Download Lecture 3 - Department of Mathematics and Statistics

MATH 2560 C F03
Elementary Statistics I
LECTURE 3: The Normal Distributions.
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Outline.
⇒ density curves;
⇒ center & spread for density curve;
⇒ normal distributions;
⇒ the 68 − 95 − 99.7 rule;
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Density Curves
We have already known about graphical (Lecture 1) and numerical (Lecture 2) tools for describing distributions. We also know how to explore data
on a single quantitative variable.
Sometimes the overall pattern of a distribution can often be described
compactly by a smooth curve.
The smooth curve drawn through the histogram (Figure 1.19) gives a
compact discription of the overall pattern. We draw a smooth curve through
a histogram to describe the shape of the distribution without the lumpiness
of the histogram.
The curve is a mathematical model (idealized mathematical description) for the distribution.
⇒Area of the Bars in Histograms=Proportions of the
Observations
⇒ To make such curves easier to compare for different distributions, we adjust the sacle so that the total area under the curve is
exactly 1.
Areas under the curve then represent proportions of the observations.
⇒ Area=Relative Frequency.
⇒ The curve is then a Density Curve
Figure 1.20 (a) and Figure 1.20 (b) are copies of Figure 1.19 with the
leftmost bars shaded.
The area of the shaded bars in Figure 1.20(a) represents the students
with vocabulary scores 6.0 or lower. There are 287 such students, who make
up the proportion 287/947 = 0.303 of all Gary seventh graders.
The shaded area in Figure 1.20 (b) under the smooth curve represents
the students with vocabulary scores 6.0 or lower. This area is 0.293, only
0.010 away from the histogram result.
Density Curve
A density curve is a curve:
1. Is always on or above the horizontal axis;
2. Has area exactly 1 underneath it.
3. The area under the curve and above any range of values
is the relative frequency of observations that fall in that range.
Remarks
1. The density curve in Figures 1.19 and 1.20 is Normal Curve.
2. Density curves, like distributions, come in many
shapes.
Figure 1.21 (a) shows symmetric density curve.
Figure 1.21 (b) shows skewed density curve.
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3.1
Measuring Center and Spread for Density
Curve
Skewed Distributions
⇒ MODE of distribution described by density curve is a peak
point of the curve, the location where the curve is highest.
⇒ MEDIAN of distribution
is the point with half the total area on each side.
⇒ QUARTILES
divide the area under the curve into quarters as accurately as possible by eye.
⇒ IQR
is the distance between the first and third quartiles.
3.2
Symmetric Distribution
⇒ MEAN
is the point at which the curve would balance if it were made out of solid
material.
(See Figure 1.22)
⇒ STANDARD DEVIATION
can not be located by eye on most density curves.
Remarks.
1. The median and mean are the same for a symmetric
density curve.
They both lie at the center of the curve.
2. The mean of the of the skewed curve is pulled away from the median
in the direction of the long tail.
3. The density curve is an idealized mathematical model for a distribution
of data.
4. The usual notation for the mean of an idealized distribution is µ.
The usual notation for the standard deviation is σ.
5. Abbreviation for the normal distribution with mean µ and standard
deviation σ is N (µ, σ).
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Normal Distributions (ND)
⇒ Normal Curves: symmetric, unimodal, and bell-shaped.
The Noraml Curves describe Normal Distributions.
Examples of Normal Curves: Figures 1.20 (a), (b) and 1.21 (a) are
the examples of normal curves.
⇒ exact normal curve is determined completely by mean µ and
standard deviation σ.
⇒ µ is located at the center of the symmetric curve and is the same as
the median.
Changing µ without changing σ moves the normal curve along the horizontal
axis without changing the spread.
⇒ σ controls the spread of a normal curve.
Figure 1.23 shows two normal curves with different values of σ.
The height of the normal density curve at any point z is given by
1 z−µ 2
1
√ e− 2 ( σ ) .
σ 2π
Here: −∞ < µ < +∞ and σ > 0. We can see that the expression of the
normal density curve is completely determined by µ and σ.
Why are the Noramal Distributions (ND)
Important in Statistics
⇒ ND are good descriptions for some distributions of real data;
⇒ ND are good approximations to the results of many kinds of
chance outcomes;
⇒ ND are bases for many statistical inference procedures;
⇐ However, most income distributions are skewed to the right and so are
not normal.
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The 68 − 95 − 99.7 Rule
All the normal curves have common properties.
The 68 − 95 − 99.7 Rule
In the Normal Distribution with mean µ and standard deviation σ:
⇒ 68 persents of the observations fall within σ of the mean µ,
i.e., in [µ − σ, µ + σ];
⇒ 95 percents of the observations fall within 2σ of µ,
i.e., in [µ − 2σ, µ + 2σ];
⇒ 99.7 percents of the observations fall within 3σ of µ,
i.e., in [µ − 3σ, µ + 3σ].
Figure 1.24 illustrate the 68 − 95 − 99.7 rule. Here: µ = 0 and σ = 1.
68 percents are in [−1, 1], 95 percents are in [−2, 2], and 99.7 percents are in
[−3, 3].
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Example 1.22
We consider an example to demonstrate the 68 − 95 − 99.7 rule.
Distribution of Heights of Young Women
The distribution of heights of young women aged 18 to 24 is approvimately
normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
Figure 1.25 shows what the 68 − 95 − 99.7 rule says about this distribution.
⇒ 68 rule: 68 percents of heights of the young women fall in the interval:
[µ − σ, µ + σ] = [64.5 − 2.5, 64.5 + 2.5] = [62, 67];
⇒ 95 rule: 95 percents of heights of the young women fall in the interval:
[µ − 2σ, µ + 2σ] = [64.5 − 5, 64.5 + 5] = [59.5, 69.5];
⇒ 99.7 rule: 99.7 percents of heights of the young women fall in the
interval:
[µ − 3σ, µ + 3σ] = [64.5 − 7.5, 64.5 + 7.5] = [57, 72].
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Summary
The mean µ (balance point), the median (equal-areas point), and the quartiles can be approximately located by eye on a density curve.
The standard deviation σ cannot be located by eye on most density
curve.
The mean and median are equal for symmetric density curves, but the
mean of a skewed curve is located farther toward the long tail than is the
meadian.
The normal distributions N (µ, σ) are described by bell-shaped, symmetric
unimodal density curves.
The mean µ and standard deviation σ completely specify the normal
distribution N (µ, σ).
The mean µ is the center of symmetry and σ is the distance from µ to
the change-of-curvature points on either side.