Numerical Descriptive Measures

\(\Large \sf To \ clean \ the \ environment \ variables\)

rm(list=ls(all=T))

\(\Large \sf Defining \ a \ Vector\)

vec <- c(20,34,56,23,45,22,60,23,56,78,23,45)

\(\Large \sf Mean \ of \ the \ vector\)

\(\frac{\Large\sum_{\Large i=1}^{\Large n}{\Large x_i}}{\Large n}\)

mean(vec)

## [1] 40.41667

\(\Large \sf Median\ of \ the \ vector\)

\(\Large \frac{(n+1)}{2} \normalsize \sf \ ranked \ value\)

median(vec)

## [1] 39.5

• For symmetrical distributed data mean, median and mode are almost equal in value
• For asymmetrical distributed data, following relationship holds good approximately
• Mode = 3 * Median - 2 * mean (or)
• Mean - Mode = 3 * (Mean - Median)
• above realtion is called as empirical relation. Using this if two measures are known, it is easy to find out the third measure

\(\Large \sf Harmonic\ Mean\ of\ a\ vector\)

\(\Large\frac {\Large n}{\Large \sum_{\Large i=1}^{\Large n}\Large \frac{\Large 1}{\Large x_i}}\)

hm_ <- function(vcetor) {
  return(length(vec)/sum(1/vec))
}

hm_(vec)

## [1] 32.88149

\(\Large \sf Geometric\ Mean\ of\ a\ vector\)

\(\large \bar X_g = (X_1*X_2*X_3 * \ .... \ X_n )^\frac{1}{n}\)

gm_ <- function(vcetor) {
  return(prod(vcetor)^(1/length(vcetor)))
}

gm_(vec)

## [1] 36.39962

\(\sf \huge Variation\ and\ shape\)

\(\sf \Large Range\ of\ a\ vector\)

\(\sf \normalsize Range = (\large x_{max}-\large x_{min})\)

range_ <- function(vcetor) {
  return(max(vcetor)-min(vcetor))
}

range_(vec)

## [1] 58

\(\sf \Large Variance\ of\ a\ vector\)

\(\Large \sigma^2 \small =\Large \frac{\sum_{i=1}^{n} (x_i-\bar X)^2}{n}\)

variance_ <- function(vcetor){
    var_ <-  0
    for (ele in 1:length(vcetor)){
      var_ = var_ + (vcetor[ele] - mean(vcetor))^2
    }
    return (var_/length(vcetor)-1)
}

variance_(vec)

## [1] 334.9097

\(\sf \Large Standard\ deviation\ of\ a\ vector\)

\(\Large \sigma \small =\sqrt{\Large \frac{\sum_{i=1}^{n} (x_i-\bar X)^2}{n}}\)

stddev_ <- function(vcetor){
  return (variance_(vcetor) ^ 0.5)
}

stddev_(vec)

## [1] 18.30054

\(\sf \Large Coefficient\ of\ Variation\)

It measures the scatter in the data with respect to the mean

\(\Large (\frac {\sigma}{\mu})*100\)

CoeffVar_ <- function(vcetor){
  return (stddev_(vcetor)/mean(vcetor)) *100
}

CoeffVar_(vec)

## [1] 0.4527968

\(\sf \Large Z-Score\)

\(\Large Z = \Large(\frac{x-\mu}{\sigma})\)

• a Z score of 0 indicates that the value is same as the mean
• Helps in identifying outliers, less than -3 and greater than +3 are considered to be outliers

zscore <- function(vcetor){
  temp_ <- c()
  for (ele in 1:length(vcetor)){
    temp_ <- append(temp_, (vcetor[ele]-mean(vcetor))/stddev_(vcetor))
  }
  return (temp_)
}

zscore(vec)

##  [1] -1.1156320 -0.3506272  0.8515232 -0.9517024  0.2504480 -1.0063456
##  [7]  1.0700960 -0.9517024  0.8515232  2.0536736 -0.9517024  0.2504480

\(\Large Skewness\)

• Mean < Median —-> left skewed or negative skew
• Mean = Median —-> symmetrical distribution (zero skewness)
• Mean > Median —-> right skewed or positive skew

\(\Large Kurtosis\)

• A distribution that has a sharper-rising center peak than the peak of a normal distribution has positive kurtosis, a kurtosis value that is greater than zero, and is called lepokurtic
• A distribution that has a slower-rising (flatter) center peak than the peak of a normal distribution has negative kurtosis, a kurtosis value that is less than zero, and is called platykurtic
• A lepokurtic distribution has a higher concentration of values near the mean of the distribution compared to a normal distribution, while a platykurtic distribution has a lower concentration compared to a normal distribution
  
  

\(\Large Quartiles\)

\(\normalsize \sf Quartiles\ split\ the\ data\ into\ 4\ equal\ parts\)

\(\normalsize Q1 = \frac {n+1}{4} \ \ \ \ \ \ \ \ Q2 = 2(\frac {n+1}{4})\)
\(\normalsize Q3 = 3(\frac {n+1}{4}) \ \ \ \ Q4 = 4(\frac {n+1}{4})\)

\(\Large Percentile\)

\(\normalsize \sf \text{Percentile divides the data into 100 equal parts}\)

\(\Large \sf \text{The Interquartile Range (IQR)}\)

\(\sf \normalsize \text{The interquartile range is the difference between third quartile }\textbf{(Q3) }\text{and first quartile }\textbf{(Q1)}\)

\(\sf \ \ \ \ \ \ \ \ \ \ \ \textbf{IQR} = \textbf{Q3 - Q1}\)

\(\Large \sf \text{The Empirical rule}\)

\(\sf \normalsize \text{The Empirical rule states that :}\)

\(\ \ \ \ \ \sf \normalsize \text{In a Normal distribution,}\)

• Approximately, 68% of the values are within \(\bf{\pm1\sigma}\)
• Approximately, 95% of the values are within \(\bf{\pm2\sigma}\)
• Approximately, 99.7% of the values are within \(\bf{\pm3\sigma}\)
  
  

\(\sf \Large \text{Chebyshev's theorem}\)

\(\sf \normalsize \text{For heavily skewed datasets that do not appear to be normally distributed, you should use chebyshev's theorem:}\)

\(\sf \normalsize \text{Regardless of the shape, the percentage of values that are found within distances of k standard deviations from the mean must be at least}\)

\(\Large (1-\frac{1}{k^2}) * 100\)

\(\sf \LARGE \text{The Covariance and the Coefficient of Correlation}\)

\(\large \sf \text{The Covariance}\)

\(\sf \normalsize \text{It measures the strength of a linear relationship between two numerical variables}\)

\(Sample,\large \sf \ cov(X,Y) = \frac{\sum_{i=1}^{n}(X_i-\bar X)(Y_i-\bar Y)}{n-1}\)

\(Population,\large \sf \ cov(X,Y) = \frac{\sum_{i=1}^{n}(X_i-\bar X)(Y_i-\bar Y)}{n}\)

vec1 <- c(23,45,34,23,34,56,78,65,45,34)
vec2 <- c(65,54,34,45,23,45,67,88,96,33)

cov_ <- function(vcetor_x,vcetor_y,sample){
  numerator <- 0
  for (i in 1:length(vcetor_x)){
    numerator <- numerator + (vcetor_x[i]-mean(vcetor_x))*(vcetor_y[i]-mean(vcetor_y))
  }
  if (sample==T){
    return (numerator/(length(vcetor_x)-1)) 
  }else if (sample==F){
    return (numerator/length(vcetor_x))
  }else{
    return ('Check your parameters !!!')
  }
}

cov_(vec1,vec2,T) # sample Variance

## [1] 196.7778

cov_(vec1,vec2,F) # Population Variance

## [1] 177.1

\(\sf \large \text{The Coefficient of Correlation}\)

\(\sf \text{It measures the relative strength of a linear relationship between two numerical variables}\)

• Values ranges between -1 to +1
• -1, perfect negative correlation
• +1, perfect positive correlation

\(\large sample\ correlation,\ \ \bf r = \frac {Cov(X,Y)}{S_XS_Y}\)

vec1 <- c(23,45,34,23,34,56,78,65,45,34)
vec2 <- c(65,54,34,45,23,45,67,88,96,33)

corr_ <- function(vcetor_x,vcetor_y){
  return(cov_(vcetor_x,vcetor_y,F)/(stddev_(vcetor_x)*stddev_(vcetor_y)))
}

corr_(vec1,vec2)

## [1] 0.4569768