AP Statistics Unit 1 Notes

Statistics and Data Concepts

  • Statistics is the science of collecting, organizing, analyzing, and interpreting data.

  • Data contains information about a group of individuals.

  • Individuals are objects described by a set of data.

  • Variables are characteristics of individuals; they may take on different values.

  • Variables can be split into two types:

    • Categorical variables: place individuals into specific groups.

    • Quantitative variables: take on numerical values for which arithmetic operations are meaningful.

  • Quantitative variables fall into two categories:

    • Discrete variables: numerical values where counting makes sense, and decimals are generally not appropriate (e.g., number of siblings).

    • Continuous variables: numerical values where decimals are appropriate, usually involving some form of measuring (e.g., height, weight).

  • Note: Just because a value is a number does not automatically make it quantitative (e.g., ZIP code is categorical).

Representing Categorical Data: Tables

  • One of the easiest ways to display categorical data is with a table.

  • One-way tables (for a single categorical variable) include:

    • Category

    • Count (Frequency)

    • Relative Count (Relative Frequency)

  • Relative Frequency is the proportion of observations in each category:
    Relative Frequency=count in categoryn\text{Relative Frequency} = \frac{\text{count in category}}{n}
    where nn is the total number of observations.

  • Two-way tables (for two categorical variables) show the joint distribution of the variables.

Bar Graphs for Categorical Data

  • Used for graphing Categorical Variables.

  • Important characteristics:

    • Label each axis clearly.

    • The x-axis contains the categorical variable, and the y-axis displays counts (or percentages).

    • Each category has its own bar, and the bars CANNOT touch.

    • The order of categories on the x-axis is not important.

Graphing Quantitative Variables: Histograms

  • The distribution of a variable tells us what values the variable takes and how often it takes these values.

  • Used for graphing Quantitative Variables.

  • Histogram construction steps:

    • Group data into bins (even intervals).

    • Determine Lowest Value (min⁡x<em>i\min{x<em>i}) and Highest Value (max⁡x</em>i\max{x</em>i}).

    • Choose a Bin Width (ww) and determine the number of bins.

    • For each bin, count how many data scores fall into that interval.

    • Draw rectangles for each bin with height representing the count.

    • Bars MUST TOUCH – NO GAPS!

    • Label the x-axis with the lower bound values of your bins.

  • Unlike bar graphs, histograms are used for quantitative data and represent continuous intervals with touching bars.

Quick Reference Formulas

  • Relative Frequency (for a category):
    Relative Frequency=count in categoryn\text{Relative Frequency} = \frac{\text{count in category}}{n}

  • Percent (conversion from relative frequency):
    Percent=100⋅Relative Frequency\text{Percent} = 100 \cdot \text{Relative Frequency}