1.2 Power Point
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Describing DistributionsDescribing Distributionswith Numberswith Numbers
Section 1.2
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What Does That Mean?What Does That Mean?
We are about to learn specific ways to calculateWe are about to learn specific ways to calculate
center and spread of a distribution. You cancenter and spread of a distribution. You can
calculate these numerical values for anycalculate these numerical values for any
quantitative variable. But to interpret thesequantitative variable. But to interpret these
measures of center and spread, and to choosemeasures of center and spread, and to choose
among the several methods you will learn, youamong the several methods you will learn, you
must think about the shape of the distributionmust think about the shape of the distributionand the meaning of the data. The numbers, likeand the meaning of the data. The numbers, like
the graphs, are aids to understanding, not ³thethe graphs, are aids to understanding, not ³the
answer´ in themselves.answer´ in themselves.
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Measures of Center Measures of Center
MeanMean::
of a sample:of a sample:
MedianMedian::the value that divides the data intothe value that divides the data into
equal halves (*it may or may notequal halves (*it may or may not
be a value in the data set)be a value in the data set)
i x
xn
!§ i
x
n Q !
§
of apopulation:
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The mean isthe balance
point of the
distribution
The median
divides the
distribution into
two equal areas.
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1. Find the mean and median for 1. Find the mean and median for
each list and contrast their each list and contrast their
behavior:behavior:1.1. 1, 2, 61, 2, 6
2.2. 1, 2, 91, 2, 93.3. 1, 2, 2971, 2, 297
Practice:Practice:
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Measures of SpreadMeasures of Spread
RangeRange: maximum: maximum -- minimumminimum
PercentilesPercentiles: the: the p pthth percentile of a distribution is thepercentile of a distribution is thevalue such thatvalue such that p p percent of the observationspercent of the observations
that fall at or below it (the median is the 50that fall at or below it (the median is the 50thth
percentile)percentile)
Quartiles:Quartiles: the lower quartile (Qthe lower quartile (Q11) is the 25) is the 25thth
percentile (or the median of the lower half) andpercentile (or the median of the lower half) andthe upper quartile (Qthe upper quartile (Q33) is the 75) is the 75thth percentile (or percentile (or the median of the upper half)the median of the upper half)
Interquartile Range (IQR)Interquartile Range (IQR) = Q= Q11
--QQ33
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Another Measure of Spread Another Measure of Spread
Standard Deviation (Std Dev)Standard Deviation (Std Dev)::
of a sample:of a sample:
of a population:of a population:
21( )
1 x i
s x xn
!
§
21( )
x i x xW !
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More About Standard DeviationMore About Standard Deviation
The differences of each value fromThe differences of each value fromthe mean are deviations:the mean are deviations:
Since the mean is the balance pointSince the mean is the balance pointof the distribution, the set of allof the distribution, the set of alldeviations from the mean will alwaysdeviations from the mean will alwaysadd to zero:add to zero:
TheThe VarianceVariance is:is:
TheThe Standard DeviationStandard Deviation is:is:
x x
§ ! 0)( x x
21( )
1 x i
s x x
n
!
§
2 21
( )1 x i s x x
n! §
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Practice:Practice:
2. For the sample: 1, 2, 4, 6, 92. For the sample: 1, 2, 4, 6, 9
a. Verify that the sum of the deviations froma. Verify that the sum of the deviations from
the mean is 0.the mean is 0.
b. Find the standard deviation by hand.b. Find the standard deviation by hand.
c. Find the standard deviation on thec. Find the standard deviation on the
graphing calculator.graphing calculator.
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3. Without computing, match each list of numbers3. Without computing, match each list of numbers
on the left, with its SD on the right:on the left, with its SD on the right:
a.a. 1, 1, 1, 11, 1, 1, 1 i. 0i. 0
b.b. 1, 2, 21, 2, 2 ii. 0.058ii. 0.058
c.c. 1, 2, 3, 4, 51, 2, 3, 4, 5 iii. 0.577iii. 0.577
d.d. 10, 20, 2010, 20, 20 iv. 1.581iv. 1.581e.e. 0.1, 0.2, 0.20.1, 0.2, 0.2 v. 3.162v. 3.162
f.f. 0, 2, 4, 6, 80, 2, 4, 6, 8 vi. 3.606vi. 3.606
g.g. 0, 0, 0, 0, 5, 6, 6, 8, 80, 0, 0, 0, 5, 6, 6, 8, 8 vii. 5.774vii. 5.774
Practice:Practice:
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The FiveThe Five--Number SummaryNumber Summary
TheThe FiveFive--Number SummaryNumber Summary includes:includes:
minimum, Qminimum, Q11, median, Q, median, Q33, maximum, maximum
It is used to createIt is used to create BoxplotsBoxplots..
The fiveThe five--number summary is usually better thannumber summary is usually better than
the mean and std dev for describing a skewedthe mean and std dev for describing a skeweddistribution or a distribution with strong outliers.distribution or a distribution with strong outliers.
UseUse x x --bar andbar and ss x x only for reasonably symmetriconly for reasonably symmetric
distributions that are free of outliers.distributions that are free of outliers.
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Danger Will Robinson, danger!!!Danger Will Robinson, danger!!!
While all of the methodWhile all of the method
discussed to compute numericaldiscussed to compute numerical
measures are very useful, theymeasures are very useful, they
should not be applied blindly.should not be applied blindly.
Statistical measures andStatistical measures and
methods based on them aremethods based on them aregenerally meaningful only for generally meaningful only for
distributions of sufficientlydistributions of sufficiently
regular shape.regular shape.
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What Happened to the Whiskers?What Happened to the Whiskers?
SideSide--byby--SideSide BoxplotsBoxplots::
maximum
minimum
medianQ1
Q3
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Calculating OutliersCalculating Outliers
An observation is considered an An observation is considered an Outlier Outlier if it fallsif it fallsoutside the interval:outside the interval:
(Q(Q11
-- 1.51.5 �� IQR, QIQR, Q33
+ 1.5+ 1.5 �� IQR)IQR)
In general, it is not a good idea toIn general, it is not a good idea to
just ignore or delete outliers, but just ignore or delete outliers, but
they do have a strong influence onthey do have a strong influence onthe data so sometimes calculationsthe data so sometimes calculations
are done with and without theare done with and without the
outliers and then compared.outliers and then compared.
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The Influence of OutliersThe Influence of Outliers
ResistantResistant ± ± a summary statistic is resistanta summary statistic is resistant
to outliers if it is not changed very much if to outliers if it is not changed very much if
the outlier is removed from the data set:the outlier is removed from the data set:
median, IQRmedian, IQR
SensitiveSensitive ± ± a summary statistic is sensitivea summary statistic is sensitive
to outliers if it tends to be affected byto outliers if it tends to be affected byoutliersoutliers
mean, range, standard deviationmean, range, standard deviation
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Give Me a Graph, Baby!Give Me a Graph, Baby!
Remember that a graph gives the bestRemember that a graph gives the best
overall picture of a distribution. Numericaloverall picture of a distribution. Numerical
measures of center and spread reportmeasures of center and spread report
specific facts about a distribution, but theyspecific facts about a distribution, but they
do not describe its entire shape.do not describe its entire shape. Always Always
plot your dataplot your data!!
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Changing the Unit of MeasurementChanging the Unit of Measurement
A change in the measurement unit is called A change in the measurement unit is called
aa Linear TransformationLinear Transformation..
A linear transformation changes the A linear transformation changes the
original variableoriginal variable x x into the new variableinto the new variable
x x new new by using the equation:by using the equation:
x x new new == aa ++ bx bx
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So What Does That Mean?So What Does That Mean?
Adding the constant Adding the constant aa shifts allshifts all
of the values of of the values of x x left or right byleft or right by
the same amount (the data isthe same amount (the data is
recenteredrecentered.).)
Multiplying by the positiveMultiplying by the positive
constantconstant bb changes the size of changes the size of
the unit of measurement (thethe unit of measurement (the
data isdata is rescaledrescaled.).)
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What Measures Are Effected?What Measures Are Effected?
Adding Adding aa (recentering) changes the mean,(recentering) changes the mean,median, and quartiles bymedian, and quartiles by aa. However,. However,none of the measures of spread change.none of the measures of spread change.
Multiplying byMultiplying by bb (rescaling) multiplies both(rescaling) multiplies boththe measures of center and spread bythe measures of center and spread by bb..
Linear transformations do not change theLinear transformations do not change theshape of a distribution!shape of a distribution!
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PracticePractice
4. The mean height of a class of 15 children4. The mean height of a class of 15 childrenis 48 inches, the median is 45 inches, theis 48 inches, the median is 45 inches, the
standard deviation is 2.4 inches, and thestandard deviation is 2.4 inches, and the
IQR is 3 inches. Find the mean, median,IQR is 3 inches. Find the mean, median,
standard deviation, and IQR if«standard deviation, and IQR if«
a. you convert each height to feet.a. you convert each height to feet.
b. each child grows 2 inches.b. each child grows 2 inches.c. each child grows 4 inches and youc. each child grows 4 inches and you
convert their heights to feet.convert their heights to feet.