Handout KOM 204 Kuliah 2
KULIAH II
STATISTIK
TABEL DATA YANG BERTUJUAN UNTUK SIMPLIFIKASI
DATA.
ANALISIS STATISTIK BERURUSAN DENGAN DATA
MENTAH.
BAGAIMANA MELAKUKAN STRUKTURISASI SEHINGGA
DATA TERSEBUT DAPAT DIKOMUNIKASI SECARA
EFEKTIF.
TABEL DAN DIAGRAM HARUS JELAS DAN AKURAT DAN
BERKOMUNIKASI SECARA CEPAT.
STATISTIK DESKRIPTIF
D e s c r i p t i v e s t a t i s t i c s a re , b y a n d l a rg e , re l a t i v e l y s i m p l e v i s u a l a n d
numerical techniques
f o r d e s c r i b i n g t h e m a j o r f e a t u re s o f o n e ’ s d a t a . Re s e a rc h e r s m a y p ro d u c e
descriptive
s t a t i s t i c s i n o rd e r t o c o m m u n i c a t e t h e m a j o r c h a r a c t e r i s t i c s o f t h e d a t a t o
others,
b u t i n t h e fi r s t i n s t a n c e t h e y a re u s e d b y re s e a rc h e r s t h e m s e l v e s i n o r d e r
to understand
t h e d i s t r i b u t i o n o f p a r t i c i p a n t s ’ re s p o n s e s i n t h e re s e a rc h . N e v e r re g a rd
descriptive
s t a t i s t i c a l a n a l y s i s a s a n u n n e c e s s a r y o r t r i v i a l s t a g e i n re s e a rc h . I t i s
p ro b a b l y m o re
i n f o rm a t i v e t h a n a n y o t h e r a s p e c t o f d a t a a n a l y s i s . B ox 2 . 2 ex p l a i n s t h e
c r u c i a l ro l e o f
d e s c r i p t i v e s t a t i s t i c s i n re s e a rc h f u r t h e r.
T h e d i s t i n c t i o n b e t w e e n n o m i n a l ( c a t e g o r y ) d a t a a n d n u m e r i c a l s c o re s
discussed in the
p re v i o u s c h a p t e r i s i m p o r t a n t i n t e rm s o f t h e a p p ro p r i a t e t a b l e s a n d
diagrams to use.
2.2 Choosing tables and diagrams
So long as you are able to decide whether your data
are either numerical scores or
nominal (category) data, there are few other choices
to be made since the available tables
and diagrams are essentially dependent upon this
distinction. Figure 2.1 gives some of
the key steps when considering tables and diagrams.
■ Tables and diagrams for nominal (category) data
One of the main characteristics of tables and
diagrams for nominal (category) data is
that they have to show the frequencies of cases in
each category used. While there may
be as many categories as you wish, it is not the
function of statistical analysis to communicate
all of the data’s detail; the task is to identify the
major trends. For example,
imagine you are researching the public’s attitudes
towards private health care. If you ask
participants in your research their occupations then
you might fi nd that they mention
t e n s i f n o t h u n d re d s o f d i ff e r e n t j o b t i t l e s – n e w s a g e n t s , h o m e m a ke r s , c o m p a n y
e xe c u t i v e s
a n d s o f o r t h . S i m p l y c o u n t i n g t h e f re q u e n c i e s w i t h w h i c h d i ff e re n t j o b t i t l e s a re
m e n t i o n e d re s u l t s i n a v a s t n u m b e r o f c a t e g o r i e s . Yo u n e e d t o t h i n k o f re l e v a n t a n d
m e a n i n g f u l w a y s o f re d u c i n g t h i s v a s t n u m b e r i n t o a s m a l l e r n u m b e r o f m u c h b ro a d e r
c a t e g o r i e s t h a t m i g h t r e v e a l i m p o r t a n t t r e n d s . Fo r e x a m p l e , s i n c e t h e re s e a rc h i s
about
a health issue you mi ght wi sh to form a categ ory made up of those invol ved in health
work – some mi ght be dentis ts , some nur ses, s ome doctors, some paramedi cs and so
f o r t h . I n s t e a d o f ke e p i n g t h e s e a s d i ff e r e n t c a t e g o r i e s , t h e y m i g h t b e c o m b i n e d i n t o a
c a t e g o r y ‘ h e a l t h w o r ke r ’ . T h e re a r e n o h a r d - a n d - f a s t r u l e s a b o u t c o m b i n i n g t o f o r m
broader categ ori es. The following mi ght be useful rule s of thumb:
1 . Ke e p y o u r n u m b e r o f c a t e g o r i e s l o w , e s p e c i a l l y w h e n y o u h a v e o n l y s m a l l n u m b e r s
o f p a r t i c i p a n t s i n y o u r r e s e a rc h .
2 . Tr y t o m a ke y o u r ‘ c o m b i n e d ’ c a t e g o r i e s m e a n i n g f u l a n d s e n s i b l e i n t h e l i g h t o f t h e
p u r p o s e s o f y o u r r e s e a rc h . I t w o u l d b e n o n s e n s e , f o r e x a m p l e , t o c a t e g o r i s e j o b s b y
the l etter of the alp habet with whi ch they s tart – nurs es, nuns, nurser y teachers and
national football ers. All of these have jobs beg inning with the same l etter, but it i s
v e r y d i ffi c u l t t o s e e a n y o t h e r c o m m o n t h re a d w h i c h a l l o w s t h e m t o b e c o m b i n e d
m e a n i n g f u l l y.
Slices for a pie diagram
There is n othin g diffi cult in constructing a pie diagram. Our
recommendation is that you turn each of your freq uencies
in to a percentage frequency. Since there are 360 d egrees in a
circle, if you mu ltiply each percentage frequency by 3.6
you will obtain the angle (in degrees) of the slice of the pie
w hich you need to mark out. In order to create the diagram,
you will requ ire a protractor to measu re the angles. H ow ever,
compu ter graph packages are standard at any u niversity
or college and do an impressive job.
In Table 2.1, 25.00% of cases were stu dents. In ord er to turn
this into the correct angle for the slice of th e pie, you
simp ly n eed to multiply 25.00 by 3.6 to give an an gle of 90
degrees.
FIGURE 2.2 A s imple pie diagram
One advan tage of u sing compute rs is that the y e nable
expe rime n tation w ith diff e re nt
sche mes of cate gorising data in orde r to de cide w hic h is be st for
you r p urpose s. In this
cas e, you would use in itially narrow c ate gor ie s for coding your
data. The n you can te ll
th e compu te r which of the s e to combine into broad er cate g orie s.
Some times it is pre fe rable to tu rn fre qu en cy tab le s into
diagr ams. Good d iagrams
are q uic kly unde rs tood and add var ie ty to the p re se ntation. The
main type s of diagr am
for nominal (c ate gory) data are pie diagrams and bar charts . A
pie diagram is a very
familiar form of pre se n tation – it simply expres se s e ach cate gory
as a s lice of a pie which
re pre se nts all case s (s e e Figu re 2.2).
Notice that the number of slices is small – a multitude of slices can be
confusing. Each
s l i c e i s c l e a r l y m a r ke d w i t h i t s c a t e g o r y n a m e , a n d t h e p e rc e n t a g e
f re q u e n c y i n e a c h
category also appears.
C H A P T E R 2 D E S C R I B I N G VA R I A B L E S 1 7
l e g e n d o r ke y t o i d e n t i fy t h e c o m p o n e n t s t o h e l p c o p e w i t h t h e o v e rc ro w d i n g
p ro b l e m . I n o t h e r w o rd s , t o o m a n y c a t e g o r i e s h a v e re s u l t e d i n a d i a g r a m
which is
f a r f ro m e a s y t o re a d – a c a rd i n a l s i n i n a n y s t a t i s t i c a l d i a g r a m .
A s i m p l e f re q u e n c y t a b l e m i g h t b e m o re e ff e c t i v e i n t h i s c a s e .
A n o t h e r v e r y f a m i l i a r f o rm o f s t a t i s t i c a l d i a g r a m f o r n o m i n a l ( c a t e g o r y ) d a t a
is
t h e b a r c h a r t . A g a i n t h e s e c h a r t s a re v e r y c o m m o n i n t h e m e d i a . B a s i c a l l y
t h e y a re
d i a g r a m s i n w h i c h b a r s re p re s e n t t h e s i z e o f e a c h c a t e g o r y. A n ex a m p l e i s
shown in
Fi g u re 2 . 4 .
T h e re l a tiv e len g t h s ( o r h e i g h t s ) o f t h e b a r s q u i ck l y re v e a l t h e m a i n
t re n d s in t h e d a t a .
Wit h a b a r c h a r t , t h ere i s v e r y l i t t l e t o re m e m b e r o t h e r t h a n t h a t t h e
b a r s h a ve a s t a n d a rd
s p a c e s e p a r a t in g t h em . T h e s p a c e s in d i c a t e t h a t t h e c a t e g o r i e s a re
n o t i n a n u m er ic a l
o rd e r ; t h ey a re f req u e n ci e s o f ca t e g o r i e s , no t s c o re s .
I t is h a rd t o g o w ro n g w i t h a b a r ch a r t ( t h a t i s n o t a ch a l l e n g e ! ) s o
lo n g a s y o u
re m e m b er t h e f o llo w in g :
1 . T h e h eig h ts o f t h e b a r s re p re s e n t f re q u e n ci e s ( n u m b e r o f c a s e s ) i n
a ca t e g o r y.
2 . E a ch b a r s h o u ld b e cl e a r ly l a b e l l e d a s t o t h e c a t e g o r y i t re p re s e n t s .
3 . To o m a n y b a r s m a ke b a r ch a r t s h a rd t o f o l lo w.
F I G U R E 2 . 3 A p o o r p ie d i a g r a m
F I G U R E 2 . 4 B a r ch a r t s h o w i n g o c cu p a t i o n a l ca t e g o r i e s i n Ta b l e 2 . 1
1 8 PA RT 1 D E S C R I PT I V E S TAT I S T I C S
4 . Av o i d h a v i n g m a n y e m p t y o r n e a r- e m p t y c a t e g o r i e s w h i c h r e p r e s e n t v e r y f e w c a s e s .
Gener al ly, the inform ati on about s ubs tantial c a tegor ie s is the m os t im port ant . (Sm al l
c a t e g o r i e s c a n b e c o m b i n e d t o g e t h e r a s a n ‘ o t h e r ’ c a t e g o r y. )
5 . N e ve rt he l e s s, i f i m po rta nt c ate gor i e s h ave ve r y fe w e ntr i e s th e n th i s n e e d s re c ord i ng .
So , f or exam p l e , a re se arc h e r wh o i s p ar ti c u l ar l y i n te re st e d i n op p or tun i t i e s for
w o m e n s u r v e y s p e o p l e i n t o p m a n a g e m e n t a n d fi n d s v e r y f e w w o m e n e m p l o y e d i n
suc h jobs. It i s im por tant to dra w atte nt ion to thi s in the ba r c har t of m ales and
fe m al e s i n top m a na ge m e n t. On c e ag ai n , th e re are no h ard - an d- fas t r ul e s to gu i d e
y o u – c o m m o n s e n s e w i l l t a ke y o u a l o n g w a y.
6 . M a ke s u re t h a t t h e v e r t i c a l a x i s ( t h e h e i g h t s o f t h e b a r s ) i s c l e a r l y m a r ke d a s b e i n g
fre qu e n c i e s or pe rc e nta ge fre qu e n c i e s.
7 . The bar s s ho ul d be of e qual wi dth.
I n n e w s p a p e r s a n d o n t e l e v i s i o n y o u a r e l i ke l y t o c o m e a c r o s s a v a r i a n t o f t h e b a r
c ha rt c al l e d t he p i c t ogr am . In thi s, th e b ars of the bar c h art a re re p l a c e d by var yi ng si ze d
dr awi n g s of s o m e th i ng e ye -c atc h i n g to d o wi t h you r c a te g or i e s . T hu s , p i c t ure s of m e n
o r w o m e n o f v a r y i n g h e i g h t s , f o r e x a m p l e , r e p l a c e t h e b a r s . Pi c t o g r a m s a r e r a r e l y u s e d
i n p rof e s si o na l pre se nt ati o ns . T he m ai n re as on i s th at p i c tu re s of thi ng s g e t wi d e r as we l l
as ta l l e r as th e y i n c re as e i n s i ze . T hi s c a n m i s re p re s e n t th e re l ati ve si z e s of t he
categories,
gi ve n th at re ad e r s f org e t th at i t i s on l y th e h e i g h t of th e p i c tu re th at c oun ts .
■ Ta b l e s a n d d i a g r a m s f o r n u m e r i c a l s c o r e d a t a
O ne cr ucia l co ns idera tio n w hen deciding w ha t ta bles a nd diagrams to use f or sc ore
data
i s t h e n u m b e r o f s e p a r a t e s c o re s r e c o r d e d f o r t h e v a r i a b l e i n q u e s t i o n . T h i s c a n
vary
m a r ke d l y. S o , f o r e x a m p l e , a g e i n t h e g e n e r a l p o p u l a t i o n c a n r a n g e f r o m n e w l y b o r n
t o o v e r 1 0 0 y e a r s o f a g e . I f w e m e r e l y re c o r d e d a g e s t o t h e n e a r e s t w h o l e y e a r
then a
t a b l e o r d i a g r a m m a y h a v e e n t r i e s f o r 1 0 0 d i ff e r e n t a g e s . S u c h a t a b l e o r d i a g r a m
would
loo k horrendo us . If w e reco rded a ge to the nea res t month, then w e could multiply
this
number of ages by 12! Such s cores can be grouped into bands or r anges o f s cores
to
a l l o w e ff e c t i v e t a b u l a t i o n ( Ta b l e 2 . 2 ) .
Many psychological variables have a much smaller range of numerical values. So,
f or exa mple, it is f airly co mmo n to use ques tio ns which pre-spec ify just a few
respons e
a l t e r n a t i v e s . T h e s o - c a l l e d L i ke r t - t y p e q u e s t i o n n a i r e i t e m i s a g o o d c a s e i n p o i n t .
Ty p i c a l l y t h i s l o o k s s o m e t h i n g l i ke t h i s :
Statistics is my favourite university subject:
A h i s t o g r a m m i g h t b e t h e b e s t f o r m o f s t a t i s t i c a l d i a g r a m t o re p r e s e n t t h e s e d a t a . A t
fi r s t s i g h t , h i s t o g r a m s l o o k v e r y m u c h l i ke b a r c h a r t s b u t w i t h o u t g a p s b e t w e e n t h e
bars.
Th is i s becau se the hi sto gram do es not represe nt dis tin ct un rela ted cate go ries but
d i ff e r e n t
p o i n t s o n a n u m e r i c a l m e a s u re m e n t s c a l e . S o a h i s t o g r a m o f t h e a b o v e d a t a m i g h t
l o o k l i ke Fi g u r e 2 . 5 .
B u t w h a t i f y o u r d a t a h a v e n u m e r o u s d i ff e re n t p o s s i b l e v a l u e s o f t h e v a r i a b l e i n
question?
O n e c o m m o n d i ffi c u l t y f o r m o s t p s y c h o l o g i c a l r e s e a rc h i s t h a t t h e n u m b e r o f
r e s p o n d e n t s t e n d s t o b e s m a l l . T h e l a r g e n u m b e r o f p o s s i b l e d i ff e r e n t s c o re s o n t h e
v a r i a b l e i s t h e r e f o r e s h a r e d b e t w e e n v e r y f e w r e s p o n d e n t s . Ta b l e s a n d d i a g r a m s
should
p r e s e n t m a j o r f e a t u re s o f y o u r d a t a i n a s i m p l e a n d e a s i l y a s s i m i l a t e d f o r m . S o ,
sometimes
y ou wi l l h av e to us e ban ds of s cores rath er tha n in div i du al score v a lu es , j us t as
y o u d i d f o r Ta b l e 2 . 2 . S o , i f w e a s ke d 1 0 0 p e o p l e t h e i r a g e s w e c o u l d c a t e g o r i s e t h e i r
repli es in to ban ds su ch as 0–9 y ear s, 10–19 y ea rs, 30–39 y ea rs, 40–49 ye ars an d a
fi n a l
Ta b l e 2 . 3 D i s t r i b u t i o n o f s t u d e n t s ’ a t t i t u d e s t o w a r d s s t a t i s t i c s
Re s p o n s e c a t e g o r y Va l u e Fre q u e n c y
S t ro n g l y a g r e e 1 1 7
A g re e 2 1 4
N e i t h e r a g r e e n o r d i s a g re e 3 6
Di sa gree 4 2
S t ro n g l y d i s a g r e e 5 1
FIGURE 2.5 Histogram of s tudents’ a ttitudes to wards s tatis tics
2 0 PA RT 1 D E S C R I P T I V E S TAT I S T I C S
c a t e g o r y o f t h o s e 5 0 y e a r s a n d o v e r. B y u s i n g b a n d s w e re d u c e t h e r i s k o f e m p t y
parts
o f t h e t a b l e a n d a l l o w a n y t re n d s t o b e c o m e c l e a r ( Fi g u r e 2 . 6 ) .
How one cho oses the bands to use is an impo rtant question. The answ er is a bit
o f l u c k a n d j u d g e m e n t , a n d a l o t o f t r i a l a n d e r r o r. I t i s v e r y t i m e - c o n s u m i n g t o re j i g
the ra nges of the bands when one is analysing the data by hand. One big a dvantage
of
c o m p u t e r s i s t h a t t h e y w i l l re c o d e y o u r s c o r e s i n t o b a n d s re p e a t e d l y u n t i l y o u h a v e
tables w hich seem to do the job as well as pos sible. The criter io n is still whether the
table
c o m m u n i c a t e s i n f o r m a t i o n e ff e c t i v e l y.
The one rule is that the bands ought to be of the same size – that is cover, for
ex ample, equal ranges of scores. Generally this is easy except at the upper and
lower
e n d s o f t h e d i s t r i b u t i o n . Pe r h a p s y o u w i s h t o u s e ‘ o v e r 7 0 ’ a s y o u r u p p e r r a n g e .
This,
in modern practice, can be done as a bar of the same width as the others, but
must be
v e r y c a re f u l l y m a r ke d . ( S t r i c t l y s p e a k i n g , t h e w i d t h o f t h e b a n d s h o u l d re p re s e n t
the
r a n g e o f s c o re s i n v o l v e d a n d t h e h e i g h t re d u c e d i n t h e l i g h t o f t h i s . H o w e v e r ,
this is
r a re l y d o n e i n m o d e r n p s y c h o l o g i c a l s t a t i s t i c s . )
O n e m i g h t re d e fi n e t h e b a n d s o f s c o re s a n d g e n e r a t e a n o t h e r h i s t o g r a m b a s e d o n
i d e n t i c a l d a t a b u t a d i ff e re n t s e t o f b a n d s ( Fi g u r e 2 . 7 ) .
F I G U R E 2 . 6 U s e o f b a n d s o f s c o re s t o e n a b l e s i m p l e p re s e n t a t i o n
FIGURE 2.7 Histogram showing ‘collapsed’ categories
C H A P T E R 2 D E S C R I B I N G VA R I A B L E S 2 1
I t re q u i re s s o m e t h o u g h t t o d e c i d e w h i c h o f t h e d i a g r a m s i s b e s t f o r a p a r t i c u l a r
p u r p o s e . T h e r e a re n o h a r d - a n d - f a s t r u l e s .
2 . 3 E rro r s t o a v o i d
T h e re a re a c o u p l e o f m i s t a ke s t h a t y o u c a n m a ke i n d r a w i n g u p t a b l e s a n d d i a g r a m s :
1 . D o n o t f o rg e t t o h e a d t h e t a b l e o r d i a g r a m w i t h a s u c c i n c t d e s c r i p t i o n o f w h a t i t
c o n c e r n s . Yo u w i l l n o t i c e t h a t w e h a v e d o n e o u r b e s t t h r o u g h o u t t h i s c h a p t e r t o
s u p p l y e a c h t a b l e a n d d i a g r a m w i t h a c l e a r t i t l e .
2 . L a b e l e v e r y t h i n g o n t h e t a b l e o r d i a g r a m a s c l e a r l y a s p o s s i b l e . W h a t t h i s m e a n s i s
t h a t y o u h a v e t o m a r k y o u r b a r c h a r t s a n d h i s t o g r a m s i n a w a y t h a t t e l l s t h e re a d e r
w h a t e a c h b a r m e a n s . T h e n y o u m u s t i n d i c a t e w h a t t h e h e i g h t o f t h e b a r re f e r s t o –
p r o b a b l y e i t h e r f r e q u e n c y o r p e r c e n t a g e f r e q u e n c y.
N o t e t h a t t h i s c h a p t e r h a s c o n c e n t r a t e d o n d e s c r i b i n g a s i n g l e v a r i a b l e a s c l e a r l y a s p o s s i b l e .
I n C h a p t e r 6 , m e t h o d s o f m a k i n g t a b l e s a n d d i a g r a m s s h o w i n g t h e re l a t i o n s h i p s
b e t w e e n t w o o r m o re v a r i a b l e s a re d e s c r i b e d .
Tr y t o m a k e y o u r t a b l e s a n d d i a g r a m s u s e f u l . I t i s n o t u s u a l l y t h e i r p u r p o s e t o r e c o r d t h e d a t a a s
y o u c o l l e c t e d i t i n y o u r re s e a rc h . O f c o u r s e y o u c a n l i s t y o u r d a t a i n t h e a p p e n d i x o f p ro j e c t s t h a t
y o u c a rr y o u t , b u t t h i s i s n o t u s e f u l a s a w a y o f i l l u s t r a t i n g t re n d s . I t i s p a r t o f a re s e a rc h e r ’ s j o b
t o m a ke t h e d a t a a c c e s s i b l e t o t h e re a d e r i n a s t r u c t u re d f o rm t h a t i s e a s i l y u n d e r s t o o d b y t h e
r e a d e r.
E s p e c i a l l y w h e n u s i n g c o m p u t e r s , i t i s v e r y e a s y t o g e n e r a t e u s e l e s s t a b l e s a n d d i a g r a m s . T h i s i s
u s u a l l y b e c a u s e c o m p u t e r a n a l y s i s e n c o u r a g e s y o u n o t t o ex a m i n e y o u r r a w d a t a i n a n y d e t a i l . T h i s
i m p l i e s t h a t y o u s h o u l d a l w a y s r e g a r d y o u r fi r s t a n a l y s e s a s t e n t a t i v e a n d m e r e l y a s t e p t o w a r d s
s o m e t h i n g b e t t e r.
I f a t a b l e i s n o t c l e a r t o y o u , i t i s u n l i ke l y t o b e a n y c l e a re r t o a n y o n e e l s e .
C h e c k e a c h t a b l e a n d d i a g r a m f o r c l e a r a n d f u l l l a b e l l i n g o f e a c h p a r t . E s p e c i a l l y , c h e c k t h a t
frequencies
a re c l e a r l y m a r ke d a s s u c h .
C h e c k t h a t t h e re i s a c l e a r , h e l p f u l t i t l e t o e a c h t a b l e a n d d i a g r a m .
STATISTIK
TABEL DATA YANG BERTUJUAN UNTUK SIMPLIFIKASI
DATA.
ANALISIS STATISTIK BERURUSAN DENGAN DATA
MENTAH.
BAGAIMANA MELAKUKAN STRUKTURISASI SEHINGGA
DATA TERSEBUT DAPAT DIKOMUNIKASI SECARA
EFEKTIF.
TABEL DAN DIAGRAM HARUS JELAS DAN AKURAT DAN
BERKOMUNIKASI SECARA CEPAT.
STATISTIK DESKRIPTIF
D e s c r i p t i v e s t a t i s t i c s a re , b y a n d l a rg e , re l a t i v e l y s i m p l e v i s u a l a n d
numerical techniques
f o r d e s c r i b i n g t h e m a j o r f e a t u re s o f o n e ’ s d a t a . Re s e a rc h e r s m a y p ro d u c e
descriptive
s t a t i s t i c s i n o rd e r t o c o m m u n i c a t e t h e m a j o r c h a r a c t e r i s t i c s o f t h e d a t a t o
others,
b u t i n t h e fi r s t i n s t a n c e t h e y a re u s e d b y re s e a rc h e r s t h e m s e l v e s i n o r d e r
to understand
t h e d i s t r i b u t i o n o f p a r t i c i p a n t s ’ re s p o n s e s i n t h e re s e a rc h . N e v e r re g a rd
descriptive
s t a t i s t i c a l a n a l y s i s a s a n u n n e c e s s a r y o r t r i v i a l s t a g e i n re s e a rc h . I t i s
p ro b a b l y m o re
i n f o rm a t i v e t h a n a n y o t h e r a s p e c t o f d a t a a n a l y s i s . B ox 2 . 2 ex p l a i n s t h e
c r u c i a l ro l e o f
d e s c r i p t i v e s t a t i s t i c s i n re s e a rc h f u r t h e r.
T h e d i s t i n c t i o n b e t w e e n n o m i n a l ( c a t e g o r y ) d a t a a n d n u m e r i c a l s c o re s
discussed in the
p re v i o u s c h a p t e r i s i m p o r t a n t i n t e rm s o f t h e a p p ro p r i a t e t a b l e s a n d
diagrams to use.
2.2 Choosing tables and diagrams
So long as you are able to decide whether your data
are either numerical scores or
nominal (category) data, there are few other choices
to be made since the available tables
and diagrams are essentially dependent upon this
distinction. Figure 2.1 gives some of
the key steps when considering tables and diagrams.
■ Tables and diagrams for nominal (category) data
One of the main characteristics of tables and
diagrams for nominal (category) data is
that they have to show the frequencies of cases in
each category used. While there may
be as many categories as you wish, it is not the
function of statistical analysis to communicate
all of the data’s detail; the task is to identify the
major trends. For example,
imagine you are researching the public’s attitudes
towards private health care. If you ask
participants in your research their occupations then
you might fi nd that they mention
t e n s i f n o t h u n d re d s o f d i ff e r e n t j o b t i t l e s – n e w s a g e n t s , h o m e m a ke r s , c o m p a n y
e xe c u t i v e s
a n d s o f o r t h . S i m p l y c o u n t i n g t h e f re q u e n c i e s w i t h w h i c h d i ff e re n t j o b t i t l e s a re
m e n t i o n e d re s u l t s i n a v a s t n u m b e r o f c a t e g o r i e s . Yo u n e e d t o t h i n k o f re l e v a n t a n d
m e a n i n g f u l w a y s o f re d u c i n g t h i s v a s t n u m b e r i n t o a s m a l l e r n u m b e r o f m u c h b ro a d e r
c a t e g o r i e s t h a t m i g h t r e v e a l i m p o r t a n t t r e n d s . Fo r e x a m p l e , s i n c e t h e re s e a rc h i s
about
a health issue you mi ght wi sh to form a categ ory made up of those invol ved in health
work – some mi ght be dentis ts , some nur ses, s ome doctors, some paramedi cs and so
f o r t h . I n s t e a d o f ke e p i n g t h e s e a s d i ff e r e n t c a t e g o r i e s , t h e y m i g h t b e c o m b i n e d i n t o a
c a t e g o r y ‘ h e a l t h w o r ke r ’ . T h e re a r e n o h a r d - a n d - f a s t r u l e s a b o u t c o m b i n i n g t o f o r m
broader categ ori es. The following mi ght be useful rule s of thumb:
1 . Ke e p y o u r n u m b e r o f c a t e g o r i e s l o w , e s p e c i a l l y w h e n y o u h a v e o n l y s m a l l n u m b e r s
o f p a r t i c i p a n t s i n y o u r r e s e a rc h .
2 . Tr y t o m a ke y o u r ‘ c o m b i n e d ’ c a t e g o r i e s m e a n i n g f u l a n d s e n s i b l e i n t h e l i g h t o f t h e
p u r p o s e s o f y o u r r e s e a rc h . I t w o u l d b e n o n s e n s e , f o r e x a m p l e , t o c a t e g o r i s e j o b s b y
the l etter of the alp habet with whi ch they s tart – nurs es, nuns, nurser y teachers and
national football ers. All of these have jobs beg inning with the same l etter, but it i s
v e r y d i ffi c u l t t o s e e a n y o t h e r c o m m o n t h re a d w h i c h a l l o w s t h e m t o b e c o m b i n e d
m e a n i n g f u l l y.
Slices for a pie diagram
There is n othin g diffi cult in constructing a pie diagram. Our
recommendation is that you turn each of your freq uencies
in to a percentage frequency. Since there are 360 d egrees in a
circle, if you mu ltiply each percentage frequency by 3.6
you will obtain the angle (in degrees) of the slice of the pie
w hich you need to mark out. In order to create the diagram,
you will requ ire a protractor to measu re the angles. H ow ever,
compu ter graph packages are standard at any u niversity
or college and do an impressive job.
In Table 2.1, 25.00% of cases were stu dents. In ord er to turn
this into the correct angle for the slice of th e pie, you
simp ly n eed to multiply 25.00 by 3.6 to give an an gle of 90
degrees.
FIGURE 2.2 A s imple pie diagram
One advan tage of u sing compute rs is that the y e nable
expe rime n tation w ith diff e re nt
sche mes of cate gorising data in orde r to de cide w hic h is be st for
you r p urpose s. In this
cas e, you would use in itially narrow c ate gor ie s for coding your
data. The n you can te ll
th e compu te r which of the s e to combine into broad er cate g orie s.
Some times it is pre fe rable to tu rn fre qu en cy tab le s into
diagr ams. Good d iagrams
are q uic kly unde rs tood and add var ie ty to the p re se ntation. The
main type s of diagr am
for nominal (c ate gory) data are pie diagrams and bar charts . A
pie diagram is a very
familiar form of pre se n tation – it simply expres se s e ach cate gory
as a s lice of a pie which
re pre se nts all case s (s e e Figu re 2.2).
Notice that the number of slices is small – a multitude of slices can be
confusing. Each
s l i c e i s c l e a r l y m a r ke d w i t h i t s c a t e g o r y n a m e , a n d t h e p e rc e n t a g e
f re q u e n c y i n e a c h
category also appears.
C H A P T E R 2 D E S C R I B I N G VA R I A B L E S 1 7
l e g e n d o r ke y t o i d e n t i fy t h e c o m p o n e n t s t o h e l p c o p e w i t h t h e o v e rc ro w d i n g
p ro b l e m . I n o t h e r w o rd s , t o o m a n y c a t e g o r i e s h a v e re s u l t e d i n a d i a g r a m
which is
f a r f ro m e a s y t o re a d – a c a rd i n a l s i n i n a n y s t a t i s t i c a l d i a g r a m .
A s i m p l e f re q u e n c y t a b l e m i g h t b e m o re e ff e c t i v e i n t h i s c a s e .
A n o t h e r v e r y f a m i l i a r f o rm o f s t a t i s t i c a l d i a g r a m f o r n o m i n a l ( c a t e g o r y ) d a t a
is
t h e b a r c h a r t . A g a i n t h e s e c h a r t s a re v e r y c o m m o n i n t h e m e d i a . B a s i c a l l y
t h e y a re
d i a g r a m s i n w h i c h b a r s re p re s e n t t h e s i z e o f e a c h c a t e g o r y. A n ex a m p l e i s
shown in
Fi g u re 2 . 4 .
T h e re l a tiv e len g t h s ( o r h e i g h t s ) o f t h e b a r s q u i ck l y re v e a l t h e m a i n
t re n d s in t h e d a t a .
Wit h a b a r c h a r t , t h ere i s v e r y l i t t l e t o re m e m b e r o t h e r t h a n t h a t t h e
b a r s h a ve a s t a n d a rd
s p a c e s e p a r a t in g t h em . T h e s p a c e s in d i c a t e t h a t t h e c a t e g o r i e s a re
n o t i n a n u m er ic a l
o rd e r ; t h ey a re f req u e n ci e s o f ca t e g o r i e s , no t s c o re s .
I t is h a rd t o g o w ro n g w i t h a b a r ch a r t ( t h a t i s n o t a ch a l l e n g e ! ) s o
lo n g a s y o u
re m e m b er t h e f o llo w in g :
1 . T h e h eig h ts o f t h e b a r s re p re s e n t f re q u e n ci e s ( n u m b e r o f c a s e s ) i n
a ca t e g o r y.
2 . E a ch b a r s h o u ld b e cl e a r ly l a b e l l e d a s t o t h e c a t e g o r y i t re p re s e n t s .
3 . To o m a n y b a r s m a ke b a r ch a r t s h a rd t o f o l lo w.
F I G U R E 2 . 3 A p o o r p ie d i a g r a m
F I G U R E 2 . 4 B a r ch a r t s h o w i n g o c cu p a t i o n a l ca t e g o r i e s i n Ta b l e 2 . 1
1 8 PA RT 1 D E S C R I PT I V E S TAT I S T I C S
4 . Av o i d h a v i n g m a n y e m p t y o r n e a r- e m p t y c a t e g o r i e s w h i c h r e p r e s e n t v e r y f e w c a s e s .
Gener al ly, the inform ati on about s ubs tantial c a tegor ie s is the m os t im port ant . (Sm al l
c a t e g o r i e s c a n b e c o m b i n e d t o g e t h e r a s a n ‘ o t h e r ’ c a t e g o r y. )
5 . N e ve rt he l e s s, i f i m po rta nt c ate gor i e s h ave ve r y fe w e ntr i e s th e n th i s n e e d s re c ord i ng .
So , f or exam p l e , a re se arc h e r wh o i s p ar ti c u l ar l y i n te re st e d i n op p or tun i t i e s for
w o m e n s u r v e y s p e o p l e i n t o p m a n a g e m e n t a n d fi n d s v e r y f e w w o m e n e m p l o y e d i n
suc h jobs. It i s im por tant to dra w atte nt ion to thi s in the ba r c har t of m ales and
fe m al e s i n top m a na ge m e n t. On c e ag ai n , th e re are no h ard - an d- fas t r ul e s to gu i d e
y o u – c o m m o n s e n s e w i l l t a ke y o u a l o n g w a y.
6 . M a ke s u re t h a t t h e v e r t i c a l a x i s ( t h e h e i g h t s o f t h e b a r s ) i s c l e a r l y m a r ke d a s b e i n g
fre qu e n c i e s or pe rc e nta ge fre qu e n c i e s.
7 . The bar s s ho ul d be of e qual wi dth.
I n n e w s p a p e r s a n d o n t e l e v i s i o n y o u a r e l i ke l y t o c o m e a c r o s s a v a r i a n t o f t h e b a r
c ha rt c al l e d t he p i c t ogr am . In thi s, th e b ars of the bar c h art a re re p l a c e d by var yi ng si ze d
dr awi n g s of s o m e th i ng e ye -c atc h i n g to d o wi t h you r c a te g or i e s . T hu s , p i c t ure s of m e n
o r w o m e n o f v a r y i n g h e i g h t s , f o r e x a m p l e , r e p l a c e t h e b a r s . Pi c t o g r a m s a r e r a r e l y u s e d
i n p rof e s si o na l pre se nt ati o ns . T he m ai n re as on i s th at p i c tu re s of thi ng s g e t wi d e r as we l l
as ta l l e r as th e y i n c re as e i n s i ze . T hi s c a n m i s re p re s e n t th e re l ati ve si z e s of t he
categories,
gi ve n th at re ad e r s f org e t th at i t i s on l y th e h e i g h t of th e p i c tu re th at c oun ts .
■ Ta b l e s a n d d i a g r a m s f o r n u m e r i c a l s c o r e d a t a
O ne cr ucia l co ns idera tio n w hen deciding w ha t ta bles a nd diagrams to use f or sc ore
data
i s t h e n u m b e r o f s e p a r a t e s c o re s r e c o r d e d f o r t h e v a r i a b l e i n q u e s t i o n . T h i s c a n
vary
m a r ke d l y. S o , f o r e x a m p l e , a g e i n t h e g e n e r a l p o p u l a t i o n c a n r a n g e f r o m n e w l y b o r n
t o o v e r 1 0 0 y e a r s o f a g e . I f w e m e r e l y re c o r d e d a g e s t o t h e n e a r e s t w h o l e y e a r
then a
t a b l e o r d i a g r a m m a y h a v e e n t r i e s f o r 1 0 0 d i ff e r e n t a g e s . S u c h a t a b l e o r d i a g r a m
would
loo k horrendo us . If w e reco rded a ge to the nea res t month, then w e could multiply
this
number of ages by 12! Such s cores can be grouped into bands or r anges o f s cores
to
a l l o w e ff e c t i v e t a b u l a t i o n ( Ta b l e 2 . 2 ) .
Many psychological variables have a much smaller range of numerical values. So,
f or exa mple, it is f airly co mmo n to use ques tio ns which pre-spec ify just a few
respons e
a l t e r n a t i v e s . T h e s o - c a l l e d L i ke r t - t y p e q u e s t i o n n a i r e i t e m i s a g o o d c a s e i n p o i n t .
Ty p i c a l l y t h i s l o o k s s o m e t h i n g l i ke t h i s :
Statistics is my favourite university subject:
A h i s t o g r a m m i g h t b e t h e b e s t f o r m o f s t a t i s t i c a l d i a g r a m t o re p r e s e n t t h e s e d a t a . A t
fi r s t s i g h t , h i s t o g r a m s l o o k v e r y m u c h l i ke b a r c h a r t s b u t w i t h o u t g a p s b e t w e e n t h e
bars.
Th is i s becau se the hi sto gram do es not represe nt dis tin ct un rela ted cate go ries but
d i ff e r e n t
p o i n t s o n a n u m e r i c a l m e a s u re m e n t s c a l e . S o a h i s t o g r a m o f t h e a b o v e d a t a m i g h t
l o o k l i ke Fi g u r e 2 . 5 .
B u t w h a t i f y o u r d a t a h a v e n u m e r o u s d i ff e re n t p o s s i b l e v a l u e s o f t h e v a r i a b l e i n
question?
O n e c o m m o n d i ffi c u l t y f o r m o s t p s y c h o l o g i c a l r e s e a rc h i s t h a t t h e n u m b e r o f
r e s p o n d e n t s t e n d s t o b e s m a l l . T h e l a r g e n u m b e r o f p o s s i b l e d i ff e r e n t s c o re s o n t h e
v a r i a b l e i s t h e r e f o r e s h a r e d b e t w e e n v e r y f e w r e s p o n d e n t s . Ta b l e s a n d d i a g r a m s
should
p r e s e n t m a j o r f e a t u re s o f y o u r d a t a i n a s i m p l e a n d e a s i l y a s s i m i l a t e d f o r m . S o ,
sometimes
y ou wi l l h av e to us e ban ds of s cores rath er tha n in div i du al score v a lu es , j us t as
y o u d i d f o r Ta b l e 2 . 2 . S o , i f w e a s ke d 1 0 0 p e o p l e t h e i r a g e s w e c o u l d c a t e g o r i s e t h e i r
repli es in to ban ds su ch as 0–9 y ear s, 10–19 y ea rs, 30–39 y ea rs, 40–49 ye ars an d a
fi n a l
Ta b l e 2 . 3 D i s t r i b u t i o n o f s t u d e n t s ’ a t t i t u d e s t o w a r d s s t a t i s t i c s
Re s p o n s e c a t e g o r y Va l u e Fre q u e n c y
S t ro n g l y a g r e e 1 1 7
A g re e 2 1 4
N e i t h e r a g r e e n o r d i s a g re e 3 6
Di sa gree 4 2
S t ro n g l y d i s a g r e e 5 1
FIGURE 2.5 Histogram of s tudents’ a ttitudes to wards s tatis tics
2 0 PA RT 1 D E S C R I P T I V E S TAT I S T I C S
c a t e g o r y o f t h o s e 5 0 y e a r s a n d o v e r. B y u s i n g b a n d s w e re d u c e t h e r i s k o f e m p t y
parts
o f t h e t a b l e a n d a l l o w a n y t re n d s t o b e c o m e c l e a r ( Fi g u r e 2 . 6 ) .
How one cho oses the bands to use is an impo rtant question. The answ er is a bit
o f l u c k a n d j u d g e m e n t , a n d a l o t o f t r i a l a n d e r r o r. I t i s v e r y t i m e - c o n s u m i n g t o re j i g
the ra nges of the bands when one is analysing the data by hand. One big a dvantage
of
c o m p u t e r s i s t h a t t h e y w i l l re c o d e y o u r s c o r e s i n t o b a n d s re p e a t e d l y u n t i l y o u h a v e
tables w hich seem to do the job as well as pos sible. The criter io n is still whether the
table
c o m m u n i c a t e s i n f o r m a t i o n e ff e c t i v e l y.
The one rule is that the bands ought to be of the same size – that is cover, for
ex ample, equal ranges of scores. Generally this is easy except at the upper and
lower
e n d s o f t h e d i s t r i b u t i o n . Pe r h a p s y o u w i s h t o u s e ‘ o v e r 7 0 ’ a s y o u r u p p e r r a n g e .
This,
in modern practice, can be done as a bar of the same width as the others, but
must be
v e r y c a re f u l l y m a r ke d . ( S t r i c t l y s p e a k i n g , t h e w i d t h o f t h e b a n d s h o u l d re p re s e n t
the
r a n g e o f s c o re s i n v o l v e d a n d t h e h e i g h t re d u c e d i n t h e l i g h t o f t h i s . H o w e v e r ,
this is
r a re l y d o n e i n m o d e r n p s y c h o l o g i c a l s t a t i s t i c s . )
O n e m i g h t re d e fi n e t h e b a n d s o f s c o re s a n d g e n e r a t e a n o t h e r h i s t o g r a m b a s e d o n
i d e n t i c a l d a t a b u t a d i ff e re n t s e t o f b a n d s ( Fi g u r e 2 . 7 ) .
F I G U R E 2 . 6 U s e o f b a n d s o f s c o re s t o e n a b l e s i m p l e p re s e n t a t i o n
FIGURE 2.7 Histogram showing ‘collapsed’ categories
C H A P T E R 2 D E S C R I B I N G VA R I A B L E S 2 1
I t re q u i re s s o m e t h o u g h t t o d e c i d e w h i c h o f t h e d i a g r a m s i s b e s t f o r a p a r t i c u l a r
p u r p o s e . T h e r e a re n o h a r d - a n d - f a s t r u l e s .
2 . 3 E rro r s t o a v o i d
T h e re a re a c o u p l e o f m i s t a ke s t h a t y o u c a n m a ke i n d r a w i n g u p t a b l e s a n d d i a g r a m s :
1 . D o n o t f o rg e t t o h e a d t h e t a b l e o r d i a g r a m w i t h a s u c c i n c t d e s c r i p t i o n o f w h a t i t
c o n c e r n s . Yo u w i l l n o t i c e t h a t w e h a v e d o n e o u r b e s t t h r o u g h o u t t h i s c h a p t e r t o
s u p p l y e a c h t a b l e a n d d i a g r a m w i t h a c l e a r t i t l e .
2 . L a b e l e v e r y t h i n g o n t h e t a b l e o r d i a g r a m a s c l e a r l y a s p o s s i b l e . W h a t t h i s m e a n s i s
t h a t y o u h a v e t o m a r k y o u r b a r c h a r t s a n d h i s t o g r a m s i n a w a y t h a t t e l l s t h e re a d e r
w h a t e a c h b a r m e a n s . T h e n y o u m u s t i n d i c a t e w h a t t h e h e i g h t o f t h e b a r re f e r s t o –
p r o b a b l y e i t h e r f r e q u e n c y o r p e r c e n t a g e f r e q u e n c y.
N o t e t h a t t h i s c h a p t e r h a s c o n c e n t r a t e d o n d e s c r i b i n g a s i n g l e v a r i a b l e a s c l e a r l y a s p o s s i b l e .
I n C h a p t e r 6 , m e t h o d s o f m a k i n g t a b l e s a n d d i a g r a m s s h o w i n g t h e re l a t i o n s h i p s
b e t w e e n t w o o r m o re v a r i a b l e s a re d e s c r i b e d .
Tr y t o m a k e y o u r t a b l e s a n d d i a g r a m s u s e f u l . I t i s n o t u s u a l l y t h e i r p u r p o s e t o r e c o r d t h e d a t a a s
y o u c o l l e c t e d i t i n y o u r re s e a rc h . O f c o u r s e y o u c a n l i s t y o u r d a t a i n t h e a p p e n d i x o f p ro j e c t s t h a t
y o u c a rr y o u t , b u t t h i s i s n o t u s e f u l a s a w a y o f i l l u s t r a t i n g t re n d s . I t i s p a r t o f a re s e a rc h e r ’ s j o b
t o m a ke t h e d a t a a c c e s s i b l e t o t h e re a d e r i n a s t r u c t u re d f o rm t h a t i s e a s i l y u n d e r s t o o d b y t h e
r e a d e r.
E s p e c i a l l y w h e n u s i n g c o m p u t e r s , i t i s v e r y e a s y t o g e n e r a t e u s e l e s s t a b l e s a n d d i a g r a m s . T h i s i s
u s u a l l y b e c a u s e c o m p u t e r a n a l y s i s e n c o u r a g e s y o u n o t t o ex a m i n e y o u r r a w d a t a i n a n y d e t a i l . T h i s
i m p l i e s t h a t y o u s h o u l d a l w a y s r e g a r d y o u r fi r s t a n a l y s e s a s t e n t a t i v e a n d m e r e l y a s t e p t o w a r d s
s o m e t h i n g b e t t e r.
I f a t a b l e i s n o t c l e a r t o y o u , i t i s u n l i ke l y t o b e a n y c l e a re r t o a n y o n e e l s e .
C h e c k e a c h t a b l e a n d d i a g r a m f o r c l e a r a n d f u l l l a b e l l i n g o f e a c h p a r t . E s p e c i a l l y , c h e c k t h a t
frequencies
a re c l e a r l y m a r ke d a s s u c h .
C h e c k t h a t t h e re i s a c l e a r , h e l p f u l t i t l e t o e a c h t a b l e a n d d i a g r a m .