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TULANE / Math / MATH 1140 / What is the good way to calculate skew?

What is the good way to calculate skew?

What is the good way to calculate skew?

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Ch#2#(Handout/#Week#1)


What is the good way to calculate skew?



Wednesday,) August)31,)2016 2:15)PM

2.1$Qualitative$Data$ 

Example:)An)instructor)has)given)an)exam)to)30)students.)The)letter)grades)of)the)exams)are) recorded)below.)

X)=)grade=)A,)B,)A,)B,)C,)C,)C,)C,)D,)F,)D,)F,)A,)A,)B,)D,)A,)C,)C,)C,)B,)B,)B,)B,)D,)C,)C,)C,)C,)F)Give) the)

Frequency)distribution:)simply)count)the)occurrence)of)each)value)

! A:)5

! B:)7

! C:)11

! D:)4)

! F:)3)

Relative)Frequency:frequency)divide)by)the)number)of)values;)these)are)often)better)than) the)frequency)distribution)because,)if)these)are)only)samples,)the)percentages)can)easily)be) applied)to)the)population)for)a)reliable)estimate


Does the data set have a mode?



! A:)5/30=0.1667)

! B:)7/30=0.2333)

! C:)11/30=0.3667)

! D:)4/30=0.1333) We also discuss several other topics like What is the purpose of social work?

! F:)3/30)=)0.1)

Give)bar)graph)for)the)grades

! Credo)diagram:)put)the)most)important)values)first)(would)be)C,)B,)A,)D,)then)F) ! Pareto)diagram:)natural)

Give)a)pie)chart)for)the)grades.)


Is the data set skewed?



! Angle)will)be)Relative)Frequency)x)360)

We also discuss several other topics like What if the data does not support the theory?
If you want to learn more check out What is the meaning of double-stranded dna?

Give)a)Pareto)diagram)for)the)grades.)

2.2.2.5$Quantitative$Data$ 

Data)Set)A:)x)=

3,4,5,5,6,7,7,8,8,8,9,9,9,10,11,11,12,13,13,13,14,15,17,19,20,25,29,34,17,37,18,41) Don't forget about the age old question of How has evolution shaped the living world?

Classes:)intervals)that)must)be)set)up)so)that)every)element)of)the)data)lies)in)one) ! Classes)of)numbers))that)are)"A<))but))≤B") Don't forget about the age old question of What freedoms did the colonists have?

! Class)width)could)be)BYA

!

You)don't)want)to)make)the)classes)too)narrow)(too)many)classes)and)too)much)info))or) too)broad)(all)data)falls)within)one)class)and)that)doesn't)tell)you)anything)) !

LLCL:)Least)lower)class)limit:)left)endpoint)of)the)leftmost)class)aka)starting)point;)must) be)smaller)than)the)smallest)element)in)the)data) If you want to learn more check out What are the different methods of measurement?

! Example:

Relative)Frequency

! Example:

))2<))))≤9

))9<))))≤16

16<))))≤23

23<))))≤30

30<))))≤37

37<))))≤44

13 9

5

2

2

1

Relative)Frequency 

.41

.28

.16

.06

.06

.03

○ Choice)LLCL:)2

○ Choice)class)width:)7)

What)do)we)learn?)Data)is)clumped)up)between)2)and)16)

!

Histogram:)same)as)bar)graph)but)does)it)with)numerical)data,)and)all) the)rectangles) touch)each)other

In)this)example)it)is)right9skewed:)a)few)highly)unusual)elements)in)the)data;)

"globbed)towards)the)left";)because)most)elements)are)below)the)value)of)16

!

Bellshapeddata)("normal)data"):)the)values)are)clumped)towards)the)middle;)we) LOVE)it)in)stats)

! You)can)relativize) the)frequency)tables)(frequency(/(n(where)n is)the)number)of)data)) ○ Take)all)of)the)numbers)and)divide)by)32)

○ They)will)add)up)to)100%)/)1.0

Summarize)data)set)A)with)a)stem)and)leaf)display.)

! Stems)=)10s)

! Leaves)=)1s

! Our)stems:)0,)1,)2,)3,)4

Stem))))Leaves

0

3,)4,)5,)6,)7,)7,)8,)8,)8,)9,)9,)9

1

0,)1,)1,)2,)3,)3,)3,)4,)5,)7,)7,)8),9 2

0,)5,)9

3

4,)7

4

1

Give)a)frequency)distribution)for)the)data)set)(occurrence)of)each)value))

3 4

4,)7 1

Give)a)frequency)distribution)for)the)data)set)(occurrence)of)each)value))

! 3:)1 ! 4:)1 ! 5:)1 ! 6:)1 ! 7:)2 ! 8:)3 ! 9:)3 ! 10:)1

! 11:)2 ! 12:)1 ! 13:)3

! 14:)1 ! 15:)1 ! 17:)2 ! 18:)1 ! 19:)1 ! 20:)1 ! 25:)1 ! 29:)1 ! 34:)1 ! 37:)1 ! 41:)1

Give)a)histogram)for)the)data

Give)a)relative)frequency)distribution)for)the)data)(frequency)divided)by)number)of)values)

! 3:)1/32=0.0313

! 14:)1/32=0.0313 !

Give)a)relative)frequency)distribution)for)the)data)(frequency)divided)by)number)of)values)

! 3:)1/32=0.0313 ! 4:)1/32=0.0313 ! 5:)1/32=0.0313 ! 6:)1/32=0.0313 ! 7:)2/32=0.0625 ! 8:)3/32=0.0938 ! 9:)3/32=0.0938 ! 10:)1/32=0.0313

! 11:)2/32=0.0625 ! 12:)1/32=0.0313 ! 13:)3/32=0.0938

! 14:)1/32=0.0313 ! 15:)1/32=0.0313 ! 17:)2/32=0.0625 ! 18:)1/32=0.0313 ! 19:)1/32=0.0313 ! 20:)1/32=0.0313 ! 25:)1/32=0.0313 ! 29:)1/32=0.0313 ! 34:)1/32=0.0313 ! 37:)1/32=0.0313 ! 41:)1/32=0.0313

Give)a)relative)histogram)for)the)data:

Measures)of)Central)Tendency:

!

Mode:)most)frequently)occurring)element)of)the)data)set,)if)there)is)such;)there)can) onl)be)one)mode

Measures)of)Central)Tendency:

!

Mode:)most)frequently)occurring)element)of)the)data)set,)if)there)is)such;)there)can) only)be)one)mode

○ Corresponds)to)the)highest)point)in)the)histogram)

! Average/Mean:)find)the)sum)of)all)numbers)/)amount)of)numbers ! Median:)known)as)the)50th)percentile)or)as)the)2nd)quartile)(Q2)

Where)at)least)50)percent)of)the)elements)are)≤ the)value)and)50)percent)are)≥

the)value

○ Odd)number)of)elements=)a)single)median)value

○ Even)number)of)elements=)must)find)average)between)the)two)middle)numbers) ! Good)way)to)calculate)skew:)

○ If)the)mean)value)is)larger) than)median=)it)is)right)skewed

○ If)the)mean)value)is)less)than)the)median=)it)is)left)skewed

○ If)the)mean)value)is)equal)to)the)median=)it)is)symmetrical)

Does)Data)Set)A)have)a)mode?)

! Our)data)has)no)mode)because)there)is)not)a)single)value)which)occurs)the)most

Find)the)mean)value)of)Data)Set)A.)

! 14.28

Find)the)median)of)data)Set)A.)

! If)there)are)32)elements,)you)want)16)to)be)≤ value)and)16)to)be)≥ value ???? In)this)example,)any)number)between)11)and)12)will)satisfy)the)median) ! Q2)will)be)the)average)of)11)and)12)

! Q2)is)11.5

Does)the)median)or)the)mean)better)describe)the)“)middle”)of)the)data?) ! The)mean)value)is)deceptive)because)it)is)rightYskewed)so)strongly

! The)median)offers)a)better)description)of)the)"middle"

Is)Data)Set)A)skewed?)

! The)data)is)rightYskewed)

! The)mean)>)median)

Measures)of)Consistency

Parameters:)μ,$σ 2 !

Statistics:)x,)s̅ 2 !

! You)can)measure)by)Range=)the)largest)element)in)the)data)– smallest)element) ???? Ex:)41–3=)a)range)of)38

Or)by)Population)Variance=)σ 2 ! ,)average)square)deviation)from)the)mean)value ???? Population)average)parameter=μ 

???? All)elements)in)population)added/number)of)elements) ???? Estimated)by x,)which)is)the)mean) ̅ 

σ 2=)(Σxi–μ)2 ???? /)N

???? Where)the)sum)is)taken)over)all)values)

???? N=)number)of)elements)in)the)population)

????

Population)standard)deviation:)a)measure)that)is)used)to)quantify)the)amount)of) variation)or)dispersion)of)a)set)of)data)values.

???? Ex:)population)N=5

???? 1,)2,)6,)10,)11

???? Σxi /)N=)μ

???? =(1)+)2)+)6)+)10)+)11))/)5 

???? =6)=μ 

))))))X ))))))xYμ (xY μ)2 

1 Y5 ))))))))25

Y4

0

4

2 ))))))))16

6 0

10 16

11 5 25

=Sum)of)Deviations=)82

???? Average)square)deviations)=)82)/)5)=)16.4) 

=Sum)of)Deviations=)82

???? Average)square)deviations)=)82)/)5)=)16.4) 

???? Ex)2:)Sample)of)elements)of)1,)6,)7,)8,)9,)10,)15 

???? n=)7 

An)estimate)of)σ 2) ???? is)

□ x̅)=)1)+)6)+)7)+)8)+)9)+)10)+)15)/)7)=)8

???? Sigma)(xYxbar)squared)=)49/108

???? BOARD

Find)the)sample)variance of)the)data:)measures)the)consistency)of)the)variability)of)data;)the) sum)of)the)squared)deviations)from)the)mean)divided)by)(nY1).)The)symbol)S2 is)used)to) represent)this)

1. Calculate)average)of)data

2. Calculate)variation)of)each)element)of)data)away)from)the)mean)value) 3. N=)number)of)elements)in)a)population:)32)

S2) 4. =)92.46673

5. S=)9.61516

Find)the)sample)standard)deviationof)the)data:)s,)is)defined)by)the)positive)square)root)of) the)sample)variance,)s2) 

Ch 2 (Handout/ Week 2)  

Monday, September  5, 2016 11:55 AM

2.8  

Percentiles(p): a pth percentile is a value so that at least p% of the values in the data  are ≤ the value and at least 100–p% are ≥ the value.  

- 0 < p ≤ 100

- The percentile always exists, but need not be unique

- Ex: if you choose the 60th percentile  of a value set of 1, 3, 4, 6, 10, 12… ○ At least 60% have to be ≤ the value

○ At least 40% have to be ≥ the value (it ends up being 50%)  

○ 6 is the 60th percentile

- Ex2: if you choose the 50th percentile of a value set of 1, 2, 6, 8, 9, 12… ○ Any number between 6 and 8 works  

- In excel: percentile.exc

○ Arrange the data in ascending order

○ Calculate (n+1)p

▪ This formula gives you the POSITION of the value of the percentile

P= written decimal, rounded up and down to give you approximate  position of the percentile in the ascending array  

○ Interpolate between the values by the fractional part  

The decimal point will give you the percentage between the two  elements  

○ Ex: 80th percentile of 1, 3, 4, 6, 10, 12

Estimate= (n+1)p = (7)(.8)=  5.6 with the .6 going up and down  because there is no element at 5.6!

▪ 12-10=2

▪ 2(.6)=1.2

▪ 10 + 1.2 = 11.2  

- Q1 is the 1st quartile, the 25th percentile  

- Q2 is the 2nd quartile, median, and 50th percentile

- Q3 is the 3rd quartile, the 75th percentile  

Handout cntd…  

Data Set A: x =

3,4,5,5,6,7,7,8,8,8,9,9,9,10,11,11,12,13,13,13,14,15,17,19,20,25,29,34,17,37,18,41  

- Q3 is te 3r quartie, te 75t percentie  

Handout cntd…  

Data Set A: x =

3,4,5,5,6,7,7,8,8,8,9,9,9,10,11,11,12,13,13,13,14,15,17,19,20,25,29,34,17,37,18,41  

Find the 25th percentile of the data. We will use Excel percentiles  - Q1 = 8

- Work:

○ (n+1)(.25) = (33)(.25)=8.25  

○ It's 25% between elements 8 and 9

○ Which is between 8 and 8  

Find the 75th percentile of the data. Find the interquartile range of the data.  - Q3 = 17.75

- Work:  

○ (n+1)(.75) = (33)(.75) = 24.75

○ Look 75% of the way between elements 24 and 25  

○ Which is between 17 and 18

Z score: to measure distances in units of standard deviations

- Population z score: z = (x – μ) / σ

- Sample z score: z = (x - x̅) / s  

-

Ex: population with a μ of 10 and a σ of 3, what is the distance from 16 to the  mean value?

○ z = (x – μ) / σ

○ Z = (16 - 10) / 3

○ Z= 2 standard deviations

Give the five number summary of the data.  

How many standard deviations is the value 29 from the mean value.  - x̅: 14.28

- So z=(29 – 14.28)/ 9.615962

- 1.54 standard deviations

Find the interquartile range  

Interquartile range (IQR)= Q3 -Q1 = 17.75 – 8 = 9.75 9.75 Give the value of the upper fence  

Upper fence: upper quartile (Q3) plus 1.5 interquartile ranges

Interquartile range (IQR)= Q3 -Q1 = 17.75 – 8 = 9.75 9.75

Give the value of the upper fence  

Upper fence: upper quartile (Q3) plus 1.5 interquartile ranges  

- Q3 + 1.5 IQR  

- 17.75 + (1.5)(9.75)

- =32.375

Give the value of the lower fence  

- Q1 – 1.5 IQR

- 8 – 1.5(9.75)

- = -6.625

Give the value which corresponds to the upper whisker  

Upper whisker: largest element in the data that is less than or equal to the upper  fence  

- In the data set, it's 29

Give the value which corresponds to the lower whisker  

Lower whisker: smallest element in the data that is greater than or equal to the lower  fence

- In the data set, it's 3  

Are there any outliers in the data?  

Outliers: elements that are outside the fences  

- In the data set, 34, 37, 41

- If (x - x̅) / s > 3, then x is an outlier

Bell-shaped(mound-shaped) data: no skew; perfectly symmetrical

Empirical Rule: suppose data is bell shaped…

- 67-68% of the values will lie within 1 standard deviation of the mean value

Empirical Rule: suppose data is bell shaped…

- 67-68% of the values will lie within 1 standard deviation of the mean value - 95% of the values will fall within 2 standard deviations of the mean value  - 99.7% of the values will fall within 3 standard deviations of the mean value - Example: Suppose population has μ= 20 and σ= 2

○ What fraction lies within 16 and 24?

○ Should describe interval using standard deviations

○ Z= (24 - 20) / 2  

▪ Z= 2  

○ Z = (16 - 20) / 2  

▪ Z= -2  

Thus, [16,24] is the interval of the value which lies within 2 sd of the  

mean; so about 95%

Cheyhyshev's Theorem

-

Suppose a population has a mean value μ and standard deviation σ, then at  least (1 - 1/k2) x 100% of the values in the population lie within k standard  deviations of the mean value

-

Ex: μ=15 and σ=2… What is the minimal fraction of the data which lie within the  range 11 to 19  NOT assuming that the population is bell-shaped  ○ (19-15)/2 = 2  

○ (11-15)/2=-2  

○ So K is 2  

At least (1 - 1/22 ○ ) x 100% = 75% of the values lie within11 to 19  

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