,\mu=50,\sigma=10\), \(z=1.5\). For a given area, use the inverse normal function on the GDC with \(\mu\) and \(\sigma\). Sketch the curve to check which tail.
How is a \(\chi^2\) test for independence carried out? Find the expected value for a cell with row total 40, column total 30, grand total 120.
H₀: the variables are independent. Expected \(=\dfrac{\text{row total}\times\text{column total}}{\text{grand total}}=\dfrac{40\times30}{120}=10\). Degrees of freedom \(=(r-1)(c-1)\).
What are the steps and conditions for a \(\chi^2\) goodness-of-fit test?
H₀: the data fit the stated distribution. Use \(\chi^2=\sum\frac{(O-E)^2}{E}\) with degrees of freedom = number of categories − 1. Expected frequencies should be at least 5.
What does a two-sample t-test compare, and what does it assume?
It tests whether two population means differ. It assumes approximately normal data and (for the pooled test) equal variances. Enter the lists in the GDC and read the p-value.
How do you conclude a hypothesis test using a p-value?
Compare the p-value with the significance level \(\alpha\). If \(p<\alpha\), reject H₀; otherwise do not reject it. State the conclusion in context, and never say H₀ is \"proved\".
How do you decide between the mean and the median as the typical value?
Use the median when data is skewed or has outliers, since it is not pulled by extreme values. Use the mean for roughly symmetric data, as it uses every value.
(HL) When is a Poisson model \(X\sim Po(m)\) appropriate, and what are its mean and variance? Find \(P(X=0)\) for \(Po(2)\).
Events occur independently at a constant average rate. Both mean and variance equal \(m\). \(P(X=0)=e^{-2}\approx0.135\).
(HL) What does a 95% confidence interval for a mean mean, and how is it found?
The method captures the true mean in 95% of repeated samples. For a mean with unknown \(\sigma\), use \(\bar x\pm t\dfrac s{\sqrt n}\) through the GDC t-interval function; a larger sample gives a narrower interval.
(HL) What is an unbiased estimate of variance? If \(n=10\) and the divide-by-\(n\) variance is 9, what is it?
The sample variance dividing by \(n-1\): \(s_{n-1}^2=\dfrac n{n-1}\sigma_n^2\). Here \(\tfrac{10}{9}\times9=10\). The sample mean is an unbiased estimate of \(\mu\).
(HL) What are Type I and Type II errors?
Type I: rejecting H₀ when it is true; its probability is the significance level \(\alpha\). Type II: failing to reject H₀ when it is false.
How do you find the expected value of a discrete random variable? Use \(x=1,2,3\) with probabilities 0.2, 0.5, 0.3.
\(E(X)=\sum xP(X=x)=1(0.2)+2(0.5)+3(0.3)=2.1\). The probabilities must sum to 1.
(HL) How do \(E(aX+b)\) and \(\mathrm{Var}(aX+b)\) behave? Find the variance of \(3X+1\) when \(\mathrm{Var}(X)=4\).
\(E(aX+b)=aE(X)+b\) and \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\); adding \(b\) does not change spread. So \(\mathrm{Var}(3X+1)=9\times4=36\).
(HL) What is the variance of \(X+Y\) or \(X-Y\) for independent variables?
\(\mathrm{Var}(X\pm Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)\): variances add even when subtracting. Means: \(E(X\pm Y)=E(X)\pm E(Y)\).
(HL) What does the central limit theorem say?
For large samples of size \(n\), the sample mean \(\bar X\) is approximately \(N\!\left(\mu,\dfrac{\sigma^2}n\right)\), whatever the shape of the original population.
(HL) What is a transition matrix, and how do you find the state after \(n\) steps and the long-term state?
A square matrix \(T\) whose columns each sum to 1. State after \(n\) steps: \(\mathbf s_n=T^n\mathbf s_0\). The steady state solves \(T\mathbf s=\mathbf s\) with entries summing to 1 (eigenvalue 1).
(HL) What do \(R^2\) and regression of \(x\) on \(y\) tell you?
\(R^2\) measures the proportion of variation in \(y\) explained by the fitted model; closer to 1 is a better fit. Regression of \(x\) on \(y\) is the correct line when you wish to predict \(x\).

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