McKay's approximation for the coefficient of variation

In statistics, McKay's approximation of the coefficient of variation is a statistic based on a sample from a normally distributed population. It was introduced in 1932 by A. T. McKay. Statistical methods for the coefficient of variation often utilizes McKay's approximation.

Let

      x
      
        i
      
    
  

{\displaystyle x_{i}}

,

    i
    =
    1
    ,
    2
    ,
    …
    ,
    n
  

{\displaystyle i=1,2,\ldots ,n}

be

    n
  

{\displaystyle n}

independent observations from a

    N
    (
    μ
    ,
    
      σ
      
        2
      
    
    )
  

{\displaystyle N(\mu ,\sigma ^{2})}

normal distribution. The population coefficient of variation is

      c
      
        v
      
    
    =
    σ
    
      /
    
    μ
  

{\displaystyle c_{v}=\sigma /\mu }

. Let

          x
          ¯
        
      
    
  

{\displaystyle {\bar {x}}}

and

    s
    
  

{\displaystyle s\,}

denote the sample mean and the sample standard deviation, respectively. Then

            c
            ^
          
        
      
      
        v
      
    
    =
    s
    
      /
    
    
      
        
          x
          ¯
        
      
    
  

{\displaystyle {\hat {c}}_{v}=s/{\bar {x}}}

is the sample coefficient of variation. McKay's approximation is

K =

      (
      
        1
        +
        
          
            1
            
              c
              
                v
              
              
                2
              
            
          
        
      
      )
    
     
    
      
        
          (
          n
          −
          1
          )
           
          
            
              
                
                  c
                  ^
                
              
            
            
              v
            
            
              2
            
          
        
        
          1
          +
          (
          n
          −
          1
          )
           
          
            
              
                
                  c
                  ^
                
              
            
            
              v
            
            
              2
            
          
          
            /
          
          n
        
      
    
  

{\displaystyle K=\left(1+{\frac {1}{c_{v}^{2}}}\right)\ {\frac {(n-1)\ {\hat {c}}_{v}^{2}}{1+(n-1)\ {\hat {c}}_{v}^{2}/n}}}

Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When

      c
      
        v
      
    
  

{\displaystyle c_{v}}

is smaller than 1/3, then

    K
  

{\displaystyle K}

is approximately chi-square distributed with

    n
    −
    1
  

{\displaystyle n-1}

degrees of freedom. In the original article by McKay, the expression for

    K
  

{\displaystyle K}

looks slightly different, since McKay defined

      σ
      
        2
      
    
  

{\displaystyle \sigma ^{2}}

with denominator

    n
  

{\displaystyle n}

instead of

    n
    −
    1
  

{\displaystyle n-1}

. McKay's approximation,

    K
  

{\displaystyle K}

, for the coefficient of variation is approximately chi-square distributed, but exactly noncentral beta distributed .