Section 2.1Density Curves & the Normal DistributionsAP StatisticsBerkley High School/CASA

AP Statistics, Section 2.1, Part 1 2

Density Curve

“Mathematical Models” The area under the

curve is related to the distribution of values

“Idealized description”

AP Statistics, Section 2.1, Part 1 3

Density Curves

Characteristics Always above the x-

axis Area always equal to 1

The area under the curve and above any range of values is the proportion of all observations that fall in that range.

AP Statistics, Section 2.1, Part 1 4

Median and Mean Since the area

represents portions of the population, the Median is the spot where the area to the left is the same as the area to the right.

AP Statistics, Section 2.1, Part 1 5

Median and Mean

The Mean represents the “balance point”.

Imagine a “see-saw”

AP Statistics, Section 2.1, Part 1 6

Median and Mean

The Mean is always pulled towards the tail in a skewed distribution

AP Statistics, Section 2.1, Part 1 7

“Idealized Distributions” We use different

notation for density curve (which represent entire populations) as compared to data sets (which represent samples)

population a ofdeviation standardpopulation a ofmean

sample a ofdeviation standard sample a ofmean

sx

AP Statistics, Section 2.1, Part 1 8

AP Statistics, Section 2.1, Part 1 9

AP Statistics, Section 2.1, Part 1 10

AP Statistics, Section 2.1, Part 1 11

Normal Distribution

A special, bell shaped, symmetric, single-peaked distribution

AP Statistics, Section 2.1, Part 1 12

Normal Distribution

Because of the symmetry, the mean and median are the same and at the line symmetry

AP Statistics, Section 2.1, Part 1 13

Normal Distribution

The inflection points (where the curve starts to flatten out) represent the width of the standard deviation

μ μ+σμ-σ

AP Statistics, Section 2.1, Part 1 14

68-95-99.7 Rule

AP Statistics, Section 2.1, Part 1 15

68-95-99.7 Rule with N(64.5,2.5)

AP Statistics, Section 2.1, Part 1 16

Assignment

Exercises 2.1 – 2.15 odd, The Practice of Statistics