PRIMJENJENA STATISTIKA

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# PRIMJENJENA STATISTIKA
## Predavanje 3: Deskriptivna statistika
### Luka Sikic, PhD
### Fakultet hrvatskih studija | <a href="https://lusiki.github.io/WebStatistika/">Github PS</a>

---

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# PREGLED PREDAVANJA

---
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# PREGLED PREDAVANJA
---
<br>
<br>
<br>
- Upoznavanje sa podatcima
<br>
- Mjere centralne tendencije
<br>
- Mjere varijabilnosti
<br>
- Mjere asimetrije i zaobljenosti
<br>
- Pregled varijabli i podatkovnih okvira
<br>
- Standardizirane vrijednosti
<br>
- Korelacija

---
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# PODATCI
---

.hi[Učitaj podatke]

```r
# Učitaj paket
library(lsr)
# Definiraj put do podataka setwd()
# Provjera getwd() 
load("../Podatci/aflsmall.Rdata") # Učitaj podatke u radni prostor
#who() # Pregledaj učitane podatke
str(afl.finalists) # Struktura podataka
```

```
#>  Factor w/ 17 levels "Adelaide","Brisbane",..: 9 10 3 10 9 3 10 3 9 10 ...
```

```r
str(afl.margins)   # Struktura podataka
```

```
#>  num [1:176] 56 31 56 8 32 14 36 56 19 1 ...
```
---
layout: true
# PODATCI
---

.hi[Pregledaj podatke]
<br>
<br>
<br>

```r
# Pregledaj podatke
print(afl.margins[1:11])
```

```
#>  [1] 56 31 56  8 32 14 36 56 19  1  3
```

```r
# Pregledaj podatke
print(afl.finalists[1:5])
```

```
#> [1] Hawthorn  Melbourne Carlton   Melbourne Hawthorn 
#> 17 Levels: Adelaide Brisbane Carlton Collingwood Essendon Fitzroy ... Western Bulldogs
```

---

.hi[Vizualizacija]
<br>
<img src="03_DESKRIPTIVNA_STATISTIKA_xar_files/figure-html/histogram1-1.svg" style="display: block; margin: auto;" />

.footnote[[*]Histogram pobjedničkih bodova(`afl.margins`) iz AFL 2010 lige američkog nogometa. Grafikon prikazuje da se broj pobjeda uz veću razliku rijeđe pojavljuje.]
---
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# MJERE CENTRALNE TENDENCIJE
---
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# MJERE CENTRALNE TENDENCIJE
---
<br>
<br>
<br>
<br>
- Aritmetička sredina
<br>
- Medijan
<br>
- Mod

---
layout: true
#ARITMETIČKA SREDINA

---

.hi[Definicija]

`$$\bar{X} = \frac {X_{1}+X_{2}+\cdots +X_{N}}{N}$$`

.hi[Sumiranje]

$$
\sum_{i=1}^5 X_i
$$
--

.hi[Skraćeni zapis]

$$
\bar{X} = \frac{1}{N} \sum_{i=1}^N X_i 
$$
--

.hi[Izračun rukom]

$$
\frac{56 + 31 + 56 + 8 + 32}{5} = \frac{183}{5} = 36.60
$$
---
<br>
<br>
.hi[Kalkulator]

```r
(56 + 31 + 56 + 8 + 32) / 5
```

```
#> [1] 36.6
```

--
<br>
<br>
.hi[Funkcija]

```r
sum( afl.margins[1:5]) / 5
```

```
#> [1] 36.6
```

---
layout: true
# MEDIJAN
---
<br>
<br>
.hi[Za neparni niz]
$$
8, 31, \mathbf{32}, 56, 56
$$
--

.hi[Za parni niz]
$$
8, 14, \mathbf{31}, \mathbf{32}, 56, 56
$$
--

.hi[Funkcija]

```r
#  Izračunaj median putem funkcije
median( x = afl.margins ) # Cijeli podatkovni skup
```

```
#> [1] 30.5
```

---
layout: true
# EKSTREMNE VRIJEDNOSTI
---

```r
# Definiraj vektor od 10 brojeva
vektor_10 <- c( -15,2,3,4,5,6,7,8,9,12 )
```
--

```r
mean( x = vektor_10 ) # Izračunaj AS
```

```
#> [1] 4.1
```
--

```r
median( x = vektor_10 ) # Izračunaj medijan
```

```
#> [1] 5.5
```

--
.hi[Korekcija]

```r
# Ukloni 10% ekstremnih vrijednosti
mean( x = vektor_10, trim = .1)
```

```
#> [1] 5.5
```
--

```r
# Ukloni 5% ekstremnih vrijednosti
mean( x = afl.margins, trim = .05)  
```

```
#> [1] 33.75
```

---
layout: true
# MOD
---
<br>
<br>

```r
# Pogledaj frekvenciju podataka
table(afl.finalists)
```

```
#> afl.finalists
#>         Adelaide         Brisbane          Carlton      Collingwood 
#>               26               25               26               28 
#>         Essendon          Fitzroy        Fremantle          Geelong 
#>               32                0                6               39 
#>         Hawthorn        Melbourne  North Melbourne    Port Adelaide 
#>               27               28               28               17 
#>         Richmond         St Kilda           Sydney       West Coast 
#>                6               24               26               38 
#> Western Bulldogs 
#>               24
```
---

```r
# Izračunaj modalnu vrijednost
modeOf( x = afl.finalists )
```

```
#> [1] "Geelong"
```
--

```r
# Izračunaj modalnu frekvenciju
maxFreq(x = afl.finalists)
```

```
#> [1] 39
```
--

```r
# Izaračun za afl.margins podatke
modeOf(afl.margins)  # Mod
```

```
#> [1] 3
```
--

```r
maxFreq(afl.margins) # Modalna frekvencija
```

```
#> [1] 8
```

---
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# MJERE VARIJABILNOSTI
---
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# MJERE VARIJABILNOSTI
---
<br>
<br>
<br>
<br>
- Raspon/Min-Max
<br>
- Kvartili
<br>
- Prosječno apsolutno odstupanje
<br>
- Varijanca
<br>
- Standardna devijacija 
<br>
- Srednje apsolutno odstupanje

---
layout: true
# RASPON/MIN-MAX
---

```r
# Maksimalna vrijednost
max(afl.margins)
```

```
#> [1] 116
```
--

```r
# Minimalna vrijednost
min(afl.margins)
```

```
#> [1] 0
```
--

```r
# Raspon podataka
range(afl.margins)
```

```
#> [1]   0 116
```
---
layout: true
# KVARTILI 
---

```r
# Izračunaj pedeseti (50i) kvartil/percentil
quantile(x = afl.margins, probs = .5)
```

```
#>  50% 
#> 30.5
```
--

```r
# Izračunaj 25i i 75i kvartil/percentil
quantile(afl.margins, probs = c(.25,.75))
```

```
#>   25%   75% 
#> 12.75 50.50
```
--

```r
# Izračunaj interkvartilni raspon
IQR(x = afl.margins)
```

```
#> [1] 37.75
```

---
layout:true
# PROSJEČNO APSOLUTNO ODSTUPANJE 
---

.hi[Formula]

$$
\mbox{}(X) = \frac{1}{N} \sum_{i = 1}^N |X_i - \bar{X}|
$$

.hi[Tablica za ručni izračun prosječnog apsolutnog odstupanja]
<table>
<caption></caption>
 <thead>
  <tr>
   <th style="text-align:right;"> `$i$` </th>
   <th style="text-align:right;"> `$X_i$` </th>
   <th style="text-align:right;"> `$X_i - \bar{X}$` </th>
   <th style="text-align:right;"> `$(X_i - \bar{X})$` </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 56 </td>
   <td style="text-align:right;"> 19.4 </td>
   <td style="text-align:right;"> 19.4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 31 </td>
   <td style="text-align:right;"> -5.6 </td>
   <td style="text-align:right;"> 5.6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 56 </td>
   <td style="text-align:right;"> 19.4 </td>
   <td style="text-align:right;"> 19.4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> -28.6 </td>
   <td style="text-align:right;"> 28.6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 32 </td>
   <td style="text-align:right;"> -4.6 </td>
   <td style="text-align:right;"> 4.6 </td>
  </tr>
</tbody>
</table>

---

.hi[Izračun rukom]

$$
\frac{19.4 + 5.6 + 19.4 + 28.6 + 4.6}{5} = 15.52
$$
--

.hi[Izračun pomoću funkcija]

```r
X <- c(56, 31,56,8,32) # Napravi vektor
X.bar <- mean( X )     # Korak 1. Izračunaj AS 
AD <- abs( X - X.bar ) # Korak 2. Uzmi aps vrijednost 
AAD <- mean( AD )      # Korak 3. Izračunaj AS devijacija
```
--

```r
print( AAD )           # Pogledaj rezultate
```

```
#> [1] 15.52
```

---
layout:true
# VARIJANCA 
---
<br>
<br>
.hi[Formula 1]
$$
\mbox{Var}(X) = \frac{1}{N} \sum_{i=1}^N \left( X_i - \bar{X} \right)^2
$$
<br>
<br>
.hi[Formula 2]

`$$\mbox{Var}(X) = \frac{\sum_{i=1}^N \left( X_i - \bar{X} \right)^2}{N}$$`
---

.hi[Ručni izračun varijance]

.hi[Kalkulatorski izračun]

```r
# Kalkulator
(376.36 + 31.36 + 376.36 + 817.96 + 21.16 ) / 5
```

```
#> [1] 324.64
```

---

.hi[Izračun putem funkcija]

```r
# Izračunaj varijancu pomoću funkcija
mean( (X - mean(X) )^2)
```

```
#> [1] 324.64
```
--

```r
var(X) # Skrati postupak
```

```
#> [1] 405.8
```

```r
## Isti primjer sa svim podatcima
# Izračunaj varijancu pomoću funkcija
mean( (afl.margins - mean(afl.margins) )^2)
```

```
#> [1] 675.9718
```
--

```r
var( afl.margins ) # Skrati postupak
```

```
#> [1] 679.8345
```

---
layout:true
# STANDARDNA DEVIJACIJA 
---
<br>
<br>
.hi[Formula 1]
$$
s = \sqrt{ \frac{1}{N} \sum_{i=1}^N \left( X_i - \bar{X} \right)^2 }
$$
<br>
<br>
.hi[Formula 2]

$$
\hat\sigma = \sqrt{ \frac{1}{N-1} \sum_{i=1}^N \left( X_i - \bar{X} \right)^2 }
$$
--

```r
# Izračunaj pomoću funkcije
sd( afl.margins ) 
```

```
#> [1] 26.07364
```

---
layout:true
# APSOLUTNO ODSTUPANJE OD MEDIJANA 
---
<br>
<br>

```r
# Prosječno apsolutno odstupanje od prosjeka
mean( abs(afl.margins - mean(afl.margins)) )
```

```
#> [1] 21.10124
```
--

```r
# *Medijansko*  apsolutno odstupanje od *medijana*:
median( abs(afl.margins - median(afl.margins)) )
```

```
#> [1] 19.5
```
--

```r
# Izračun putem funkcije
mad( x = afl.margins, constant = 1 )
```

```
#> [1] 19.5
```
---
layout:true
# KOEFICIJENT ASIMETRIJE 
---
<br>
<br>
<br>
<img src="03_DESKRIPTIVNA_STATISTIKA_xar_files/figure-html/skewness-1.svg" style="display: block; margin: auto;" />
---

.hi[Formula]
$$
\mbox{skewness}(X) = \frac{1}{N \hat{\sigma}^3} \sum_{i=1}^N (X_i - \bar{X})^3
$$
--
.hi[Funkcijski izračun]

```r
# Izračunaj na stvarnim podatcima
skew( x = afl.margins )
```

```
#> [1] 0.7671555
```

---
layout:true
# KOEFICIJENT ZAOBLJENOSTI 
---
<br>
<br>
<br>
<br>
<img src="03_DESKRIPTIVNA_STATISTIKA_xar_files/figure-html/kurtosis-1.svg" style="display: block; margin: auto;" />
---
<br>
<br>
.hi[Formula]
<br>
<br>
$$
\mbox{kurtosis}(X) = \frac{1}{N \hat\sigma^4} \sum_{i=1}^N \left( X_i - \bar{X} \right)^4  - 3
$$
--
<br>
<br>
.hi[Funkcijski izračun]

```r
# Izračunaj na stvarnim podatcima
kurtosi( x = afl.margins )
```

```
#> [1] 0.02962633
```

---
layout:true
# DESKRIPTIVNA STATISTIKA NA VARIJABLI 
---

.hi[Numerička varijabla]

```r
# Pregled numeričke varijable
summary( object = afl.margins ) # Deskriptivna stat 
```

```
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>    0.00   12.75   30.50   35.30   50.50  116.00
```
--
.hi[Logička varijabla]

```r
# Pregled logičke varijable
ekstremi <-  afl.margins > 50 # Stvori log varijablu
```
--

```r
head(ekstremi,5) # Pogledaj podatke
```

```
#> [1]  TRUE FALSE  TRUE FALSE FALSE
```

```r
summary(ekstremi) # Deskriptivna stat
```

```
#>    Mode   FALSE    TRUE 
#> logical     132      44
```
---
.hi[Faktorska varijabla]

```r
# Pregled faktorske varijable
summary(object = afl.finalists) # Deskriptivna stat
```

```
#>         Adelaide         Brisbane          Carlton      Collingwood 
#>               26               25               26               28 
#>         Essendon          Fitzroy        Fremantle          Geelong 
#>               32                0                6               39 
#>         Hawthorn        Melbourne  North Melbourne    Port Adelaide 
#>               27               28               28               17 
#>         Richmond         St Kilda           Sydney       West Coast 
#>                6               24               26               38 
#> Western Bulldogs 
#>               24
```
--

```r
# Pregled tekstualne varijable
txt <- as.character( afl.finalists ) # Stvori txt var
summary( object = txt ) # Deskriptivna stat
```

```
#>    Length     Class      Mode 
#>       400 character character
```
---
layout:true
# NOVI PODATKOVNI SKUP 
---
<br>
<br>
.hi[Podatci]
<br>

```r
rm(list = ls()) # Očisti radni prostor
load("../Podatci/clinicaltrial.Rdata") # Učitaj podatke
who(TRUE) # Pregled podataka 
```

```
#>    -- Name --    -- Class --   -- Size --
#>    clin.trial    data.frame    18 x 3    
#>     $drug        factor        18        
#>     $therapy     factor        18        
#>     $mood.gain   numeric       18
```

---
layout:true
# DESKRIPTIVNA STATISTIKA DF 
---
<br>
<br>
.hi[Obični pregled]
<br>

```r
# Deksriptivna statistika na podatkovnom okviru
summary(clin.trial) # Desktiptivna stat
```

```
#>        drug         therapy    mood.gain     
#>  placebo :6   no.therapy:9   Min.   :0.1000  
#>  anxifree:6   CBT       :9   1st Qu.:0.4250  
#>  joyzepam:6                  Median :0.8500  
#>                              Mean   :0.8833  
#>                              3rd Qu.:1.3000  
#>                              Max.   :1.8000
```
---
<br>
<br>
.hi[Alternativna funkcija]
<br>

```r
# Deksriptivna statistika na podatkovnom okviru
describe(clin.trial) # Desktiptivna stat/ druga funkcija
```

```
#>           vars  n mean   sd median trimmed  mad min max range skew kurtosis
#> drug*        1 18 2.00 0.84   2.00    2.00 1.48 1.0 3.0   2.0 0.00    -1.66
#> therapy*     2 18 1.50 0.51   1.50    1.50 0.74 1.0 2.0   1.0 0.00    -2.11
#> mood.gain    3 18 0.88 0.53   0.85    0.88 0.67 0.1 1.8   1.7 0.13    -1.44
#>             se
#> drug*     0.20
#> therapy*  0.12
#> mood.gain 0.13
```
---

.hi[Grupirani pregled]

```r
# Pregledaj grupirano prema terapiji
by(data = clin.trial, # Izvor podataka
   INDICES = clin.trial$therapy, # Odredi grupiranje
   FUN = summary) # Odredi funkciju
```

```
#> clin.trial$therapy: no.therapy
#>        drug         therapy    mood.gain     
#>  placebo :3   no.therapy:9   Min.   :0.1000  
#>  anxifree:3   CBT       :0   1st Qu.:0.3000  
#>  joyzepam:3                  Median :0.5000  
#>                              Mean   :0.7222  
#>                              3rd Qu.:1.3000  
#>                              Max.   :1.7000  
#> ------------------------------------------------------------ 
#> clin.trial$therapy: CBT
#>        drug         therapy    mood.gain    
#>  placebo :3   no.therapy:0   Min.   :0.300  
#>  anxifree:3   CBT       :9   1st Qu.:0.800  
#>  joyzepam:3                  Median :1.100  
#>                              Mean   :1.044  
#>                              3rd Qu.:1.300  
#>                              Max.   :1.800
```
---

.hi[Grupirani pregled]
<br>
<br>

```r
# Pregledaj grupirano prema razlici u raspoloženju
aggregate(formula = mood.gain ~ drug + therapy, # Prikaz
          data = clin.trial, # Podatci
          FUN = mean) # AS
```

```
#>       drug    therapy mood.gain
#> 1  placebo no.therapy  0.300000
#> 2 anxifree no.therapy  0.400000
#> 3 joyzepam no.therapy  1.466667
#> 4  placebo        CBT  0.600000
#> 5 anxifree        CBT  1.033333
#> 6 joyzepam        CBT  1.500000
```
---

.hi[Grupirani pregled]
<br>

```r
# Pregledaj grupirano prema razlici u raspoloženju
aggregate(mood.gain ~ drug + therapy,  # Prikaz
          clin.trial, # Podatci
          sd) # Standardna devijacija
```

```
#>       drug    therapy mood.gain
#> 1  placebo no.therapy 0.2000000
#> 2 anxifree no.therapy 0.2000000
#> 3 joyzepam no.therapy 0.2081666
#> 4  placebo        CBT 0.3000000
#> 5 anxifree        CBT 0.2081666
#> 6 joyzepam        CBT 0.2645751
```

---
layout:true
# STANDARDNE VRIJEDNOSTI 
---

.hi[Formula]
$$
\mbox{standardna vrijednost} = \frac{\mbox{vrijednost opservacije} - \mbox{prosjek}}{\mbox{standardna devijacija}}
$$

.hi[Z-score]

$$
z_i = \frac{X_i - \bar{X}}{\hat\sigma}
$$
--

.hi[Izračun rukom]

$$
z = \frac{35 - 17}{5} = 3.6
$$
--

.hi[Distribucija]

```r
# Vidi dio u distribuciji
pnorm( 3.6 ) 
```

```
#> [1] 0.9998409
```

---
layout:true
# NOVI PODATKOVNI SKUP
---

```r
rm(list = ls()) # Očisti radni prostor
 # Učitaj podatke
load("../Podatci/parenthood.Rdata")
who(TRUE) # Pregled podataka 
```

```
#>    -- Name --     -- Class --   -- Size --
#>    parenthood     data.frame    100 x 4   
#>     $dan.sleep    numeric       100       
#>     $baby.sleep   numeric       100       
#>     $dan.grump    numeric       100       
#>     $day          integer       100
```
--

```r
# Pregledaj podatke
head(parenthood, 7) # Prvih 7 redova
```

```
#>   dan.sleep baby.sleep dan.grump day
#> 1      7.59      10.18        56   1
#> 2      7.91      11.66        60   2
#> 3      5.14       7.92        82   3
#> 4      7.71       9.61        55   4
#> 5      6.68       9.75        67   5
#> 6      5.99       5.04        72   6
#> 7      8.19      10.45        53   7
```
---
<br>
<br>

```r
# Pogledaj deskriptivnu statistiku
describe(parenthood)
```

```
#>            vars   n  mean    sd median trimmed   mad   min    max range  skew
#> dan.sleep     1 100  6.97  1.02   7.03    7.00  1.09  4.84   9.00  4.16 -0.29
#> baby.sleep    2 100  8.05  2.07   7.95    8.05  2.33  3.25  12.07  8.82 -0.02
#> dan.grump     3 100 63.71 10.05  62.00   63.16  9.64 41.00  91.00 50.00  0.43
#> day           4 100 50.50 29.01  50.50   50.50 37.06  1.00 100.00 99.00  0.00
#>            kurtosis   se
#> dan.sleep     -0.72 0.10
#> baby.sleep    -0.69 0.21
#> dan.grump     -0.16 1.00
#> day           -1.24 2.90
```
---

.hi[Vizualizacija]

.footnote[[*]Grafički prikaz varijabli u `parenthood` podatkovnom skupu.]
---

layout:true
# KORELACIJA 
---

.hi[Prikaz podataka (*još jednom*)]

```r
head(parenthood[,c("dan.sleep", "baby.sleep", "dan.grump")],10)
```

```
#>    dan.sleep baby.sleep dan.grump
#> 1       7.59      10.18        56
#> 2       7.91      11.66        60
#> 3       5.14       7.92        82
#> 4       7.71       9.61        55
#> 5       6.68       9.75        67
#> 6       5.99       5.04        72
#> 7       8.19      10.45        53
#> 8       7.19       8.27        60
#> 9       7.40       6.06        60
#> 10      6.58       7.09        71
```

.footnote[[*]Pogledaj [tutorial](http://www.sthda.com/english/wiki/correlation-test-between-two-variables-in-r) o korelacijama!]

---

.hi[Grafički prikaz korelacije]
<br>
<img src="03_DESKRIPTIVNA_STATISTIKA_xar_files/figure-html/scatterparent1a-1.svg" style="display: block; margin: auto;" />

.footnote[[*]Dijagram rasipanja za varijable `Sati spavanja/roditelj` i `Raspoloženje`.]

---

.hi[Grafički prikaz korelacije]

.footnote[[*]Dijagram rasipanja za varijable `Sati spavanja/dijete` i `Sati spavanja/roditelj`.]

---
<br>
<br>
<br>
<br>
.hi[Kovarijanca]

$$
\mbox{Cov}(X,Y) = \frac{1}{N-1} \sum_{i=1}^N \left( X_i - \bar{X} \right) \left( Y_i - \bar{Y} \right)
$$
--
<br>
<br>
<br>
.hi[Personov korelacijski koeficijent; standardizacija kovarijance]

$$
r_{XY}  = \frac{\mbox{Cov}(X,Y)}{ \hat{\sigma}_X \ \hat{\sigma}_Y}
$$
.footnote[[*] [Pogledaj](https://datascience.stackexchange.com/questions/64260/pearson-vs-spearman-vs-kendall/64261) za raspravu: Pearson vs Spearman vs Kendall.]
---
layout:true
# SMJER I INTENZITET KORELACIJE 
---

---
layout:true
# IZRAČUN KORELACIJE U R
---

.hi[Funkcijski izračun; pojedinačno]

```r
# Izračunaj korelaciju između spavanja i raspoloženja
cor(x = parenthood$dan.sleep, y = parenthood$dan.grump)
```

```
#> [1] -0.903384
```
--

.hi[Funkcijski izračun; cijeli df]

```r
# Izračunaj korelacijsku tablicu
cor(x = parenthood)
```

```
#>              dan.sleep  baby.sleep   dan.grump         day
#> dan.sleep   1.00000000  0.62794934 -0.90338404 -0.09840768
#> baby.sleep  0.62794934  1.00000000 -0.56596373 -0.01043394
#> dan.grump  -0.90338404 -0.56596373  1.00000000  0.07647926
#> day        -0.09840768 -0.01043394  0.07647926  1.00000000
```
---
layout:true
# INTERPRETACIJA KORELACIJE
---

.hi[Okvirne smjernice za interpretaciju korelacije]

<table>
<caption></caption>
 <thead>
  <tr>
   <th style="text-align:left;"> Korelacija </th>
   <th style="text-align:left;"> Snaga </th>
   <th style="text-align:left;"> Smjer </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> -1.0 to -0.9 </td>
   <td style="text-align:left;"> Izrazito jaka </td>
   <td style="text-align:left;"> Negativna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -0.9 to -0.7 </td>
   <td style="text-align:left;"> Jaka </td>
   <td style="text-align:left;"> Negativna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -0.7 to -0.4 </td>
   <td style="text-align:left;"> Umjerena </td>
   <td style="text-align:left;"> Negativna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -0.4 to -0.2 </td>
   <td style="text-align:left;"> Slaba </td>
   <td style="text-align:left;"> Negativna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -0.2 to 0 </td>
   <td style="text-align:left;"> Zanemariva </td>
   <td style="text-align:left;"> Negativna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 0 to 0.2 </td>
   <td style="text-align:left;"> Zanemariva </td>
   <td style="text-align:left;"> Pozitivna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 0.2 to 0.4 </td>
   <td style="text-align:left;"> Slaba </td>
   <td style="text-align:left;"> Pozitivna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 0.4 to 0.7 </td>
   <td style="text-align:left;"> Umjerena </td>
   <td style="text-align:left;"> Pozitivna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 0.7 to 0.9 </td>
   <td style="text-align:left;"> Jaka </td>
   <td style="text-align:left;"> Pozitivna </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 0.9 to 1.0 </td>
   <td style="text-align:left;"> Izrazito jaka </td>
   <td style="text-align:left;"> Pozitivna </td>
  </tr>
</tbody>
</table>

---
layout:true
# NOVI PODATKOVNI SKUP
---
<br>
<br>
<br>

```r
rm(list=ls()) # Očisti radni prostor
load("../Podatci/effort.Rdata") # Učitaj podatke
who(TRUE) # Pregledaj podatke
```

```
#>    -- Name --   -- Class --   -- Size --
#>    effort       data.frame    10 x 2    
#>     $hours      numeric       10        
#>     $grade      numeric       10
```
---

.hi[Pregled podataka]

```r
head(effort, 10) #Pregledaj podatke
```

```
#>    hours grade
#> 1      2    13
#> 2     76    91
#> 3     40    79
#> 4      6    14
#> 5     16    21
#> 6     28    74
#> 7     27    47
#> 8     59    85
#> 9     46    84
#> 10    68    88
```
--

```r
cor(effort$hours, effort$grade) # Izračunaj korelaciju
```

```
#> [1] 0.909402
```
---

.hi[Vizualizacija]

.footnote[[*]Odnos između sati studiranja i ocjene (svaka točka predstavlja jednog studenta). Isprekidana linija prikazuje linearni odnos. Korelacija između ove dvije varijable je visoka `$r = .91$`. Valja primjetiti da više sati učenja uvijek dodnosi veću ocjenu što se odražava u visokom Spearman koeficijentu korelacije of `$rho = 1$`.]

---
layout:true
# SPEARMANOVA KORELACIJA 
---

```r
korelacije <- data.frame(
  sati = effort$hours,
  ocjena = effort$grade,
  satiRang = rank(effort$hours), # Rang sati
  ocjenaRang = rank(effort$grade)) # Rang ocjena

korelacije # Pregledaj podatke (+ rang varijable)
```

```
#>    sati ocjena satiRang ocjenaRang
#> 1     2     13        1          1
#> 2     6     14        2          2
#> 3    16     21        3          3
#> 4    27     47        4          4
#> 5    28     74        5          5
#> 6    40     79        6          6
#> 7    46     84        7          7
#> 8    59     85        8          8
#> 9    68     88        9          9
#> 10   76     91       10         10
```

.footnote[[*]Spearmanova korelacijska metoda računa korelaciju između ranga dvije varijable.]

---
<br>
<br>
.hi[Funkcijski izračun]

```r
cor(korelacije$sati,korelacije$ocjena, method = "pearson") # Izračunaj korelaciju
```

```
#> [1] 0.909402
```
--

```r
cor(korelacije$satiRang,korelacije$ocjenaRang, method = "pearson") # Izračunaj korelaciju
```

```
#> [1] 1
```
--

```r
# Dodaj argument "spearman"
cor(korelacije$satiRang,korelacije$ocjenaRang, method = "spearman")
```

```
#> [1] 1
```

---
layout:true
# KENDALLOVA KORELACIJA 
---

```r
# Dodaj argument "spearman"
cor(korelacije$satiRang,korelacije$ocjenaRang, method = "kendall")
```

```
#> [1] 1
```
.footnote[[*]Kendallova korelacijska metoda računa korespondenciju između rangiranja dvije varijable.]

---
layout:true
# KORELACIJA (još jedan primjer) 
---

.hi[Podatci]

```
#>                    mpg cyl  disp  hp drat    wt  qsec vs am gear carb
#> Mazda RX4         21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4
#> Mazda RX4 Wag     21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4
#> Datsun 710        22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1
#> Hornet 4 Drive    21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1
#> Hornet Sportabout 18.7   8 360.0 175 3.15 3.440 17.02  0  0    3    2
#> Valiant           18.1   6 225.0 105 2.76 3.460 20.22  1  0    3    1
#> Duster 360        14.3   8 360.0 245 3.21 3.570 15.84  0  0    3    4
#> Merc 240D         24.4   4 146.7  62 3.69 3.190 20.00  1  0    4    2
```

---

.hi[Vizualizacija]

---

.hi[Testovi distribucije]

```r
# Shapiro-Wilk test normalnosti za mpg
shapiro.test(my_data$mpg) # => p = 0.1229
```

```
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  my_data$mpg
#> W = 0.94756, p-value = 0.1229
```

```r
# Shapiro-Wilk test normalnosti za wt
shapiro.test(my_data$wt) # => p = 0.09
```

```
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  my_data$wt
#> W = 0.94326, p-value = 0.09265
```

---

.hi[Testovi distribucije]

---

.hi[Testovi distribucije]

---

.hi[Korelacijski test]

```r
pers <- cor.test(my_data$wt, my_data$mpg, 
                    method = "pearson")
pers
```

```
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  my_data$wt and my_data$mpg
#> t = -9.559, df = 30, p-value = 1.294e-10
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  -0.9338264 -0.7440872
#> sample estimates:
#>        cor 
#> -0.8676594
```

---

.hi[Korelacijski test]

```r
sper <- cor.test(my_data$wt, my_data$mpg, 
                    method = "spearman")
sper
```

```
#> 
#> 	Spearman's rank correlation rho
#> 
#> data:  my_data$wt and my_data$mpg
#> S = 10292, p-value = 1.488e-11
#> alternative hypothesis: true rho is not equal to 0
#> sample estimates:
#>       rho 
#> -0.886422
```

---

.hi[Korelacijski test]

```r
kend <- cor.test(my_data$wt, my_data$mpg, 
                    method = "kendall")
kend
```

```
#> 
#> 	Kendall's rank correlation tau
#> 
#> data:  my_data$wt and my_data$mpg
#> z = -5.7981, p-value = 6.706e-09
#> alternative hypothesis: true tau is not equal to 0
#> sample estimates:
#>        tau 
#> -0.7278321
```

---

layout:false
class: middle, inverse
# HVALA NA PAŽNJI! 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Sljedeće predavanje: Grafikoni i vizualizacije)