variance & std deviation

[bluefrog]

I've come across two methods for calculating variance & std deviation.
When applying a single method to a set of test figures specific to each, neither comes out correct.
But when applying the test figures specific to each method, both come out correct.

The 2 methods are outlined as follows:
method 1: http://www.mathsisfun.com/data/standard-deviation.html
method 2: http://www.statisticshowto.com/how-to-find-the-sample-variance-and-standard-deviation-in-statistics/

Expected results are:
method 1: test{600,470,170,430, 300} variance = 21,704 std dev = 147.32..
method 2: test{3,21,98,203,17,9} variance = 6,219.9 std dev = 78.86..

Could somebody suggest which method to use?

code:

#include <iostream>     // std::cou
#include <algorithm>    // std::transform

double calc_sum_of_squares1(std::vector<double> & vld);
double calc_sum_of_squares2(std::vector<double> & vld);
int sq(int c, int d) {return c+d*d;}

int main () {
  std::vector<double> vld1 { 600,470,170,430, 300};
  std::vector<double> vld2 { 3, 21, 98, 203, 17, 9};

  double sum_of_squares = calc_sum_of_squares1(vld1); // http://www.mathsisfun.com/data/standard-deviation.html
  std::cout << "Variance 1: " << sum_of_squares/vld1.size() << " Std Deviation 1: " << sqrt(sum_of_squares/vld1.size()) << std::endl;

  sum_of_squares = calc_sum_of_squares2(vld2); // http://www.statisticshowto.com/how-to-find-the-sample-variance-and-standard-deviation-in-statistics/
  std::cout << "Variance 2: " << sum_of_squares/(vld2.size()-1) << " Std Deviation 2: " << sqrt(sum_of_squares/(vld2.size()-1)) << std::endl;

  return 0;
}

// http://www.mathsisfun.com/data/standard-deviation.html
double calc_sum_of_squares1(std::vector<double> & vld) {
  double mean = std::accumulate(begin(vld), end(vld), 0.0) / vld.size();
  std::vector<double> diff(vld.size());
  std::transform(begin(vld), end(vld), begin(diff), std::bind2nd(std::minus<double>(), mean));
  return std::inner_product(begin(diff), end(diff), begin(vld), 0.0);
}

// http://www.statisticshowto.com/how-to-find-the-sample-variance-and-standard-deviation-in-statistics/
double calc_sum_of_squares2(std::vector<double> & vld) {
  double sum = std::accumulate(begin(vld), end(vld), 0.0); // step1
  sum = (sum*sum) / vld.size(); // step 2
  sum = std::accumulate(begin(vld), end(vld), 0.0,  sq) - sum; // step3&4
  return sum; // step5&6 occur after return
}

[lastchance]

By the look of it, variance 1 is the actual SAMPLE variance (which involves dividing by N).
Variance 2 is an unbiased estimate of the POPULATION variance (which involves dividing by N-1).

This is a statistics issue - not a C++ one: it depends what you are trying to do. If you intend to do any of the standard statistical tests (normal distribution or t-test) you will probably be using the latter.

Either method of calculating the sum of squared deviations about the mean will do. It's whether you subsequently divide by N or N-1 that produces the difference. If your sample consisted of the whole population then you would divide by N. If, as is more usual, you have a representative sample of a much larger population then you would divide by N-1.

If you live in the United States then, as found out earlier this month, most statistical theory might as well be thrown out of the window ...

[bluefrog]

yes, you are correct. I've just re-verified the figures using https://www.easycalculation.com/statistics/standard-deviation.php and both methods are in fact correct, but the second method applies to sample and the first to the whole population.

Thanks