Compute the first four sample moments about the mean (and the standard deviation) iteratively from a stream of samples

moments.awk

BEGIN { OFS = "\t" }

{
  accs[0] += 1
  accs[1] += $1
  accs[2] += (val2 = $1   * $1)
  accs[3] += (val3 = val2 * $1)
  accs[4] += (val4 = val3 * $1)

  mean  = accs[1] / accs[0]
  mean2 = mean    * mean
  mean3 = mean2   * mean
  var  = (accs[2] - accs[1] * mean) / accs[0]
  std  = sqrt(var)

  if (var > 0) {
    skew = (     accs[3] \
           - 3 * accs[2] * mean \
           + 2 * accs[1] * mean2 \
           ) / (var * std * accs[0])
    kurt = (     accs[4] \
           - 4 * accs[3] * mean \
           + 6 * accs[2] * mean2 \
           - 3 * accs[1] * mean3 \
           ) / (var * var * accs[0])
  } else {
    skew = kurt = ""
  }

  print $1, mean, std, var, skew, kurt
}