AP Statistics2.1 Density Curves and the Normal

Distribution

Differentiate between a density curve and a histogram

Understand where mean and median lie on curves that are symmetric, skewed right, and skewed left.

Use a normal distribution to calculate the area under a curve

Learning Objective

1-Always plot your data: make a graph, usually a histogram or a stem plot.

2-Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers.

3-Calculate a numerical summary to briefly describe the center and spread.

median-5 # summary mean-μ and σ

We now add one more step!

4-Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

How do we explore data with a single quantitative variable?

it is an idealized description. It gives a compact picture of the overall pattern of the data but ignores minor irregularities as well as many outliers.

Mathematical Model

HistogramActual count of

observations that fall within an interval

Density CurveThe proportion of

values that fall within an area under the curve.

Density Curves versus Histograms

A Density Curve is a curve that:- Is always on or above the horizontal

axis- Has an area of exactly 1 underneath it.

A density curve describes the overall pattern of a distribution. The area under the curve and any above range of values is the proportion of all observations that fall in that range.

A normal curve is one that is symmetrically skewed.

Density curves, like normal distributions come in many shapes.

The following density curve is skewed to the right.

What does the shaded area mean?

The proportion of observations taking values between 9 and 10.

The median is point where half the observations are on either side.

The quartiles divide the area under the curve into quarters .

The median of a symmetric density curve is at the center.

What do we know about the mean and median of the following 3 curves? Draw lines to represent the mean and median on each curve.

symmetrically skewed skewed to the right skewed to the left

Ex: pg. 71 2.2 a-c

2.3a-d

Normal Distributions

1- all normal dist. have the same overall shape (symmetric, single-peaked, bell shaped)

The exact density curve for a particular normal distribution is described by giving its:

1- mean (μ) and2- standard deviation (σ)

μ=mu σ=sigma

Draw a normal curve with μ=10 and σ=2 Draw a normal curve with μ=10 and σ= 5

What do you notice?

σ controls the spread. The larger σ, the more spread out the curve.

What happens to two normal curves with different standard

deviations?

In a normal distribution with mean (µ) and standard deviation (σ):

-68% of observations fall within 1σ of μ.

-95% of observations fall within 2σ of μ.

-99.7% of observations fall within 3σ of μ.

The 68-95-99.7 rule

The average height of women is 64.5 inches with σ=2.5

Draw a curve.1-What height of women do the middle 68%

fall?

2-What height is the 84th percentile?

3-What height is the highest 2.5% of women?