2.1: Density Curves, Normal Distributions, and Segmented Bar Charts

Some data can be graphed on a relatively smooth curve

Detail may be lost about some specific data, but this is outweighed by the simplicity the curve allows us to analyze with

**Mathematical model**: an idealized description of a situation

When a curve is used to approximate a distribution, the scale is adjusted to make it into a density curve

**Density curve**: a distribution which is always above the x-axis and always has an area of 1The area in any part of the curve is equal to the percent of data present in that part of the curve

**Normal curve**: a distribution which is symmetric, bell-shaped, and has a single peakDefined by their means (μ) and standard deviations (σ)

Normal curves may be analyzed further using z-scores, which can be found and interpreted in relation to probabilities using a z-chart

This chart is provided on the AP exam, along with many relevant formulas

When solving z-score problems, it is always necessary to:

Show work for getting the z-score

Write a probability statement

Give the final probability

Data may be represented as N(μ, σ)

μ: mean

σ: standard deviation

*Who is taller relative to their gender: a 76.5-inch-tall woman or a 80-inch-tall man?*Female heights are N(64, 2.5) and male heights are N(69.1, 2.8)

Female: 76.5 inches

That means that this value is 5 standard deviations above the mean

This is where the term “off the charts” comes from → if you reference what 5 standard deviations above the mean would look like on the normal curve above in the notes, 5 standard deviations to either side is virtually unheard of and statistically very rare

Male: 80 inches

The height of an 80-inch-tall man is 3.89 standard deviations above the mean

This means that statistically, the woman is taller proportionate to her gender than the man is proportionate to his.

**Segmented bar chart**: a display of categorical data showing such data as a percentage of the wholeLike a pie chart in bar form

**Mosaic plot**: segmented bar charts for comparing groups where the widths of the bars are proportional to the size of the groups

Some data can be graphed on a relatively smooth curve

Detail may be lost about some specific data, but this is outweighed by the simplicity the curve allows us to analyze with

**Mathematical model**: an idealized description of a situation

When a curve is used to approximate a distribution, the scale is adjusted to make it into a density curve

**Density curve**: a distribution which is always above the x-axis and always has an area of 1The area in any part of the curve is equal to the percent of data present in that part of the curve

**Normal curve**: a distribution which is symmetric, bell-shaped, and has a single peakDefined by their means (μ) and standard deviations (σ)

Normal curves may be analyzed further using z-scores, which can be found and interpreted in relation to probabilities using a z-chart

This chart is provided on the AP exam, along with many relevant formulas

When solving z-score problems, it is always necessary to:

Show work for getting the z-score

Write a probability statement

Give the final probability

Data may be represented as N(μ, σ)

μ: mean

σ: standard deviation

*Who is taller relative to their gender: a 76.5-inch-tall woman or a 80-inch-tall man?*Female heights are N(64, 2.5) and male heights are N(69.1, 2.8)

Female: 76.5 inches

That means that this value is 5 standard deviations above the mean

This is where the term “off the charts” comes from → if you reference what 5 standard deviations above the mean would look like on the normal curve above in the notes, 5 standard deviations to either side is virtually unheard of and statistically very rare

Male: 80 inches

The height of an 80-inch-tall man is 3.89 standard deviations above the mean

This means that statistically, the woman is taller proportionate to her gender than the man is proportionate to his.

**Segmented bar chart**: a display of categorical data showing such data as a percentage of the wholeLike a pie chart in bar form

**Mosaic plot**: segmented bar charts for comparing groups where the widths of the bars are proportional to the size of the groups