Chapter 1: The Normal
AP Statistics Standards
I. Exploring Data: (continued).
II. Anticipating Patterns:
C. The normal distribution
Properties of the normal
Using tables of the normal
The normal distribution as a
model for measurements
|Essential Question: What does it mean to be in the 95 percentile group?
- Understand the meaning of percentile.
Example: if a score of 57 is at
the 10th percentile then 10% of the observations or data points fall below a score of 57.
- Convert distribution data in various forms into a relative cumulative
frequency graph (generally expressed in %), often called an ogive.
Note: Ogives are often drawn with the otherwise abhorrent connect the
dots style. Ogives tend to have a sigmoidal shape.
Formative Assessment: create ogives out of the following items or the following items out of ogives.
- data table
- 5-number summary
- stem and leaf plot
Homefun (formative/summative assessment): prob. 1,3,5,9 pp.105-106
|Essential Question: What will changing the units of
measurement do to measures of spread and central tendency?
- What is a linear transform? (See p. 53).
xnew = a + b∙xold
Example: the linear transform
to change from Celsius to Fahrenheit
(ºF) = 32 + 1.8 (ºC)
- State the effect that multiplying each number by
a constant and/or adding a
constant to each number in a data set has on the following: (This effect is important to
know when changing the data points' units.)
point) + a
point) * b
||(mean) + a
||(mean) * b
||(median) + a
||(median) * b
| standard deviation
||(std dev) * b
||(IQR) * b
||(range) * b
|Q1 and Q3
||(Q1) + a & (Q3) + a
||(Q1) * b & (Q3) * b
Formative assessment: Using the TI-83 calculators and monthly temperature data in Fahrenheit from the Geenville-Spartenburg Airport, calculate and record the mean, median, standard
deviation, IQR, range, Q1, and Q3 for both Fahrenheit and Celsius. Convert the recorded Fahrenheit values directly to Celsius. How do the converted values compare to the ones calculated in Celsius.?
Homefun (formative/summative assessment):
prob. 19, 21 p. 107
Is the bell-curve real?
The Normal Distribution
- Define density curve.
- Models a distribution
- Above horizontal axis
- Area under it = 1
- Area under part of the curve = probability
- mean: the "balance point", similar to the center of mass in physics.
- median: the equal area point
- Describe the normal distribution.
- Bell Shaped
- Range negative infinity to positive
How to Calculate the Normal
pdf = 1/[σ(2π)] • exp[ - (x - μ)2 / (2σ2)]
Note: determining the shape of the normal
distribution requires a knowledge of both the mean and
- pdf = probability density function (height or y-axis coordinate of the N-distribution)
- x = horizontal
- μ =
- σ = the standard deviation
- Create or analyze drawings of normal distributions by knowing where where the 2 inflection points
An inflection point is the point on a curve where the curve changes from concaved upward to concaved downward
For N-distr. inflection pts = ± 1 sigma from the mean)
State the probability of getting a data point with a
specific value. zero. Why? A point has a width of
zero, hence, there's zero area under a
single point on the normal distribution
Correctly use and interpret N (mean, sigma) notation. The mean and standard deviation -- just 2 numbers -- completely describes a normal distribution. Incredibly, just these 2 numbers have the power to characterize massive sets of data points with a diversity of values.
Evaluate the effects on a
of increasing or decreasing the standard deviation.
Relevance: The normal distribution is
ubiquitous. Discussions about it pop up in all kinds of both technical, news,
and popular publications.
Homefun (formative/summative assessment): prob. 27, 29, 31, 33 pp. 108 - 109
How is the area under the
bell-curve related to the standard deviation?
Estimating Areas Under Normal
- Find probabilities (areas under the curve)
for normal distributions
(sometimes called the empirical rule).
in the 1st part of the test for this chapter, you will not be permitted to use calculators or
tables. The entire test will be based on the above rule.
Formative assessment: Using the empirical rule evaluate areas under the curve.
Relevance: If you know the empirical rule you
can easily double check your answers when solving for areas under the
n-distribution. The ability to evaluate answers is a very high level skill.
Homefun (formative/summative assessment): prob. 43, 45 p.131
Does an ogive give the
same information as a normal distribution?
Relating the Normal Distribution to Ogives
- Convert the N-distribution into an ogive (cumulative
frequency plot) using the empirical rule. (Formative assessment, group work)
- Estimate the mean, median, sigma, Q1, Q3, and IQR using an
ogive for a normal
distribution. (Formative assessment, group work)
How different are the characteristics of non-normal distributions to normal distributions?
Non-Normal vs. Normal Distributions
- Estimate the median, quartiles, and probabilities for any
distribution by estimating the area under the curve:
Be as one with the
0-75-89 Rule based on Chebyshev's Theorem,
which applies to any distribution.
Tail areas, several standard deviations from the mean, of the normal distribution are often used in decision making. Chebyshev's Theorem indicates the maximum error in calculating a tail area if a
distribution is incorrectly identified as a normal distribution.
- Chebyshev's Theorem:
- p = (1 - 1/k2)
Where: p = the lowest possible
probability of finding a value between ±k
standard deviations from the mean.
Formative Assessment: Calculate the maximum error in tail areas beyond ± 1, 2, and 3 sigma that can be caused by assuming a distribution is normally distributed when it isn't.
From the above, mathematically evaluate why Chebyshev's theorem only has value for making predictions when K > 1.
How can the area under the bell-curve
be determined more accurately than by estimating?
Determining Areas Under the Normal
Distribution By Using Tables
Derive the mathematical definition of a z-score (abbreviated as z) using its English language definition and use it for calculate z-scores.
- z-score = number of standard
deviations from the mean
Find areas under the normal distribution using
tables. Note, the tables contain a "standardized normal distribution".
Here, the mean = zero and the standard
deviation = 1.
Probability = area under curve
Formative Assessment: Find the area under the curve of the normal distribution by using tables.
|Essential Question: How does the box and whiskers plot look like for normally distributed data?
Working Area Under the Curve Problems Backwards
z-scores corresponding to LL, Q1, median, Q3, UL of a modified box and whiskers plot
Formative Assessment: Draw a box and whiskers plot of a normal distribution.
- Using a normal distribution, estimate a critical
value given the probability of finding a value as extreme or more extreme
Homefun (formative/summative assessment): prob. 47, 49, 51, 55 p.132
How can we tell if a
distribution is normal and why would we care?
Evaluate the normality of a distribution by
examining histograms, stem plots, dot plots, ogives, or box and whiskers plots.
Follows 68-95-99.7 Rule
whisker ≈ 1.5 x ( box width)
histograms, stem plots, dot plots
box & whiskers plots.
Formative Assessment: Evaluate various data sets to determine if it is reasonable to assume that they are normally distributed.
Evaluate the normality of a distribution
using a normal quantile plot on a TI-83.
n-distributions look like straight lines on normal quantile plots.
Homefun (formative/summative assessment): prob. 65, 67, 69, 75, 71 pp. 134 - 135
What is the best way to
find probabilities associated with normal distributions?
Work quality control reject rate problems.
Note: rejects occur in the tail areas.
Find areas under the normal distribution using tables
using a TI -83 calculator.
Have you taken your LUMS today?
Given a tail area, find the critical value (see
objective 15) using a TI-83 calculator.
I think therefore I AMS.
(must be a decimal fraction)
Remember, in tables for
finding standard normal distribution probabilities, the mean = zero and
the standard deviation = 1. If these values are input into
invNorm along with an area,
the result will be a z-score.
Homefun (formative/summative assessment): Work the Chapter 2 practice Test T2.1 to T2.13 138-139
How can I make an "A" on
Normal Distributions Review
- Work the practice test.
- Review the objectives.
- Look over
free response problems
from previous years.
- Master the vocabulary (see example