text: import numpy as np
  from scipy import stats
  import scipy
  import math
  
  # if data is not a frequency distribution
  x = np.array([1,2,3,4,5])
  mean = np.mean(x)
  n = x.size
  x_minus_mean= x-mean
  x_minus_mean_square = np.square(x_minus_mean)
  x_minus_mean_square_sum = np.sum(x_minus_mean_square)
  
  standard_deviation = math.sqrt(x_minus_mean_square_sum/n)
  print("standard_deviation 1 : ",standard_deviation)
  
  # if dataset is frequency distribution
  x_square = np.square(x)
  f = np.array([6,16,24,25,19])
  cum_freq = np.array([6,22,46,71,90])
  f_x_square = f*x_square
  
  sum_f_x_square = np.sum(f_x_square)
  sum_f = np.sum(f)
  mean_square = math.pow(mean,2)
  standard_deviation = math.sqrt((sum_f_x_square/sum_f) - mean_square)
  
  print("standard_deviation 2 : ",standard_deviation)