taburan normal

1. 10/3/2015
1
Taburan Normal
GB6023 Kaedah Penyelidikan II
zol@ukm.edu.my
Objektif:
 Pengenalan kpd taburan normal
 Ciri-ciri taburan normal standard.
 Penggunaan taburan normal dlm inferensi.
 Taburan t.
 Jangan mengharap semua data anda akan
tertabur secara normal.
Terms that Describe Distributions
Term Features Example
"Symmetric"
left side is mirror
image of right
side
"Positively
skewed"
right tail is longer
then the left
"Negatively
skewed"
left tail is longer
than the right
"Unimodal" one highest point
"Bimodal" two high points
"Normal"
unimodal,
symmetric,
asymptotic
4
Adakah taburan data saya normal?
 Guna kaedah pemerhatian grafik untuk
melihat taburan:
 Histogram
 Stem and leaf plot
 Box plot
 P- P plot
 Q-Q plot
Apa yg perlu diperhatikan?

2. 10/3/2015
2
Jom lihat data
sebenar!!!!
(Sila download file data spss dlm i-folio)
Taburan normal
Mengapa taburan normal sgt penting?
 Banyak pembolehubah bersandar selalunya
diandaikan tertabur secara normal dlm
populasi.
 Jika pembolehubah itu hampir tertabur secara
normal, kita boleh membuat inferensi
terhadap nilai pada pembolehubah itu.
 Cth: Taburan persampelan min.
Maka …apa? So what?
 Taburan normal dan ciri-cirinya kita ketahui, dan
jika pembolehubah yg kita minati itu tertabur
secara normal, kita boleh mengaplikasikan apa yg
kita tahu berkaitan dgn taburan normal itu dalam
setiuasi penyelidikan kita.
 Kita boleh menentukan kebarangkalian sesuatu
hasilan (outcomes)
Taburan normal
 Simetri, bentuk lengkuk loceng
 Juga dikenali sebagai taburan Gaussian
 Poin titik perbubahan = 1 sisihan piwai dari
min
 Formula matematik
f(X) 
1
 2
(e)

(X )2
2 2
 Oleh kerana kita tahu bentuk lengkung, kita boleh
kira luas kawasan dibawah lengkung.
 Peratusan kawasan dibawah lengkuk boleh
digunakan untuk menentukan kebarangkalian
sesuatu nilai yang di ambil dari taburan.
 Dengan kata lain, luas dibawah lengkung memberitahu
kita berkenaan dgn kebarangkalian (nilai-p) untuk
keputusan(data) dengan mengandaikan set data kita
tertabur secara normal.

3. 10/3/2015
3
Kawasan utama dibawah lengkung
 Utk taburan normal
+ 1 SD ~ 68%
+ 2 SD ~ 95%
+ 3 SD ~ 99.9%
Cth min IQ = 100 sisihan piwai (s) = 15
 masalah:
 Setiap taburan normal mempunyai nilai min 
dan sisihan piawai  memerlukan pengiraannya
sendiri untuk menentukan luas dibawah lengkung
pada setiap titik.
Taburan Kebarangkalian Normal
Taburan normal standard– N(0,1)
 Kita setuju guna
taburan normal
standard.
 Bentuk loceng
 =0
 =1
 Nota: tidak semua
taburan bentuk loceng
adalah taburan normal
Normal Probability Distribution
 Can take on an infinite
number of possible
values.
 Curve has area or
probability = 1
Taburan Normal
 Taburan normal standard membolehkan kita
membuat hujah (claims) berkaitan dengan
kebarangkalian sesuatu nilai berkaitan dengan
data kita.
 Bagaimana kita mengaplikasi taburan normal
standard kepada data kita?

4. 10/3/2015
4
Kurtosis
 Jika taburan adalah simetri, soalan seterusnya berkaitan
dengan puncak tengah (central peak):
 Adakah ianya tinggi dan tajam, atau pendek dan lebar? Anda
akan mendapat idea dari melihat bentuk histogram, tetapi
pengukuran bernombor adalah lebih tepat.
 Ketinggian dan ketajaman puncak relatif kepada data-data
yang lain yang diukur, dinamakan kurtosis.
Taburan normal standard mempunyai kurtosis = 3.
 A normal distribution has kurtosis exactly 3 Any distribution with kurtosis
≈3 (excess ≈0) is called mesokurtic.
 A distribution with kurtosis <3 is called platykurtic. Compared to a
normal distribution, its central peak is lower and broader, and its tails are
shorter and thinner.
 A distribution with kurtosis >3 is called leptokurtic. Compared to a
normal distribution, its central peak is higher and sharper, and its tails are
longer and fatter.
Skewness
 The first thing you usually notice about a distribution’s shape is whether it
has one mode (peak) or more than one. If it’s unimodal (has just one
peak), like most data sets, the next thing you notice is whether it’s
symmetric or skewed to one side.
 If the bulk of the data is at the left and the right tail is longer, we say that
the distribution is skewed right or positively skewed; if the peak is toward
the right and the left tail is longer, we say that the distribution is skewed
left or negatively skewed.
Skewness rule of thumb:
 If skewness = 0, the data are perfectly symmetrical.
 But a skewness of exactly zero is quite unlikely for real-world data, so
how can you interpret the skewness number?
 If skewness is less than −1 or greater than +1, the distribution is highly
skewed.
 If skewness is between −1 and −½ or between +½ and +1, the distribution
is moderately skewed.
 If skewness is between −½ and +½, the distribution is approximately
symmetric.
Skor-Z
jika kita tahu min populasi dan sisihan piawai
populasi, untuk sebarang nilai X kita boleh
mengira skor z dengan menggunakan rumus:
z 
X 

Info penting skor-z
 Skor z memberitahu kita berapa jauh samada di atas
atau di bawah min nilai tersebut dalam unit sisihan
piawai.
 Ianya adalah transformasi linear dari skor asal.
 mendarab(atau bahagi) dan tambah/tolak X dengan satu
pemalar (constant)
 Kedudukan ranking skor tidak akan berubah.
Z = (X-)/
oleh itu
X = Z + 

5. 10/3/2015
5
Kebarangkalian dan skor z: Jadual z
 Luas total = 1
 Hanya mempunyai kebarangkalian dari lebar
 Utk satu nombor skor z yang
“infinite”kebarangkalian adalah 0 ( untuk satu
titik poin)
 Dlm jadual tiada nilai negatif
 Simetri, oleh itu kawasan bawah negatif =
kawasan di atas positif.
 Lakaran sangat membantu!!!!
Kebarangkalian yg digambarkan kawasan dibawah
lengkung
 Luas keseluruhan dibawah
lengkung = 1
 Kawasan berwarna merah p(z >
1)
 Kawasan berwarna biru p(-1< z
<0)
 Oleh kerana ciri taburan normal
diketahui, kawasan2 lain boleh
ditentukan dari jadual taburan
normal secara kiraan.
Strategi mencari kebarangkalian utk
pembolehubah rawak dari taburan normal
 Lakarkan gambaran dalam kawasan yang
anda inginkan dalam taburan normal.
 Ekpresikan kawasan berdasarkan kepada
kawasan dalam jadual.
 Lihat kawasan dengan menggunakan jadual
 Jika perlu lakukan operasi + dan -
Suppose Z has standard normal distribution
Find p(0<Z<1.23)
Find p(-1.57<Z<0) Find p(Z>.78)

6. 10/3/2015
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Z is standard normal
Calculate p(-1.2<Z<.78) Example
 Data come from distribution:  = 10,  = 3
 What proportion fall beyond X=13?
 Z = (13-10)/3 = 1
 =normsdist(1) or table  0.1587
 15.9% fall above 13
Example: IQ
 A common example is IQ
 IQ scores are theoretically normally
distributed.
 Mean of 100
 Standard deviation of 15
IQ’s are normally distributed with mean 100 and standard
deviation 15. Find the probability that a randomly selected
person has an IQ between 100 and 115
(100 115)
(100 100 100 115 100)
100 100 100 115 100
(
15 15 15
(0 1) .3413
P X
P X
X
P
P Z
  
     
  
  
  
Say we have GRE scores are normally distributed with mean 500 and
standard deviation 100. Find the probability that a randomly selected
GRE score is greater than 620.
 We want to know what’s the probability of getting a
score 620 or beyond.
 p(z > 1.2)
 Result: The probability of randomly getting a score
of 620 is = 0.115
620 500
1.2
100
z

 
Work time...
 What is the area for scores less than z = -1.5?
 What is the area between z =1 and 1.5?
 What z score cuts off the highest 30% of the
distribution?
 What two z scores enclose the middle 50% of the
distribution?
 If 500 scores are normally distributed with mean =
50 and SD = 10, and an investigator throws out the
20 most extreme scores, what are the highest and
lowest scores that are retained?

7. 10/3/2015
7
Standard Scores
 Z is not the only transformation of scores to be
used
 First convert whatever score you have to a z
score.
 New score – new s.d.(z) + new mean
 Example- T scores = mean of 50 s.d. 10
 Then T = 10(z) + 50.
 Examples of standard scores: IQ, GRE, SAT
Wrap up
 Assuming our data is normally distributed
allows for us to use the properties of the
normal distribution to assess the likelihood of
some outcome
 This gives us a means by which to determine
whether we might think one hypothesis is
more plausible than another (even if we don’t
get a direct likelihood of either hypothesis)
39
The t Distributions and
Degrees of Freedom
 You can think of the t statistic as an "estimated z-
score."
 With a large sample, the estimation is very good and
the t statistic will be very similar to a z-score.
 With small samples, however, the t statistic will
provide a relatively poor estimate of z.
41
The t Distributions and
Degrees of Freedom (cont.)
 The value of degrees of freedom, df = n - 1, is used
to describe how well the t statistic represents a z-
score.
 Also, the value of df will determine how well the
distribution of t approximates a normal distribution.
 For large values of df, the t distribution will be
nearly normal, but with small values for df, the t
distribution will be flatter and more spread out than a
normal distribution.
 Sekian , Terima Kasih…………
If you worried about falling off the bike, you'd
never get on. ~ Lance Armstrong.