RED_WINE_VARS <-c("fixed.acidity","volatile.acidity","citric.acid","residual.sugar","chlorides","free.sulfur.dioxide","total.sulfur.dioxide","density","pH","sulphates","alcohol","quality")

> my_RW_stats <- function(RW, na.omit = FALSE) {

    if (na.omit)

    x <- RW[!is.na(RW)]

    m <- mean(RW)

    med <- median(RW)

    n <- length(RW)

    min <- min(RW)

    max <- max(RW)

    s <- sd(RW)

    v<- var(RW)

    quant <- quantile (RW)

    skew <- sum((RW - m)^3/s^3)/n

    kurt <- sum((RW - m)^4/s^4)/n - 3

    return(c( n = n, Min = min, Max = max, Mean= m, Median = med, Stddev = s, Var =v, Quantile = quant, Skew = skew, Kurtosis = kurt))

    }

> sapply(RW[RED_WINE_VARS],my_RW_stats)

              fixed.acidity volatile.acidity   citric.acid residual.sugar

n              1599.0000000    1599.00000000 1599.00000000    1599.000000

Min               4.6000000       0.12000000    0.00000000       0.900000

Max              15.9000000       1.58000000    1.00000000      15.500000

Mean              8.3196373       0.52782051    0.27097561       2.538806

Median            7.9000000       0.52000000    0.26000000       2.200000

Stddev            1.7410963       0.17905970    0.19480114       1.409928

Var               3.0314164       0.03206238    0.03794748       1.987897

Quantile.0%       4.6000000       0.12000000    0.00000000       0.900000

Quantile.25%      7.1000000       0.39000000    0.09000000       1.900000

Quantile.50%      7.9000000       0.52000000    0.26000000       2.200000

Quantile.75%      9.2000000       0.64000000    0.42000000       2.600000

Quantile.100%    15.9000000       1.58000000    1.00000000      15.500000

Skew              0.9809084       0.67033307    0.31774029       4.532140

Kurtosis          1.1196987       1.21268929   -0.79304553      28.485020

                   chlorides free.sulfur.dioxide total.sulfur.dioxide

n             1599.000000000         1599.000000          1599.000000

Min              0.012000000            1.000000             6.000000

Max              0.611000000           72.000000           289.000000

Mean             0.087466542           15.874922            46.467792

Median           0.079000000           14.000000            38.000000

Stddev           0.047065302           10.460157            32.895324

Var              0.002215143          109.414884          1082.102373

Quantile.0%      0.012000000            1.000000             6.000000

Quantile.25%     0.070000000            7.000000            22.000000

Quantile.50%     0.079000000           14.000000            38.000000

Quantile.75%     0.090000000           21.000000            62.000000

Quantile.100%    0.611000000           72.000000           289.000000

Skew             5.669693705            1.248222             1.512689

Kurtosis        41.525963494            2.007221             3.785676

                        density            pH     sulphates      alcohol

n             1599.000000000000 1599.00000000 1599.00000000 1599.0000000

Min              0.990070000000    2.74000000    0.33000000    8.4000000

Max              1.003690000000    4.01000000    2.00000000   14.9000000

Mean             0.996746679174    3.31111320    0.65814884   10.4229831

Median           0.996750000000    3.31000000    0.62000000   10.2000000

Stddev           0.001887333954    0.15438646    0.16950698    1.0656676

Var              0.000003562029    0.02383518    0.02873262    1.1356474

Quantile.0%      0.990070000000    2.74000000    0.33000000    8.4000000

Quantile.25%     0.995600000000    3.21000000    0.55000000    9.5000000

Quantile.50%     0.996750000000    3.31000000    0.62000000   10.2000000

Quantile.75%     0.997835000000    3.40000000    0.73000000   11.1000000

Quantile.100%    1.003690000000    4.01000000    2.00000000   14.9000000

Skew             0.071153970752    0.19332027    2.42411764    0.8592144

Kurtosis         0.922500001405    0.79591912   11.66152852    0.1916586

> library(lessR)

lessR 3.2      now RStudio compatible        www.lessRstats.com

---------------------------------------------------------------

To get started, and for help in general, enter:  > Help()

To read a text, Excel, SPSS or R data file:  > mydata <- Read()

---------------------------------------------------------------

> ?boxplot

> boxplot.stats(fixed.acidity)

$stats

[1]  4.6  7.1  7.9  9.2 12.3

$n

[1] 1599

$conf

[1] 7.817024 7.982976

$out

 [1] 12.8 12.8 15.0 15.0 12.5 13.3 13.4 12.4 12.5 13.8 13.5 12.6 12.5 12.8 12.8 14.0 13.7 13.7

[19] 12.7 12.5 12.8 12.6 15.6 12.5 13.0 12.5 13.3 12.4 12.5 12.9 14.3 12.4 15.5 15.5 15.6 13.0

[37] 12.7 13.0 12.7 12.4 12.7 13.2 13.2 13.2 15.9 13.3 12.9 12.6 12.6

> opar <- par(no.readonly=TRUE)

> par(mfrow=c(2,2))

> y <- fixed.acidity

> x <- total.sulfur.dioxide

> m <- volatile.acidity

> n <- density

> # 4 box plots with outliers more strongly highlighted

> BoxPlot(y, col.stroke="red", horiz=TRUE, col.fill="grey",xlab="_Fixed_Acidity_")

--- _Fixed_Acidity_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 4.6 

Lower Whisker: 4.6 

Lower Hinge  : 7.1 

Median       : 7.9 

Upper Hinge  : 9.2 

Upper Whisker: 12.3 

Maximum      : 15.9 

1st Quartile : 7.1 

3rd Quartile : 9.2 

IQR          : 2.1 

Number of outliers: 49 

Small: none

Large: 12.4  12.4  12.4  12.4  12.5  12.5  12.5  12.5  12.5  12.5  12.5  12.6  12.6  12.6  12.6  12.7  ...  15  15.5  15.5  15.6  15.6  15.9  

> BoxPlot(x, col.stroke="blue", horiz=1, col.fill="plum", xlab="_ total.sulfur.dioxide_")

--- _ total.sulfur.dioxide_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 6.0 

Lower Whisker: 6.0 

Lower Hinge  : 22.0 

Median       : 38.0 

Upper Hinge  : 62.0 

Upper Whisker: 122.0 

Maximum      : 289.0 

1st Quartile : 22.0 

3rd Quartile : 62.0 

IQR          : 40.0 

Number of outliers: 55 

Small: none

Large: 124  124  124  125  125  126  127  127  128  128  129  129  129  130  131  131  ...  153  155  160  165  278  289  

> BoxPlot(m, col.stroke="red", horiz=1, col.fill="grey", xlab="_Volatile_Acidity_")

--- _Volatile_Acidity_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 0.120 

Lower Whisker: 0.120 

Lower Hinge  : 0.390 

Median       : 0.520 

Upper Hinge  : 0.640 

Upper Whisker: 1.010 

Maximum      : 1.580 

1st Quartile : 0.390 

3rd Quartile : 0.640 

IQR          : 0.250 

> BoxPlot(n, col.stroke="blue", horiz=1, col.fill="plum", xlab="_ Density _")

--- _ Density _ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 0.99007 

Lower Whisker: 0.99235 

Lower Hinge  : 0.99560 

Median       : 0.99675 

Upper Hinge  : 0.99784 

Upper Whisker: 1.00100 

Maximum      : 1.00369 

1st Quartile : 0.99560 

3rd Quartile : 0.99784 

IQR          : 0.00223 

> 

> 

> opar <- par(no.readonly=TRUE)

> par(mfrow=c(2,2))

> ca <- citric.acid

> rs <- residual.sugar

> cl <- chlorides

> al <- alcohol 

> # 4 box plots with outliers more strongly highlighted

> 

> BoxPlot(ca, col.stroke="red", horiz=TRUE, col.fill="plum",xlab="_Citric.Acid_")

--- _Citric.Acid_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 0.00 

Lower Whisker: 0.00 

Lower Hinge  : 0.09 

Median       : 0.26 

Upper Hinge  : 0.42 

Upper Whisker: 0.79 

Maximum      : 1.00 

1st Quartile : 0.09 

3rd Quartile : 0.42 

IQR          : 0.33 

> BoxPlot(rs, col.stroke="blue", horiz=TRUE, col.fill="plum",xlab="_Residual.Sugar_")

--- _Residual.Sugar_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 0.90 

Lower Whisker: 0.90 

Lower Hinge  : 1.90 

Median       : 2.20 

Upper Hinge  : 2.60 

Upper Whisker: 3.65 

Maximum      : 15.50 

1st Quartile : 1.90 

3rd Quartile : 2.60 

IQR          : 0.70 

> BoxPlot(cl, col.stroke="black", horiz=TRUE, col.fill="plum",xlab="_ chlorides_")

--- _ chlorides_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 0.012 

Lower Whisker: 0.041 

Lower Hinge  : 0.070 

Median       : 0.079 

Upper Hinge  : 0.090 

Upper Whisker: 0.119 

Maximum      : 0.611 

1st Quartile : 0.070 

3rd Quartile : 0.090 

IQR          : 0.020 

> BoxPlot(al, col.stroke="black", horiz=TRUE, col.fill="plum",xlab="_ Alcohol _")

--- _ Alcohol _ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 8.400000 

Lower Whisker: 8.400000 

Lower Hinge  : 9.500000 

Median       : 10.200000 

Upper Hinge  : 11.100000 

Upper Whisker: 13.500000 

Maximum      : 14.900000 

1st Quartile : 9.500000 

3rd Quartile : 11.100000 

IQR          : 1.600000 

Number of outliers: 13 

Small: none

Large: 13.56667  13.6  13.6  13.6  13.6  14  14  14  14  14  14  14  14.9  

> 

> 

> 

> 

> RW$residual.sugar<- ifelse(residual.sugar>4.8,(RW$residual.sugar=="5"),(RW$residual.sugar<-RW$residual.sugar ))

> summary(RW$residual.sugar)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 

  0.000   1.800   2.100   2.152   2.500   4.800 

> 

> BoxPlot(RW$residual.sugar, col.stroke="blue", horiz=TRUE, col.fill="plum",xlab="_Residual.Sugar_")

Error: 

------

Data frame (table) mydata, the default data table name, does not exist

So either create the data table with the Read function, or

  specify the actual data table with the parameter: data

> RS<-RW$residual.sugar

> BoxPlot(RS, col.stroke="blue", horiz=TRUE, col.fill="plum",xlab="_Residual.Sugar_")

--- _Residual.Sugar_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 0.00 

Lower Whisker: 0.90 

Lower Hinge  : 1.80 

Median       : 2.10 

Upper Hinge  : 2.50 

Upper Whisker: 3.50 

Maximum      : 4.80 

1st Quartile : 1.80 

3rd Quartile : 2.50 

IQR          : 0.70 

> 

> 

> RW$residual.sugar<- ifelse(residual.sugar>3.65,(RW$residual.sugar=="4"),(RW$residual.sugar<-RW$residual.sugar ))

> summary(RW$residual.sugar)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 

  0.000   1.800   2.100   1.975   2.400   3.650 

> 

> RS<-RW$residual.sugar

> BoxPlot(RS, col.stroke="blue", horiz=TRUE, col.fill="plum",xlab="_Residual.Sugar_")

--- _Residual.Sugar_ --- 

Present: 1599 

Missing: 0 

Total  : 1599 

Minimum      : 0.00 

Lower Whisker: 1.00 

Lower Hinge  : 1.80 

Median       : 2.10 

Upper Hinge  : 2.40 

Upper Whisker: 3.30 

Maximum      : 3.65 

1st Quartile : 1.80 

3rd Quartile : 2.40 

IQR          : 0.60 

> 

> 

> opar <- par(no.readonly=TRUE)

> par(mfrow=c(2,2))

> with(RW,plot(alcohol,quality,main="RW-Corr_ALCOHOL_with_QUAL.",

+              col="BLACK"))

> with(RW,plot(citric.acid,quality,main="RW-Corr_CITRIC.ACID_with_QUAL.",

+              col="BLUE"))

> with(RW,plot(sulphates,quality,main="RW-Corr_SULPHATES_with_QUAL.",

+              col="RED"))

> with(RW,plot(density,quality,main="RW-Corr_DENSITY_with_QUAL.",

+              col="BROWN")) 

> 

> opar <- par(no.readonly=TRUE)

> par(mfrow=c(2,2))

> with(RW,plot(volatile.acidity,quality,main="RW-Corr_ALCOHOL_with_QUAL.",col="BLACK"))

> with(RW,plot(total.sulfur.dioxide,quality,main="RW-Corr_CITRIC.ACID_with_QUAL.",col="BLUE"))

> with(RW,plot(residual.sugar,quality,main="RW-Corr_SULPHATES_with_QUAL.",col="RED"))

> with(RW,plot(density,quality,main="RW-Corr_DENSITY_with_QUAL.",            col="BROWN")) 

> 

> opar <- par(no.readonly=TRUE)

> par(mfrow=c(2,2))

> with(RW,plot(volatile.acidity,quality,main="RW-Corr_Volatile.Acidity_with_QUAL.",col="BLACK"))

> with(RW,plot(total.sulfur.dioxide,quality,main="RW-Corr_total.sulfur.dioxide_with_QUAL.",col="BLUE"))

> with(RW,plot(residual.sugar,quality,main="RW-Corr_Residual.Sugar_with_QUAL.",col="RED"))

> with(RW,plot(density,quality,main="RW-Corr_Density_with_QUAL.",            col="BROWN")) 

> 

> ## Testing if the Correlation between Alcohol and Quality is Significant

 cor(WW$quality,WW[,1:11])

     fixed.acidity volatile.acidity  citric.acid residual.sugar  chlorides

[1,]    -0.1136628        -0.194723 -0.009209091    -0.09757683 -0.2099344

     free.sulfur.dioxide total.sulfur.dioxide    density         pH

[1,]         0.008158067           -0.1747372 -0.3071233 0.09942725

      sulphates   alcohol

[1,] 0.05367788 0.4355747

> 

We can observe – Highest Positive Correlation –

1.  Alcohol and Quality   =  .435 or 43.5%

2.  pH and Quality        =  .099 or 10%

3.  Sulphates and Quality =  .053 or 5.3%

Thus by increasing the proportion of these above mentioned three variables

we have a high probability of increasing White Wine Quality.

Highest Negative Correlation –

1.  Density and Quality          =  -0.307 or 31%

2.  Chloride and Quality         =  -0.209 or 21%

3.  Volatile Acidity and Quality =  -0.194 or 19.4%

Thus by decreasing the proportion of these above mentioned three variables

 we have a high probability of increasing White Wine Quality.

We can observe – Highest Positive Correlation –

1.  Alcohol and Quality   =  .476 or 47%

2.  Sulphates and Quality =  .251 or 25%

3.  CitricAcid and Quality=  .226 or 22%

Thus by increasing proportion of above mentioned variables we have high probability of increasing Red Wine Quality.

Highest Negative Correlation –

1.  Volatile Acidity and Quality      =  -0.390 or 39%

2.  Total Sulfur Di Oxide and Quality =  -0.185 or 18%

3.  Density and Quality               =  -0.174 or 17%

Thus by decreasing proportion of these variables we have high probability of increasing Red Wine Quality.

 Scatter plots – Correlating Red Wine Variables

with RW$QUALITY.

opar <- par(no.readonly=TRUE)

> par(mfrow=c(2,2))

> with(RW,plot(alcohol,quality,main="RW-Corr_ALCOHOL_with_QUAL.",

+              col="BLACK"))

> with(RW,plot(citric.acid,quality,main="RW-Corr_CITRIC.ACID_with_QUAL.",

+              col="BLUE"))

> with(RW,plot(sulphates,quality,main="RW-Corr_SULPHATES_with_QUAL.",

+              col="RED"))

> with(RW,plot(density,quality,main="RW-Corr_DENSITY_with_QUAL.",

+              col="BROWN"))

opar <- par(no.readonly=TRUE)

par(mfrow=c(2,2))

with(RW,plot(volatile.acidity,quality,main="RW-Corr_ALCOHOL_with_QUAL.",col="BLACK"))

with(RW,plot(total.sulfur.dioxide,quality,main="RW-Corr_CITRIC.ACID_with_QUAL.",col="BLUE"))

with(RW,plot(residual.sugar,quality,main="RW-Corr_SULPHATES_with_QUAL.",col="RED"))

with(RW,plot(density,quality,main="RW-Corr_DENSITY_with_QUAL.  col="BROWN"))

opar <- par(no.readonly=TRUE)

par(mfrow=c(2,2))

VAR1 <- fixed.acidity

h<-hist(VAR1,breaks=72,col="orange",border="red",xlab=" Fixed.Acidity",

main="Fixed.Acidity- with Normal Distribution Curve ") ### - Defined two more Variables - VAR1fit & yfit - ###

VAR1fit<-seq(min(VAR1), max(VAR1), length=40)

yfit<-dnorm(VAR1fit, mean=mean(VAR1), sd=sd(VAR1))

yfit <- yfit*diff(h$mids[1:2])*length(VAR1)

lines(VAR1fit, yfit, col="red", lwd=2)

box()

VAR2 <- pH

h<-hist(VAR2,breaks=72,col="orange",border="red",xlab="pH",

main="pH RED WINE- with Normal Distribution Curve ") ### - Defined two more Variables – VAR2fit & yfit - ###

VAR2fit<-seq(min(VAR2), max(VAR2), length=40)

yfit<-dnorm(VAR2fit, mean=mean(VAR2), sd=sd(VAR2))

yfit <- yfit*diff(h$mids[1:2])*length(VAR2)

lines(VAR2fit, yfit, col="red", lwd=2)

box()

> opar <- par(no.readonly=TRUE)

> par(mfrow=c(2,2))

> fa <- fixed.acidity

> va <- volatile.acidity

> tsd <- total.sulfur.dioxide

> den <- density

>

> BoxPlot(fa, col.stroke="red", horiz=TRUE,

+ col.fill="plum",xlab="_fixed.acidity_")

--- _fixed.acidity_ ---

Present: 4898

Missing: 0

Total  : 4898

Minimum      : 3.80

Lower Whisker: 4.80

Lower Hinge  : 6.30

Median       : 6.80

Upper Hinge  : 7.30

Upper Whisker: 8.80

Maximum      : 14.20

1st Quartile : 6.30

3rd Quartile : 7.30

IQR          : 1.00

Number of outliers: 119

Small: 3.8  3.9  4.2  4.2  4.4  4.4  4.4  4.5

 4.6  4.7  4.7  4.7  4.7  4.7 

Large: 8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9 

8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9

 ...  10.3  10.3  10.7  10.7  11.8  14.2 

> RW$residual.sugar<- ifelse(residual.sugar>3.65,

+ (RW$residual.sugar=="5"),

+ (RW$residual.sugar<-RW$residual.sugar ))

> summary(RW$residual.sugar)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.

  0.000   1.800   2.100   1.975   2.400   3.650

As a result of the capping done using the ifelse –

We see Mean has changed from earlier = 2.538 to 1.975

Median has changed from the earlier = 2.200 to 2.100

Max changed from = 15.500 to 3.650

N the sample size remains the same = 1599.

Thus this is better than the use of the SUBSET function in which observations / rows from all variables – which did not have outliers also were getting removed.

White Wine Variables are seen to have high number of Outliers – we use IFELSE to cap the OUTLIERS . We can see below the 

Summary STATS for capped and Non Capped variables. 

> library("lessR", lib.loc="~/R/win-library/3.1")

lessR 3.2      now RStudio compatible        www.lessRstats.com

---------------------------------------------------------------

To get started, and for help in general, enter:  > Help()

To read a text, Excel, SPSS or R data file:  > mydata <- Read()

---------------------------------------------------------------

> y <- WW$fixed.acidity

> BoxPlot(y, col.stroke="red", horiz=TRUE,col.fill="grey",xlab="_Fixed_Acidity_")

--- _Fixed_Acidity_ --- 

Present: 4898 

Missing: 0 

Total  : 4898 

Minimum      : 3.80 

Lower Whisker: 4.80 

Lower Hinge  : 6.30 

Median       : 6.80 

Upper Hinge  : 7.30 

Upper Whisker: 8.80 

Maximum      : 14.20 

1st Quartile : 6.30 

3rd Quartile : 7.30 

IQR          : 1.00 

Number of outliers: 119 

Small: 3.8  3.9  4.2  4.2  4.4  4.4  4.4  4.5  4.6  4.7  4.7  4.7  4.7  4.7  

Large: 8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  ...  10.3  10.3  10.7  10.7  11.8  14.2  

> 

> y <- WW$fixed.acidity

> BoxPlot(y, col.stroke="red", horiz=TRUE,col.fill="grey",xlab="_Fixed_Acidity_",main="WW$Fixed.Acidity_119_Outliers_UpperWhisker-8.80_")

--- _Fixed_Acidity_, WW$Fixed.Acidity_119_Outliers_UpperWhisker-8.80_ --- 

Present: 4898 

Missing: 0 

Total  : 4898 

Minimum      : 3.80 

Lower Whisker: 4.80 

Lower Hinge  : 6.30 

Median       : 6.80 

Upper Hinge  : 7.30 

Upper Whisker: 8.80 

Maximum      : 14.20 

1st Quartile : 6.30 

3rd Quartile : 7.30 

IQR          : 1.00 

Number of outliers: 119 

Small: 3.8  3.9  4.2  4.2  4.4  4.4  4.4  4.5  4.6  4.7  4.7  4.7  4.7  4.7  

Large: 8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  8.9  ...  10.3  10.3  10.7  10.7  11.8  14.2  

> 

> WW$fixed.acidity<- ifelse(fixed.acidity>8.80,(WW$fixed.acidity=="9"),(WW$fixed.acidity<-WW$fixed.acidity ))

> summary(WW$fixed.acidity)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 

  0.000   6.300   6.800   6.657   7.300   8.800 

> 

> y <- WW$fixed.acidity

> BoxPlot(y, col.stroke="red", horiz=TRUE,col.fill="grey",xlab="_Fixed_Acidity_",main="WW$Fixed.Acidity_119_Outliers_UpperWhisker-8.80_")

--- _Fixed_Acidity_, WW$Fixed.Acidity_119_Outliers_UpperWhisker-8.80_ --- 

Present: 4898 

Missing: 0 

Total  : 4898 

Minimum      : 0.00 

Lower Whisker: 4.80 

Lower Hinge  : 6.30 

Median       : 6.80 

Upper Hinge  : 7.30 

Upper Whisker: 8.80 

Maximum      : 8.80 

1st Quartile : 6.30 

3rd Quartile : 7.30 

IQR          : 1.00 

Number of outliers: 119 

Small: 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ...  4.6  4.7  4.7  4.7  4.7  4.7  

Large:  none

> 

> y <- WW$fixed.acidity

> BoxPlot(y, col.stroke="red", horiz=TRUE,col.fill="grey",xlab="_Fixed_Acidity_",main="WW$Fixed.Acidity_119_Small-Outliers_UpperWhisker and MAX Value-8.80_")

--- _Fixed_Acidity_, WW$Fixed.Acidity_119_Small-Outliers_UpperWhisker and MAX Value-8.80_ --- 

Present: 4898 

Missing: 0 

Total  : 4898 

Minimum      : 0.00 

Lower Whisker: 4.80 

Lower Hinge  : 6.30 

Median       : 6.80 

Upper Hinge  : 7.30 

Upper Whisker: 8.80 

Maximum      : 8.80 

1st Quartile : 6.30 

3rd Quartile : 7.30 

IQR          : 1.00 

Number of outliers: 119 

Small: 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ...  4.6  4.7  4.7  4.7  4.7  4.7  

Large:  none