Statistics

The science of collecting, organizing, analyzing, and interpreting data.

Why is statistics important?

Statistics helps us make decisions and draw conclusions from data while accounting for variability and uncertainty.

Scientific method

A process used to answer questions through observation, forming hypotheses, collecting data, analyzing data, and drawing conclusions.

Experiment

A study where researchers impose a treatment or condition on subjects and measure the response.

Observational study

A study where researchers observe and measure variables without imposing treatments.

Variable

A characteristic or measurement that can take different values for different individuals or observations.

Individual

A person, object, or case being studied.

Population

The entire group of individuals that we want information about.

Sample

A subset of the population that is actually observed or measured.

Population parameter

A numerical summary that describes an entire population.

Sample statistic

A numerical summary calculated from a sample.

Parameter vs. statistic

A parameter describes a population; a statistic describes a sample.

Descriptive statistics

Methods used to organize, summarize, and describe the data that were collected.

Inferential statistics

Methods used to use sample data to make conclusions or predictions about a larger population.

Categorical variable

A variable that places individuals into groups or categories.

Quantitative variable

A variable that takes numerical values where arithmetic makes sense.

Example of categorical data

Eye color, major, gender, type of car.

Example of quantitative data

Height, age, GPA, number of siblings.

How do you tell categorical from quantitative?

If the values are labels or groups, they are categorical. If the values represent numerical measurements or counts, they are quantitative.

Frequency

The number of times a value or category occurs.

Relative frequency

The proportion or percentage of observations in a category.

Bar graph

A graph commonly used to display categorical data.

Pie chart

A graph that displays categorical data as parts of a whole.

Histogram

A graph used to display the distribution of quantitative data.

Dotplot

A graph that displays individual quantitative observations using dots.

Boxplot

A graph that summarizes quantitative data using the five-number summary.

Five-number summary

Minimum, Q1, median, Q3, maximum.

Mean

The sum of all observations divided by the number of observations.

Median

The middle value when the data are ordered.

Mode

The value or category that occurs most often.

Range

The maximum value minus the minimum value.

Standard deviation

A measure of how spread out quantitative data are around the mean.

IQR

Interquartile range; Q3 minus Q1.

Q1

The first quartile; about 25% of observations are below it.

Q3

The third quartile; about 75% of observations are below it.

Distribution

The overall pattern of values in a dataset.

Symmetric distribution

A distribution where the left and right sides are approximately mirror images.

Right-skewed distribution

A distribution with a long tail extending to the right.

Left-skewed distribution

A distribution with a long tail extending to the left.

Which measure is usually better for skewed data?

Median, because it is less affected by extreme values.

Which measure is usually better for symmetric data?

Mean and standard deviation are commonly used.

Outlier

An observation that is unusually far from the rest of the data.

Association

A relationship or pattern between variables.

Scatterplot

A graph used to display the relationship between two quantitative variables.

Explanatory variable

The variable that may help explain or predict changes in another variable.

Response variable

The variable being explained or predicted.

Positive association

As one variable increases, the other tends to increase.

Negative association

As one variable increases, the other tends to decrease.

No association

There is no clear pattern between the two variables.

Strength of association

How closely the points follow a pattern or trend.

Direction of association

The direction of the relationship: positive or negative.

Form of association

The shape of the relationship, such as linear or curved.

Correlation

The strength and direction of a linear relationship between two quantitative variables.

Correlation symbol

r is commonly used to represent the sample correlation.

Correlation range

Correlation is always between -1 and 1.

What does r close to 1 mean?

A strong positive linear association.

What does r close to -1 mean?

A strong negative linear association.

What does r close to 0 mean?

Little or no linear association.

Can correlation prove causation?

No. Correlation does not prove that one variable causes another.

Two-way table

A table used to summarize the relationship between two categorical variables.

Marginal distribution

The distribution of one categorical variable without considering the other variable.

Conditional distribution

The distribution of one variable given a particular value or category of another variable.

Joint distribution

Information about combinations of two categorical variables.

Probability

A number between 0 and 1 that describes how likely an event is to occur.

Probability as relative frequency

Probability can be viewed as the long-run proportion of times an event occurs.

Law of Large Numbers

As the number of trials increases, the relative frequency of an event tends to get closer to its true probability.

Experiment in probability

A process that produces an outcome.

Outcome

A possible result of a probability experiment.

Sample space

The set of all possible outcomes.

Event

A set of outcomes from a sample space.

Complement of an event

The event that A does not occur, written Aᶜ.

Intersection

The outcomes that are in both events, written A ∩ B.

Union

The outcomes that are in A or B or both, written A ∪ B.

Aᶜ meaning

Not A.

A ∩ B meaning

A and B both occur.

A ∪ B meaning

A or B or both occur.

Venn diagram

A diagram that uses circles to show relationships between events.

Tree diagram

A diagram that shows possible outcomes through a sequence of events.

Contingency table

A table that displays counts or proportions for combinations of categorical variables.

Marginal probability

The probability of one event without conditioning on another event.

Joint probability

The probability that two events occur together: P(A ∩ B).

Conditional probability

The probability of an event given that another event has occurred.

Conditional probability formula

P(A|B) = P(A ∩ B) / P(B), as long as P(B) > 0.

Conditioning event

The event after the vertical bar in a conditional probability; it tells you what information is being assumed.

P(A|B)

Probability of A given B.

Complement rule

P(Aᶜ) = 1 - P(A).

General addition rule

P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

When can you add probabilities directly?

When events are mutually exclusive, P(A ∪ B) = P(A) + P(B).

Multiplication rule

P(A ∩ B) = P(A)P(B|A).

Multiplication rule for independent events

If A and B are independent, P(A ∩ B) = P(A)P(B).

Independent events

Two events are independent if knowing one occurred does not change the probability of the other.

How can you identify independence?

A and B are independent if P(A|B) = P(A), or equivalently P(A ∩ B) = P(A)P(B).

Mutually exclusive events

Two events that cannot happen at the same time.

Can mutually exclusive events occur together?

No. If A and B are mutually exclusive, P(A ∩ B) = 0.

Are mutually exclusive events independent?

Generally no, unless one of the events has probability 0.

Discrete random variable

A random variable that takes countable values, usually whole numbers.

Continuous random variable

A random variable that can take any value within an interval.

Examples of discrete variables

Number of pets, number of students, number of heads in coin flips.

Examples of continuous variables

Height, weight, temperature, time.

Random variable

A variable whose value is determined by the outcome of a random process.