Chapter 2
Density Curves and Normal
Distributions
Remember
1. ALWAYS plot data (usually histogram or
stemplot)
2. Overall Pattern (CUSS)
3. Calculate Numerical Summary
4. NEW – Sometimes overall pattern is so
regular it can be described by a smooth curve
(called a density curve) fig 2.2 pg78
Density Curve
• Mathematical Model – an idealized
description
– Always on or above horizontal axis
– Has area of exactly 1 underneath it
– Areas under the curve represent proportions of
the observations
Mean & Median of Density Curves
• Median (pg 81) – the point with half the
observations on either side or it is the equal
areas point.
• Mean (pg 82) – point where the curve would
balance if made of solid material
– Mean of density curve µ ‘mu’
– Standard deviation σ ‘sigma’
The are under the
curve is a rectangle
with height 1 and
width 1.
A uniform Distribution Exercise 2.2
• Why is the total area under
this
curve
equal to
20% , the
region
is
a rectangle with
1?
height of 1 and
base width of 0.2lie above
• What percent of the observations
0.8?
60%
• What percent of the observations lie below
0.6?
50%
• What percent of the observations lie between
Mean = ½ or 0.5,
0.25 andthe0.75?
‘balance
point’ of the
• What isdensity
the mean
µ of this distribution?
curve
Quartiles
• Divides the area under the curve into quarters
• ¼ of the area is left of Q1
• ¾ of the area is left of Q3
• Symmetric Curve
– Median and mean are equal (2.5a, pg 81)
• Skewed Curve
– Mean pulled toward the long tail of a skewed
distribution
HW Monday
• 1, 3, 4
• QUIZ BLOCK DAY first thing then 2.2
• This quiz will go in the grade book so be sure
to look over your notes and examples in the
book.
Normal Curves
• Symmetric, single peaked, and bell shaped;
describe normal distributions
Normal Distributions
• (all have the SAME overall shape) they are
described by giving the mean (µ) and the
standard deviation (σ)
• Notation is N(µ,σ)
Things to Know
• Changing µ without changing σ moves the normal
curve along the horizontal axis without changing
spread.
• Standard deviation controls the spread of a
normal curve (fig 2.10)
• Inflection points are points located at a distance
of σ on either side of µ.
– Points where there is a change of curvature.
• In general µ and σ do not alone specify shape of
most distributions, these are special properties of
normal curves.
Importance of Normal Distributions
• Good descriptions for some distributions of real
data
– EX – SAT, IQ Tests, Psychological tests, characteristics
of biological populations (lengths of cockroaches,
yields of corn)
• Good approximations to the results of many
kinds of chance outcomes
– EX – Tossing a coin
• Many statistical inference procedures (ch 6-12)
based on normal distributions work well with
other roughly symmetric distributions.
The Empirical Rule
The 68 – 95 – 99.7 Rule
Lets Draw a normal curve with
notation
N(64.5, 2.5)
Find percentiles for the following heights
1. 64.5 inches
2. 59.5 inches
3. 67 inches
4. 72 inches
QUIZ BLOCK DAY first thing, then 2.2
This quiz will go in the grade book so be sure to
look over your notes and examples in the book.
2.2 Standardizing and z-scores
• All normal distributions are the same if we
measure in units of size σ about the mean µ as
center
• Changing to these units is called
STANDARDIZING
• If x is an observation from a distribution that
has mean µ and standard deviation σ, the
standardized value of x is
• z=x-µ
σ
Standardized value is often
called the z-score
• Z-score tells you how many standard
deviations the original observation (x) falls
away from the mean (µ) and in which
direction
• Lets look back at N(64.5, 2.5) average adult
women
• Pg. 98 Example 2.7
54.5
57
59.5 62 64.5 67
69.5 72
74.5
What proportion of all young women are less that
68 inches tall? 𝑥 −µ 68−64.5
Z=
σ
=
2.5
= 1.4
How to use table A to find the proportions of
observations from the standard normal distribution
that are greater than a specific z-score:
z = -2.15
Find -2.15 on the left hand column and .05 in the top
row…. The table entry is 0.0158. This means that the
area to the left of -2.15 is 0.0158. Since the total area
under the curve is equal to 1, the area lying to the
right of -2.15 is:
1 - 0.0158 = 0.9842
Figure 2.16 on page 97 illustrates these areas.
Pg. 102 Example 2.10
How to find a value when given the proportion,
𝜇, and 𝜎.
You will need to know how to solve the problem
both ways!!!!
Finding Normal Proportions
1. State problem, draw pic, shade area of
interest
2. Standardize x, draw new pic
3. Find area under curve with Table A
4. Write conclusion in CONTEXT of problem
Example 2.8 on Pg. 99