Below is the Powershell Script in order to update the site coulmn when used in multiple content types. Hope this might be helpful to some one.
function Update-CalcField([string]$url, [string]$Field, [string]$OutputType, [string]$Formula) {
Add-PSSnapin Microsoft.SharePoint.PowerShell -erroraction SilentlyContinue
Start-SPAssignment -Global
$SPSite = New-Object Microsoft.SharePoint.SPSite($url)
$OpenWeb = $SPsite.OpenWeb()
$OpenWeb.Fields[$Field].OutputType = $OutputType
$OpenWeb.Fields[$Field].Formula = $Formula
$OpenWeb.Fields[$Field].Update($true)
$OpenWeb.Dispose()
$SPSite.Dispose()
}
#using the above function
Update-CalcField http://myserverapp.testserver.com/sites/test -Field MyCalc -OutputType Text -Formula '=IF(ISBLANK([title]),"Null","Completed")'