Extensive notes with examples and practice problems ECON10072A Advanced Statistics (ECON10072A)

Discrete dala variable of which
: X can only yield isolated values
,
some are often repeated

·
Ordinal : categories have a natural ordering

Bur Charts : firs
& Football league, Premier league the
frequencies and




·
Nominal :
No natural ordering to the categories the relative frequencies
↳ Gende : Male , female

A variable that the of certain
Count
·
:
represents count events



↳ Number of children in household : 0. 1 ,
2
,
3 etc




Continous dala : Variable Y is centinous if it can assure
any value taken from a continuum/interval/rarge of numbers


i height weight incope distribution grash
.
e
,
, government expenditure >
- Histogram : raw data grouped
into intervals &
histyram
=

& find frequery fregers
(a) (fraentr)
relable
,
and dis


Cross-section data : taken at point in time
observation on a
particular variable a
single


for example : anneal Crime figure recorded in 1999, initial salsies of DOM
grads in 2012
, Minterparate & Several cities on a .
single day
Time-series dala : fire
Observation on a
particular variable recorded over a
period of at
regular intervals


for example :
Personal Grime figures in Manchester recorded annually from 1900 -
2022
,
Monthly household expendine on food ,
oil grive over lo years



↳ can be represerved a live
graphs

↳ for daka with different axies -> on excel crak a combo live
gragh with relationship between two variables


Secudey axis steen in the scatter flot


Location Measures : Central location/Cernal Fendency/average

& what a typical value from a set of observations is



& value around which observations in the sample are distributed
.




applicable for continous dala
The Mean :& only

Arithmetic Man = )
= =
. ...
n


adding all values in a sample and dividing this by the sample Size.




Mean of binary variables :

,The I applied
for both continue and discre daca
Median : e finding the middle value of a sampl


Median value Middle value of the data in (smalles to biggest


5Menuvalve
:
set order
organized


tradaa b
ar




Medium doesn't use all data
= less influenced



If dala set is positive skence : Man) Median by outliers

4 If highes value increases - man changes ,
median stays the same


↳ Mean is allded by extreme value skewed
- right

If
*
dala

For
set


symrehic
is


data
regaliely
- mean
skewed

and
: Mean


Median
<Median

are similar M
- - -
2, 6 8, 9 100
, ,




regellin sler
weighted mean : positive Symmetric
S kew


reformulate the arithmetic mean to the following equivalent formula weighted mem :
In
= Wi + we ... twn

=
& = M
+...




= x + x + ....
+ Y



take for of the variables isn't the
& we can't the arithmetic mean certain data types if the size same




alled in the of indes
weighted Mean is used construction numbers

ie - consumer price index (CPE)


↳ eighing calegories differently like food us music




Exable question :


6 , 241
S Pop prop
=
: VE = 9 9 . % = , 0002
0 all


452, 189
11 Poy
=
D : VE = . 0 % Prop = 0 . 02132



WE = 7, 457 , 632
N : 1 % %.

Pop =
Prop = 0 .
3515


6269
G : WE = 16 . 6% pay = 13, 298, 115 Prop = 0 .




total = 71 .
2%
tolal = 21 ,
214 , 177


29 9 .
(0 0002941) . + 11(0 02132).
+ 13 . 9 (0 .
3515) +


16 . 4(0 6269) . =
15 41
.

, Percentiles : - in EXCELL PERCENTILE
EXOCAxc ercantie
.




vale
dala
(
between
smalle bigger




nI i
und
all
including
inc :
and
Smallis
bigge

!

58 99 - 99th percentile
Median exceeding
where 1% earn income

F th
rell capy who less
- ame↳: eurn
F 2
345678910
It, 18, 23 28 30 , 35 , 39 , 48 59 , 80
, , ,




50th Median t5 32.
percentile = = =




18th percentile : One observation <18 and eight <18



: I/(I + 8) : 1/9 = 0 .
111111 or 11 11 %
. of observations are lover than 18




2/2 + 7) 2/9 22 22 22 %
percentiv
0
23rd =
= = = .
.




To calculate the 20th percentin :


4
0 3 20
.
.




example : ↑
Y S 6 7 8 9 10
3
is,
18 , 23, 38
,
30
,
35
,
09, 48 , 59
, 80



27th percenble ?


/(3
2
319 0 33 = 33 %
20th percentive = 3 + 3) = = .




23rd 22 22 %
percentive
.
=




27th in between 22 22
percentile = and 33 %
.




(0 27 12/9)/(1(9) (28 23) =
2370 4)
23 + 25 15
-
.
- ·
. . 5 = .




Deviation : How much the values in a
saigh differ from one another


range =
largest
-
Smallest


↳ Not ideal due to
cuiliers and
only uses the two extremes
.




Mea



Dispersions from central location : Example where and - =



Sample x Xi di Sargle Y Yi di

I
I 6 2 8 Y
di = xi -

2 3 2 3 I di
I =
j j
-
- -




3 3 3
↳ from the
airhidic mean
-

+ 3 I deciations

observed value -
Mean

, & How for dala set is from the mean
Sample variance :
each


Mean absolute deciatio (MAD) :
E lail = x - >




Mean squared deviation [MSD) : n di = (x -
1)
↳ variance
If dala represents releas population MSD =
Population variance (02)
↳



samyl rainne (c) = (xi -
x) n-1 =
degrees of freed on

values

& Max number of logically independent


S2(x) = (2)3 ( 1)" + 1 13
Example + disadvantages for data :
-
=
variance
-
:
2



-
= (4) + 1 + 1
has interpretation in chir of variance
2
·
vanience no
easy income dala = found
6
& - what is the
2
squared of meaning ?

= 3 to
·
variances for different dala sals are almost impossible compare


( 12 ( - 33
5(y) =

= (4) + -
+




+ (1) + (9)
= (16)
I 13




Standard deciation :

Population standard deviation : 0 = = (xi - ul"

= =
S(xi
standard deviation : S
sanyle n -
1
- 2)



Example :

1 2 7 7 10 , 18
, , , ,



* = 7


S
~
= = ( 6) + ( 5) -
+ (0) + (0) (3)
+ + (832

# (36) + (25) + (9) + (64)
-



= 26 . 8