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The empirical (68-95-99.7) rule
•^
With a bell shaped distribution, ¾
about 68% of the data fall within a distance of 1 standard^ deviation from the mean. ¾
95% fall within 2 standard deviations of the mean. ¾
99.7% fall within 3 standard deviations of the mean.
-^
What if the distribution is not bell-shaped? There is another rule, named Chebyshev's Rule, that tells us that there must be at least 75% of the data within 2 standard deviations of the mean, regardless of the shape, and at least 89% within 3 standard deviations.
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Linear transformations
•^
A linear transformation changes the original value
x
into a
new variable
x
new
•^
x new
is given by an equation of the form,
•^
Example 1.19 on page 54 in IPS.(i) A distance
x
measured in km. can be expressed in
miles as follow,
(ii) A temperature x measured in degrees Fahrenheit can be
converted to degrees Celsius by
x^
a^
b x
n e w
=
0 .6 2
x^
x
n e w
=
5
160
5
(^
9
9
9
x^
x^
x
new
=^
−^
=−
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Measure
x
x
new
MeanMedian
M Mode
Range
R
IQR
IQR
Stdev
s
a +
b M
Mode
a +
b Mode
x
x
b
a
R
b^ IQRb
s
b
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Example 1
•^
A sample of 20 employees of a company was taken andtheir salaries were recorded. Suppose each employeereceives a $300 raise in the salary for the next year. State whether the following statements are true or false.
a)
The
IQR
of the salaries will
i.^
be unchanged
ii.
increase by $
iii.
be multiplied by $
b)
The mean of the salaries will i.^
be unchanged
ii.
increase by $
iii.
be multiplied by $
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Example 2 - Nonlinear transformations
0
1
2
3
4
5
6
7
8
9
10
60 50 40 30 20 10 0
ln(sales)
Frequency
Histogram for ln(sales)
0
1000 2000 3000 4000 5000 6000 7000 8000 9000
200 100 0
Sales
Frequency
Histogram for sales data
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Density curves
•^
Using software, clever algorithms can describe a distributionin a way that is not feasible by hand, by fitting a smooth curveto the data in addition to or instead of a histogram. The curvesused are called
density curves
•^
It is easier to work with a smooth curve, because histogramdepends on the choice of classes.
-^
Density Curve Density curve is a curve that^ ¾
is always on or above the horizontal axis. ¾
has area exactly 1 underneath it.
•^
A density curve describes the overall pattern of a distribution.
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Median and mean of Density Curve
•^
The
median
of a distribution described by a density curve
is the point that divides the area under the curve in half.
-^
A
mode
of a distribution described by a density curve is a
peak point of the curve, the location where the curve ishighest.
-^
Quartiles
of a distribution can be roughly located by
dividing the area under the curve into quarters asaccurately as possible by eye.
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Normal distributions
•^
An important class of density curves are the symmetricunimodal bell-shaped curves known as
normal curves
. They
describe
normal distributions
•^
All normal distributions have the same overall shape.
-^
The exact density curve for a particular normal distribution isspecified by giving its mean
μ
and its standard deviation
σ
•^
The mean is located at the center of the symmetric curve andis the same as the median and the mode.
-^
Changing
μ
without changing
σ
moves the normal curve
along the horizontal axis without changing its spread.
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•^
There are other symmetric bell-shaped density curves thatare not normal e.g.
t^
distribution.
•^
The normal density curves are specified by a particularfunction. The height of a normal density curve at any point x^
is given by
-^
Notation: A normal distribution with mean
μ
and standard
deviation
σ
is denoted by
N
(μ
,^ σ
2
1
1
2
2
x
e
μ σ
σ^
π
⎛^
⎞
⎜^
⎟
⎜^
⎟
⎜^
⎟
⎜^
⎟
⎜^
⎟
⎜^
⎟
⎝^
⎠ −
−
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14
The 68-95-99.7 rule
In the normal distribution with mean
μ
and standard deviation
σ
^
Approx. 68% of the observations fall within
σ
of the mean
μ
^
Approx. 95% of the observations fall within 2
σ
of the mean
μ
^
Approx. 99.7% of the observations fall within 3
σ
of the mean
μ
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Standardizing and
z
-scores
•^
If
x
is an observation from a distribution that has mean
μ
and
standard deviation
σ
, the standardized value of
x^
is given by
-^
A standardized value is often called a
z
-score.
•^
A
z
-score tells us how many standard deviations the original observation falls away from the mean of the distribution.
-^
Standardizing is a linear transformation that transform the datainto the standard scale of
z
-scores. Therefore, standardizing does
not change the shape of a distribution, but changes the value ofthe mean and stdev.
x
z
μ − σ
=
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Example 1.24 on p73 in IPS
•^
The heights of women is approximately normal with mean μ
= 64.5 inches and standard deviation
σ
= 2.5 inches.
•^
The standardized height is
-^
The standardized value (z-score) of height 68 inches isor 1.4 std. dev. above the mean.
-^
A woman 60 inches tall has standardized heightor 1.8 std. dev. below the mean.
6 4.
h e ig h t z^
−
=
6 8
6 4.
z^
−
=^
=
60
z^
− =^
= −
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The standard normal tables
•^
Table A
gives cumulative proportions for the standard
normal distribution. The table entry for each value
z
is the
area under the curve to the left of
z,
the notation used is
P( Z
z
e.g. P( Z
20
Standard Normal Distribution
z^
.
.
.
.
.
.
.
.
.
.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.
.5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7580 .7611 .7642 .7673 .7703 .7734 .7764 .7794 .7823 .7852 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990.
The table showsarea to left of ‘
z ’
under standardnormal curve For a negativenumber, -
z^
Area below (-
z ) =
Area above (
z ) =
1 – Area below (
z )
