AP STATISTICSSection 2.1: Measures of Relative Standing and Density Curves

Objective: To be able to recognize density curves and use them to calculate area under the curve.

Standardizing is the process by which we convert the original values to standard deviation units.

(z-scores) or

z-scores tell us how many standard deviations away from the mean the original observation falls and in what direction.• No units• Positive z-score means the observation lies above the

mean.• Negative z-score means the observation lies below the

mean.

Why do we use z-scores?

1. To express all observations on a common scale.

2. To compare relative standing of individuals on different scales.

Ex. Test Scores

Converting from a z-scores to a percentile depends on the shape of the distribution.

Percentiles (again)

Chebyshev’s Inequality: In any distribution the percentage of observations falling within k standard deviations of the mean is at least percent. (k>1)Let k=2, then ________% of all observations lie within 2 st. dev. of the mean.

Let k=3, then ________% of all observations lie within 3 st. dev. of the mean.

Let k=4, then ________% of all observations lie within 4 st. dev. of the mean.

Ex. GE 100 watt light bulbs have been tested by Consumer Reports in accelerated lifetimes. It has been found that the average life expectancy is 997 hours and the standard deviation is 74 hours. Using Chebyshev’s inequality, what can be said about the lifetimes of the light bulbs?

A mathematical model is a curve that best represents the distribution of the data.

Why do we use mathematical models?• Gives a compact picture of the distribution.• Ignores minor irregularities.• Ignores outliers.• Easier to work with.

A density curve is a curve that lies on or above the x-axis and has an area of 1 underneath it.

***The area under the curve represents the proportion of observations that fall over a given range on the x-axis.

Ex. 1. Density Curves

a. Find the height so that the curve is a density curve.

b. Find the proportion of observations that fall over the given range of X:

Median of a density curve is the point on the x-axis in which we have equal areas.

Mean of a density curve is the balance point on the x-axis (ignoring the curve).

Since density curves represent the distribution of the population, we will use μ for the mean and σ for standard deviation.

Ex. 2.

a. Confirm that this is a density curve

b. Find the proportion of observations that fall over the given range of X:

X < 0.25

X > 0.60

X = 0.60

X < 0.60

0.32 < X < 0.80

Ex. 3.

a. Graph the uniform density curve over

b. Find h so that this is a density curve.

c. Find X<0

d. Find 1<X<2

e. Find -1<X<1

Ex. 3.

a. Graph the density curve over

b. Find X<2

c. Find 2<X<3

d. Find X=2

e. Find X > 2