Statistics and Probability

The One Thing

Statistics describes patterns in data, while probability measures uncertainty. Together, they support informed inferences and decisions by requiring attention to variability, sampling, bias, relationships between variables, and the limits of available evidence.

Definitions and Results

• Population: The complete group of people, objects, or measurements being studied.
• Sample: A smaller group selected from a population to provide information about the whole population.
• Random Sample: A sample selected so that members of the population have a fair and known chance of being chosen.
• Bias: A systematic tendency in data collection or analysis that causes results to differ from the truth.
• Variable: A characteristic or measurement that can take different values.
• Categorical Variable: A variable whose values are categories, such as type of transportation or favorite subject.
• Quantitative Variable: A variable whose values are numerical measurements or counts.
• Mean: The arithmetic average, found by adding all values and dividing by the number of values:

$$\bar{x}=\frac{\text{sum of all data values}}{\text{number of data values}}.$$

• Median: The middle value when data are arranged in order. If there are two middle values, their average is used.
• Mode: The value or category that occurs most often.
• Range: The difference between the greatest and least data values:

$$\text{Range}=\text{maximum value}-\text{minimum value}.$$

• Interquartile Range: The spread of the middle 50% of the data:

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

• Standard Deviation: A measure of the typical distance of data values from the mean. For a population,

$${\sigma }^2=\frac{\sum (x-\mu {)}^2}{N},\sigma =\sqrt{{\sigma }^2}.$$ For a sample, the denominator is commonly $n-1$.

• Outlier: A data value that is unusually far from the rest of the distribution. Under the common $1.5\times \text{IQR}$ rule, values below

$$Q_1-1.5(\text{IQR})$$ or above $$Q_3+1.5(\text{IQR})$$ may be outliers.

• Distribution: The pattern of values in a data set, including its shape, center, spread, and unusual features.
• Five-number summary: The minimum, first quartile, median, third quartile, and maximum.
• Two-Way Table: A table displaying counts or relative frequencies for two categorical variables.
• Conditional Relative Frequency: A proportion calculated within a particular row or column of a two-way table.
• Association: A relationship between two variables in which knowing one variable provides information about the other.
• Correlation: A measure of the direction and strength of a linear relationship between two quantitative variables.
• Causation: A relationship in which changes in one variable directly produce changes in another.
• Random Variable: A variable whose value is determined by the outcome of a chance process.
• Theoretical Probability: A probability based on a mathematical model of equally likely outcomes.
• Experimental Probability: A probability estimated from results observed in repeated trials.
• Complement: The event that an event does not occur.
• Independent Events: Events for which the occurrence of one does not change the probability of the other.
• Conditional Probability: The probability that one event occurs given that another event has already occurred.
• Expected Value: The long-run average outcome of a random variable, found by multiplying each outcome by its probability and adding the products.
• Simulation: A model of a real situation that uses random outcomes to estimate probabilities or investigate decisions.

Worked Methods

Summarizing quantitative data

• Arrange the data when calculating the median, quartiles, or five-number summary.
• Calculate the mean by adding all data values and dividing by the number of values:

$$\bar{x}=\frac{\text{sum of all data values}}{\text{number of data values}}.$$

• Calculate the range:

$$\text{Range}=\text{maximum value}-\text{minimum value}.$$

• Identify $Q_1$, the first quartile, and $Q_3$, the third quartile.
• Calculate the interquartile range:

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

• Use the five-number summary—minimum, $Q_1$, median, $Q_3$, and maximum—to describe the distribution.
• For variance and standard deviation, calculate deviations from the mean, square them, average the squared deviations, and take the square root for standard deviation. For a population:

$${\sigma }^2=\frac{\sum (x-\mu {)}^2}{N},\sigma =\sqrt{{\sigma }^2}.$$ For a sample, use $n-1$ as the denominator commonly.

• Identify possible outliers using:

$$Q_1-1.5(\text{IQR})\text{and}{Q}_3+1.5(\text{IQR}).$$

The mean and standard deviation are especially useful for roughly symmetric distributions without strong outliers. The median and interquartile range are more resistant to outliers and skewed data.

Reading graphical displays

• Use a histogram to display the distribution of quantitative data across intervals called bins.
• Use a box plot to display the five-number summary and compare center, spread, and possible outliers.
• Use a scatter plot to display paired quantitative data.
• Describe a distribution or relationship by considering shape, center, spread, direction, form, and unusual features.
• Interpret all numerical and graphical summaries in the context of the data.

Comparing distributions

• Describe the shape of each distribution.
• Compare their centers using appropriate measures, such as the mean or median.
• Compare their spreads using the range, IQR, or standard deviation.
• Identify unusual features, including possible outliers.
• State the comparison in the context of the measured variable.

Analyzing categorical variables with a two-way table

• Organize the data in a two-way table showing counts or relative frequencies for two categorical variables.
• Select the relevant row or column as the conditioning group.
• Divide the frequency of the specified outcome by the total frequency in that row or column.
• Compare the resulting conditional relative frequencies.
• If the proportions differ, the variables may show an association.

Association does not establish causation. A lurking variable may explain an observed association.

Calculating equally likely probabilities

• Identify the event $E$.
• Count the favorable outcomes.
• Count the total number of equally likely outcomes.
• Calculate:

$$P(E)=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}.$$

• Check that:

$$0\le P(E)\le 1.$$

Using complements

• Identify the event $E$.
• Define its complement, the event that $E$ does not occur.
• Calculate:

$$P(\text{not }E)=1-P(E).$$

Applying the addition rule

• Identify events $A$ and $B$.
• Determine $P(A)$, $P(B)$, and $P(A\text{ and }B)$.
• Calculate:

$$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B).$$

• If $A$ and $B$ are mutually exclusive, then $P(A\text{ and }B)=0$, so:

$$P(A\text{ or }B)=P(A)+P(B).$$

Calculating conditional probability

• Identify the event being measured and the condition already known.
• Find the probability of their intersection.
• Divide by the probability of the conditioning event:

$$P(A\text{ given }B)=\frac{P(A\text{ and }B)}{P(B)},$$ provided $P(B)\ne 0$.

Applying the multiplication rule

• Identify the events $A$ and $B$.
• Find $P(A)$.
• Find $P(B\text{ given }A)$.
• Calculate:

$$P(A\text{ and }B)=P(A)\times P(B\text{ given }A).$$

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

$$P(A\text{ and }B)=P(A)\times P(B).$$

Calculating expected value

• List each possible outcome of the random variable.
• Assign its probability.
• Multiply each outcome by its probability.
• Add the products:

$$E(X)=\sum [\text{outcome}\times \text{probability of outcome}].$$

• Interpret the result as the long-run average outcome, not necessarily an outcome that occurs in one trial.

Using experimental probability and simulation

• Define the chance process and the outcome of interest.
• Conduct repeated trials or construct a simulation using random outcomes.
• Record the outcomes.
• Estimate the probability from the observed proportion:

$$\text{Experimental probability}=\frac{\text{number of occurrences of the event}}{\text{number of trials}}.$$

• Use the estimate to investigate probabilities or decisions when exact calculations are difficult.

Making inferences from samples

• Identify the population and the sample.
• Examine how the sample was selected.
• Determine whether it is a representative random sample.
• Consider sample size, variability, and possible bias.
• Use the sample to estimate or draw conclusions about the population.
• Interpret the margin of error as the amount by which a sample estimate may reasonably differ from the population value.
• Recognize that greater confidence usually requires a wider interval.

A representative random sample supports stronger conclusions about a population than a convenience or voluntary-response sample. Larger random samples generally produce more stable estimates, but they do not automatically remove bias from a poorly designed study.

Distinguishing surveys and experiments

• Identify whether the study observes responses or assigns treatments.
• A survey observes responses and does not assign treatments.
• An experiment assigns treatments to subjects.
• Assess whether randomization and control of alternative explanations are present.
• Treat well-designed randomized experiments as providing stronger evidence for cause-and-effect relationships.

Where It Goes Wrong

• Treating the mean and standard deviation as automatically appropriate when a distribution is skewed or contains strong outliers; the median and interquartile range are more resistant in these cases.
• Forgetting that the range is maximum minus minimum, or that the interquartile range is $Q_3-Q_1$.
• Applying the $1.5\times \text{IQR}$ outlier rule without calculating the lower and upper fences $Q_1-1.5(\text{IQR})$ and $Q_3+1.5(\text{IQR})$.
• Interpreting correlation or association as proof of causation; a lurking variable or another alternative explanation may be responsible.
• Using the addition rule without subtracting $P(A\text{ and }B)$, or failing to recognize the special mutually exclusive condition under which that intersection is zero.
• Treating events as independent without checking whether one event changes the probability of the other, or forgetting that conditional probability requires $P(B)\ne 0$.

What Gets Asked

• Calculate and interpret the mean, median, mode, range, IQR, variance, and standard deviation.
• Construct or interpret a five-number summary, box plot, histogram, or scatter plot.
• Identify possible outliers using the $1.5\times \text{IQR}$ rule.
• Compare distributions using shape, center, spread, and unusual features.
• Complete or interpret a two-way table using conditional relative frequencies.
• Determine whether data show association, and explain why association does not by itself establish causation.
• Distinguish a population from a sample and evaluate random sampling, convenience sampling, voluntary-response sampling, sample size, variability, and bias.
• Calculate theoretical and experimental probabilities.
• Apply the complement, addition, conditional probability, and multiplication rules.
• Determine whether events are mutually exclusive or independent.
• Calculate and interpret expected value.
• Design or interpret a simulation for a repeated random process.
• Explain how margin of error and confidence affect an inference.
• Distinguish surveys from experiments and identify when evidence supports cause-and-effect conclusions.
• Compare possible outcomes, their probabilities, and their consequences when evaluating a decision.