Statistics and Probability

The One Thing

Measures of dispersion quantify how widely observations vary around a central value, while probability quantifies uncertainty through sample spaces, events, and probability rules. Correct calculation depends on identifying the data structure and the relevant relationship between events.

Definitions and Results

• Dispersion: The extent to which observations are scattered or spread around a central value.
• Range: The difference between the largest and smallest observations:

$$\text{Range}=\text{Maximum value}-\text{Minimum value}.$$

• Coefficient of Range: A relative measure of dispersion:

$$\frac{\text{Maximum value}-\text{Minimum value}}{\text{Maximum value}+\text{Minimum value}}.$$

• Quartiles: Values dividing ordered data into four equal parts. $Q_1$ is the lower quartile, $Q_2$ is the median, and $Q_3$ is the upper quartile.
• Interquartile Range: The spread of the middle 50% of the data:

$$\text{IQR}=Q_3-Q_1.$$

• Quartile Deviation: Half the interquartile range:

$$\text{Quartile Deviation}=\frac{Q_3-Q_1}{2}.$$

• Coefficient of Quartile Deviation: A relative measure, provided $Q_3+Q_1\ne 0$:

$$\frac{Q_3-Q_1}{Q_3+Q_1}.$$

• Mean Deviation: The arithmetic mean of the absolute deviations from a chosen central value, usually the mean or median. About $a$:

$$\text{MD}=\frac{\sum |x-a|}{n}$$ for ungrouped data, and $$\text{MD}=\frac{\sum f|x-a|}{\sum f}$$ for frequency data.

• Variance: The average of the squared deviations from the mean.
• Standard Deviation: The positive square root of variance. Population variance and standard deviation are

$${\sigma }^2=\frac{\sum (x-\bar{x}{)}^2}{n},\sigma =\sqrt{\frac{\sum (x-\bar{x}{)}^2}{n}}.$$ For frequency data: $${\sigma }^2=\frac{\sum f(x-\bar{x}{)}^2}{\sum f}.$$

• Coefficient of Variation: A relative measure of dispersion, provided $\bar{x}\ne 0$:

$$\text{CV}=\frac{\sigma }{\bar{x}}\times 100.$$

• Random Experiment: An experiment with a clearly defined set of possible outcomes whose exact outcome cannot be predicted with certainty.
• Sample Space: The set of all possible outcomes, usually denoted by $S$.
• Event: A collection of one or more outcomes from the sample space.
• Simple Event: An event containing exactly one outcome.
• Compound Event: An event containing two or more outcomes.
• Equally Likely Outcomes: Outcomes having the same chance of occurring.
• Probability of an Event: For equally likely outcomes,

$$P(A)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}.$$

• Complementary Event: The event that $A$ does not occur, written $A^{\prime }$ or $A^c$:

$$P(A^{\prime })=1-P(A).$$

• Mutually Exclusive Events: Events that cannot occur together. Thus,

$$A\cap B=\varnothing ,P(A\cap B)=0.$$

• Exhaustive Events: Events whose union contains every outcome in the sample space.
• Independent Events: Events in which the occurrence of one does not change the probability of the other:

$$P(A\cap B)=P(A)P(B).$$ If $P(B)>0$, $$P(A\mid B)=P(A).$$

• Conditional Probability: The probability of $A$ when $B$ has already occurred:

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)},P(B)>0.$$

• Addition Rule: For any two events,

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$

• Probability Bounds and Certain Events:

$$0\le P(A)\le 1,P(S)=1,P(\varnothing )=0.$$

• Multiplication Rule:

$$P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B).$$

Worked Methods

Calculating the Mean

• For an individual or ungrouped dataset with $n$ observations, add the observations.
• Divide by the number of observations:

$$\bar{x}=\frac{\sum x}{n}.$$

• For a frequency distribution, multiply each observation or class mark $x$ by its frequency $f$.
• Divide the total of $fx$ by the total frequency:

$$\bar{x}=\frac{\sum fx}{\sum f}.$$

• For grouped continuous data, use class marks as the values of $x$.

Calculating the Range and Coefficient of Range

• Identify the largest and smallest observations.
• Subtract the smallest from the largest:

$$\text{Range}=\text{Largest observation}-\text{Smallest observation}.$$

• For the coefficient of range, divide this difference by the sum of the largest and smallest observations:

$$\text{Coefficient of Range}=\frac{\text{Maximum value}-\text{Minimum value}}{\text{Maximum value}+\text{Minimum value}}.$$

• Interpret the range as a simple measure that is highly affected by extreme values.

Calculating Quartiles and Quartile-Based Measures

• Arrange the observations in ascending order.
• Identify $Q_1$, $Q_2$, and $Q_3$, where $Q_2$ is the median.
• Calculate the interquartile range:

$$\text{IQR}=Q_3-Q_1.$$

• Calculate the quartile deviation:

$$\text{Quartile Deviation}=\frac{Q_3-Q_1}{2}.$$

• If a relative measure is required, calculate:

$$\text{Coefficient of Quartile Deviation}=\frac{Q_3-Q_1}{Q_3+Q_1},$$ provided $Q_3+Q_1\ne 0$.

• Interpret quartile deviation as a measure focused on the middle half of the data.

Calculating Mean Deviation

• Select the central value $a$, usually the mean or median.
• Subtract $a$ from each observation.
• Take the absolute value of every deviation:

$$|x-a|.$$

• For ungrouped data, add the absolute deviations and divide by $n$:

$$\text{MD}=\frac{\sum |x-a|}{n}.$$

• For frequency data, multiply each absolute deviation by its frequency and divide by total frequency:

$$\text{MD}=\frac{\sum f|x-a|}{\sum f}.$$

• Retain absolute values; positive and negative deviations must not cancel.

Calculating Variance and Standard Deviation

• Calculate the mean $\bar{x}$.
• Find each deviation $x-\bar{x}$.
• Square each deviation.
• Add the squared deviations.
• For population data, divide by $n$:

$${\sigma }^2=\frac{\sum (x-\bar{x}{)}^2}{n}.$$

• Take the positive square root:

$$\sigma =\sqrt{\frac{\sum (x-\bar{x}{)}^2}{n}}.$$

• For frequency data, use:

$${\sigma }^2=\frac{\sum f(x-\bar{x}{)}^2}{\sum f}.$$

• Alternatively, use the computational formula for ungrouped data:

$${\sigma }^2=\frac{\sum x^2}{n}-{\bar{x}}^2.$$

• For frequency data, use the corresponding formula:

$${\sigma }^2=\frac{\sum f{x}^2}{\sum f}-{\bar{x}}^2.$$

• Standard deviation is zero only when all observations are identical; otherwise it is positive.

Comparing Datasets Using the Coefficient of Variation

• Calculate the standard deviation $\sigma$ for each dataset.
• Calculate the mean $\bar{x}$ for each dataset.
• Provided the mean is non-zero, calculate:

$$\text{CV}=\frac{\sigma }{\bar{x}}\times 100.$$

• Compare the coefficients rather than the standard deviations when datasets have different means, scales, or units.
• The dataset with the smaller coefficient of variation is generally more consistent.
• A smaller standard deviation generally indicates greater consistency, while a larger standard deviation indicates greater variability.

Calculating Probability from a Sample Space

• Define the random experiment.
• List all possible outcomes to form the sample space $S$.
• Identify the event $A$, which may be simple or compound.
• Check that the outcomes are equally likely.
• Count favourable outcomes and divide by the total number of outcomes:

$$P(A)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}.$$

• Do not use this counting formula unless the outcomes are equally likely.

Using Complements

• Identify the event $A$.
• Identify its complement $A^{\prime }$ or $A^c$, representing the event that $A$ does not occur.
• Use:

$$P(A^{\prime })=1-P(A).$$

• This is often the simplest method for finding the probability of at least one occurrence or the probability that an event does not occur.

Using the Addition Rule

• Define events $A$ and $B$.
• Determine $P(A)$, $P(B)$, and $P(A\cap B)$.
• Substitute into:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$

• Subtract $P(A\cap B)$ because outcomes common to both events would otherwise be counted twice.
• If $A$ and $B$ are mutually exclusive, then $P(A\cap B)=0$, so:

$$P(A\cup B)=P(A)+P(B).$$

Using Conditional Probability and the Multiplication Rule

• Identify which event has already occurred. If $B$ has occurred, the required probability is $P(A\mid B)$.
• Ensure that $P(B)>0$.
• Use:

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$$

• Rearrange to obtain the multiplication rule:

$$P(A\cap B)=P(B)P(A\mid B).$$

• Equivalently:

$$P(A\cap B)=P(A)P(B\mid A).$$

• If $A$ and $B$ are independent, use:

$$P(A\cap B)=P(A)P(B),$$ and, when $P(B)>0$, $$P(A\mid B)=P(A).$$

Where It Goes Wrong

• Confusing range with a relative measure: the range is $\text{Maximum}-\text{Minimum}$, whereas the coefficient of range divides this difference by $\text{Maximum}+\text{Minimum}$.
• Forgetting that quartile deviation is half the interquartile range:

$$\frac{Q_3-Q_1}{2},$$ not simply $Q_3-Q_1$.

• Omitting frequencies in frequency-data calculations. The mean, mean deviation, and variance require $\sum f$, $\sum f|x-a|$, or $\sum f(x-\bar{x}{)}^2$ as appropriate.
• Using signed deviations for mean deviation instead of absolute deviations $|x-a|$.
• Confusing variance with standard deviation: variance is measured in squared units, whereas standard deviation is the positive square root of variance.
• Applying $\frac{\text{favourable outcomes}}{\text{total outcomes}}$ when outcomes are not equally likely, or confusing mutually exclusive events with independent events. Mutually exclusive events cannot occur together; independent events can occur together but do not affect one another’s probabilities.
• Forgetting the subtraction of $P(A\cap B)$ in the general addition rule, or using conditional probability when $P(B)=0$.

What Gets Asked

• Define dispersion and explain why a measure of central tendency alone cannot fully describe a dataset.
• Calculate the range and coefficient of range from given observations.
• Find $Q_1$, $Q_2$, and $Q_3$, then calculate the interquartile range, quartile deviation, and coefficient of quartile deviation.
• Calculate mean deviation about the mean or median for ungrouped or frequency data.
• Calculate the mean, variance, and standard deviation for individual, frequency, or grouped continuous data using class marks.
• Use either the deviation formula or the computational formula for variance.
• Compare two datasets using standard deviation and coefficient of variation.
• Define a random experiment, sample space, simple event, compound event, equally likely outcomes, mutually exclusive events, exhaustive events, and independent events.
• Construct a sample space and calculate an event probability by counting favourable outcomes when outcomes are equally likely.
• Calculate the probability of a complementary event.
• Apply the addition rule to overlapping events and simplify it for mutually exclusive events.
• Calculate conditional probabilities using

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$$

• Apply the multiplication rule to dependent or independent events.
• Distinguish mutual exclusivity from independence and verify probability bounds, including $P(S)=1$ and $P(\varnothing )=0$.