AP Statistics 1.5 Graphical Representations for One Quantitative Variable Study Notes - New Syllabus

1.5.A Construct Quantitative One-Variable Graphical Representations

Quantitative variables contain numerical values that can be counted or measured.

To better understand quantitative data, statisticians use graphical representations that display the distribution of the values.

These graphs help reveal:

• Patterns in the data
• Clusters of values
• Gaps in the data
• Shape of the distribution
• Spread of the values
• Possible outliers

Unlike categorical graphs, quantitative graphs maintain the natural numerical order of the data from smallest to largest.

Common graphical displays for quantitative variables include:

• Histograms
• Stem-and-leaf plots
• Dotplots

1.5.A.1 Quantitative Graphical Representations

Histograms, stem-and-leaf plots, and dotplots visually represent the distribution of quantitative data.

These graphs display:

• Frequencies
• Relative frequencies
• Intervals of values
• Individual observations

An important feature of quantitative graphs is that the data values remain ordered from smallest to largest.

This ordering allows statisticians to observe:

• Center of the distribution
• Variability
• Symmetry or skewness
• Unusual observations

Each graph displays the same quantitative data in a different way.

1.5.A.2 Histograms

A histogram is a graph that displays quantitative data using intervals called bins.

The bins are placed along the horizontal axis in numerical order.

Each bar represents a range of values, and the height of the bar shows:

• The frequency
• Or the relative frequency

of observations within that interval.

Unlike bar charts for categorical data:

• Histogram bars touch each other because quantitative intervals are continuous.
• The horizontal axis represents numerical intervals.

The appearance of a histogram depends on the chosen bin widths.

Changing the bin widths may:

• Reveal different patterns
• Hide clusters or gaps
• Change the apparent shape of the distribution

For example:

• Narrow bins show more detail.
• Wide bins produce a smoother appearance.

Histograms may also be drawn horizontally, with bins on the vertical axis.

The frequency for each interval is calculated using:

\( \mathrm{Frequency=Number\ of\ observations\ within\ a\ bin} \)

In the histogram:

• The \( \mathrm{70-79} \) interval would have the tallest bar.
• The bars would touch because the intervals are continuous.

1.5.A.3 Stem-and-Leaf Plots

A stem-and-leaf plot displays quantitative data by separating each value into two parts:

• A stem
• A leaf

The stem usually contains the leading digit or digits.

The leaf usually contains the final digit.

For example:

\( \mathrm{74} \)

can be split into:

• Stem: \( \mathrm{7} \)
• Leaf: \( \mathrm{4} \)

Stem-and-leaf plots preserve the original data values while organizing the data from smallest to largest.

This makes them useful for:

• Identifying clusters
• Finding gaps
• Detecting outliers
• Observing the shape of the distribution

Both stems and leaves are arranged in ascending order.

A key is often included to explain the meaning of the stems and leaves.

For example:

\( \mathrm{7|4=74} \)

This means:

• \( \mathrm{7} \) is the stem
• \( \mathrm{4} \) is the leaf
• The complete value is \( \mathrm{74} \)

The plot above represents the data values:

\( \mathrm{62,\ 65,\ 68,\ 70,\ 72,\ 72,\ 75,\ 78,\ 81,\ 84} \)

Stem-and-leaf plots are most effective for small or medium-sized data sets.

1.5.A.4 Dotplots

A dotplot represents each observation in a quantitative data set using a dot.

Each dot is placed directly above a value on a horizontal axis or beside a value on a vertical axis.

If multiple observations have the same value, the dots are stacked on top of each other.

Dotplots help visualize:

• Clusters
• Gaps
• Outliers
• Shape of the distribution
• Repeated values

Dotplots are especially useful for:

• Small data sets
• Comparing frequencies of individual values
• Preserving exact observations

Unlike histograms, dotplots show every individual data value directly.

For example, the data:

\( \mathrm{2,\ 3,\ 3,\ 4,\ 5,\ 5,\ 5,\ 6} \)

would place:

• One dot above \( \mathrm{2} \)
• Two dots above \( \mathrm{3} \)
• Three dots above \( \mathrm{5} \)

Stacked dots show repeated observations clearly.

Dotplots maintain the natural ordering of quantitative data from smallest to largest.