AP Statistics 1.1-1.2: Introducing Statistics and Variables

What proportion of a class is male if 22 of 40 students are male?

The proportion is $\frac{22}{40}=0.55$, or $55\%$. The female proportion is $\frac{18}{40}=0.45$, or $45\%$.

Arithmetic mean

The sum of all observed values divided by the number of values: $\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}=\frac{\sum_{i=1}^{n}x_i}{n}$ for a sample.

Statistics

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

Proportion

The fraction of observations in a category: $\text{proportion}=\frac{\text{number in category}}{\text{total number}}$. It can be expressed as a decimal or multiplied by $100\%$.

What are the broad stages of a statistical investigation?

A statistical investigation generally involves asking a question, collecting or obtaining data, analyzing the data, and interpreting the results in context.

Descriptive statistics

Methods for organizing and summarizing observed data, such as constructing graphs or calculating an average.

Inferential statistics

Methods for using sample data and probability to draw conclusions about a larger population.

How do descriptive and inferential statistics differ?

Descriptive statistics summarize the data that were collected. Inferential statistics use sample data to make probability-based conclusions about a population.

Why does statistical interpretation require more than calculations?

Calculations can be performed by a calculator or computer, but understanding what the data show requires thoughtful examination of the data, its context, and how it was produced.

Dot plot

A graph consisting of a number line with a dot above each data value; repeated values receive multiple dots.

Probability

A mathematical measure of the likelihood that a random event will occur; it is used in statistics to assess the strength of conclusions from data.

Population

The complete collection of people, objects, or observations that a statistical study aims to understand.

Sample

A subset of a population selected for observation or measurement.

Why are samples commonly used instead of studying an entire population?

Studying every member of a population may be too expensive, time-consuming, or practically impossible. A well-selected sample can provide useful information about the population.

Representative sample

A sample that reflects important characteristics of the population it is intended to represent.

How does the quality of a sample affect statistical inference?

A statistic estimates a population parameter more reliably when the sample represents the population well. A nonrepresentative sample can lead to inaccurate conclusions.

Simple random sample

A sample of size $n$ in which every possible group of $n$ individuals from the population has an equal chance of being selected.

What is the purpose of using random numbers when selecting a simple random sample?

Random numbers provide an objective chance process for selecting individuals rather than allowing personal preference or convenience to determine the sample.

Data

The observed values collected for one or more variables; data may be numerical or categorical.

Datum

A single observed value in a data set.

Variable

A characteristic or measurement that can have different values for different members of a population; variables are often represented by capital letters such as $X$ or $Y$.

What is the distinction between a variable and data?

A variable is the characteristic being measured, whereas data are the actual recorded values of that variable. For example, amount spent is a variable, while $150$, $200$, and $225$ are data values.

How should the population, sample, parameter, statistic, variable, and data be identified in a statistical study?

The population is the full group of interest; the sample is the observed subset; the parameter summarizes the population; the statistic summarizes the sample; the variable is the characteristic measured; and the data are the recorded values.

Statistic

A numerical summary calculated from sample data, such as a sample mean or sample proportion.

Parameter

A numerical characteristic of an entire population, such as a population mean or population proportion.

How are a statistic and a parameter related?

A statistic describes a sample and is often used to estimate the corresponding parameter, which describes the population. The quality of the estimate depends strongly on how representative the sample is.

In a study of the average amount first-year students spend on school supplies at a college, what is the parameter?

The parameter is the true mean amount spent by all first-year students at that college during the specified term, excluding books.

In a study of 500 randomly selected doctors, what is the statistic if the goal is to estimate the malpractice-involvement rate?

The statistic is the proportion of the 500 sampled doctors who have been involved in at least one malpractice lawsuit.

Qualitative data

Data that classify or describe attributes using categories, words, or labels; they are also called categorical data.

Quantitative data

Numerical data produced by counting or measuring a characteristic.

Can numerical-looking records ever be treated as qualitative data?

Yes. A numerical measurement can be converted into categories; for example, numerical quiz scores might be reported as letter grades A through F.

Why is calculating an arithmetic mean inappropriate for categorical variables such as blood type?

Categories such as blood type do not represent numerical quantities with meaningful arithmetic operations. They can be summarized with counts or proportions instead.

Quantitative discrete data

Numerical data produced by counting and restricted to distinct, separate values, such as the number of books or phone calls.

Quantitative continuous data

Numerical data produced by measurement that can, in principle, take any value in an interval, including decimals or fractions; examples include time, length, and weight.

How can you distinguish quantitative discrete data from quantitative continuous data?

Discrete data result from counting individual items and take separate numerical values. Continuous data result from measuring and can take arbitrarily precise values within a range.

In an automobile safety study, how could the variable 'head injury' be represented?

It is a categorical variable with possible data values such as 'yes, head injury' and 'no, no head injury.'