Transcript Section 2.1 Second Day Transforming data

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Chapter 2: Modeling Distributions of Data
Section 2.1
Describing Location in a Distribution
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE
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In Chapter 1, we developed a kit of graphical and numerical
tools for describing distributions. Now, we’ll add one more step
to the strategy.
Exploring Quantitative Data
1. Always plot your data: make a graph.
2. Look for the overall pattern (shape, center, and spread) and
for striking departures such as outliers.
3. Calculate a numerical summary to briefly describe center
and spread.
4.
Sometimes the overall pattern of a large number of
observations is so regular that we can describe it by a
smooth curve.
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Curves
Describing Location in a Distribution
 Density
Curve
A density curve is a curve that
•is always on or above the horizontal axis, and
•has area exactly 1 underneath it.
A density curve describes the overall pattern of a distribution.
The area under the curve and above any interval of values on
the horizontal axis is the proportion of all observations that fall in
that interval. AREA = PROBABILITY = PROPORTION.
The overall pattern of this histogram of
the scores of all 947 seventh-grade
students in Gary, Indiana, on the
vocabulary part of the Iowa Test of
Basic Skills (ITBS) can be described
by a smooth curve drawn through the
tops of the bars.
Describing Location in a Distribution
Definition:
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 Density
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Mean and Median of a Density
Curve
 Symmetric:
Mean = Median
 Skewed
Left: Mean < Median
 Skewed
Right: Mean > Median
 The
median of a density curve is the equal-areas
point, where ½ of the area is to the left and ½ of the
area is to the right.
 The
mean of a density curve is the balance point,
where the curve would balance if it were made of
solid material.
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Examples
For each of
these density
curves, which
line represents
the mean? The
median?
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Quartiles
For any density curve…
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How much area is to the left of the first quartile?
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How much area is to the right of the first quartile?
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How much area is between the first and third quartiles?
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The uniform distribution
 The
uniform distribution is a special kind of
density curve. It looks like a rectangle.
 It
has a constant height (i.e. horizontal line)
over some interval of values.
 So, this density curve describes a variable
whose values are distributed evenly
(UNIFORMLY) over some interval of
values.
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Example
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Accidents on a level, 3-mile bike path occur uniformly along the
length of the path.
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A) Show that this density curve satisfies the two requirements
for a density curve.
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B) The proportion of accidents that occur in the first mile of the
path is the area under the density curve between 0 miles and 1
mile. What is this area?
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C) Sue’s property adjoins the bike path between the 0.8 mile
mark and the 1.1 mile mark. What proportion of accidents
happen in front of Sue’s property?
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Normal Distributions
 Density
curves have an area = 1 and are always
positive.
 Normal curves are a special type of density curves.
 T/F All density curves are normal curves.
 T/F All normal curves are density curves.
Characteristics of Normal Curves
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Symmetric
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Single-peaked (also called
unimodal)
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Bell-shaped
μ
σ
The mean, μ, is located at the center of the
curve.
The standard deviation, σ, is located at the
inflection points of the curve.
Parameters of the Normal Curve
 The
same way a line is
defined by its slope
and y-intercept, a
normal curve is
defined by its mean
and standard
deviation.
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Why Be Normal?
 Normal
curves are good descriptions for lots
of real data: SAT test scores, IQ, heights,
length of cockroaches (yum!).
 Normal
curves approximate random
experiments, like tossing a coin many times.
 Not
all data is normal (or even approximately
normal). Income data is skewed right.
Notation
We abbreviate the Normal distribution with mean µ and
standard deviation σ as N(µ,σ).
IQ scores on the WISC-IV are distributed Normally with a
mean of 100 and a standard deviation of 15.
IQ ~ N(100,15).
Women’s heights ~ N(64.5”, 2.5”)
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Distributions
Normal Distributions
 Normal
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Remember z-scores?
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What’s the formula to standardize a value?
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If a person’s IQ has a z-score of 0, what does that mean?
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What does a z-score of -1 mean?
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On the normal curve, we draw 3 standard deviations on either
side of the mean.