deviation measures (the standard deviation and the variance)

We have seen that measures of central tendency like the mean can describe how a set of data is typical compared to other sets.
In the same way, the variance can describe the spread of a set of data.
Let’s say we have this set as example: A=[1, 3, 29] and B=[10,11,12] both with mean = 11.
These are the deviation measures:

  • deviation from the mean is the difference between the mean and a given data point.

    deviation(i) = \left | x_{i} - \mu \right |

    For the set A, they are respectively 10, 8 and 18.

  • Variance is the mean square deviation, i.e. the sum of all the deviations from the mean, squared, and divided by the number of data points:

\sigma^{2} = \sum (x_{i} - \mu )^{^{2}}\frac{1}{n}

  • As you can see, the variance is hard to interpret, being its unit a squared, therefore the standard deviation has been introduced, that is just the square root of the variance.

    \sigma = \sqrt{\sum (x_{i} - \mu )^{^{2}}\frac{1}{n}}

    For set A it would be 12.75; the high standard deviation shows that the set is quite dispersed (in this case due to the number 29).

Let’s see how to calculate the standard deviation in Python, given a list of values:

def stdDev(X):
  X: a list of values 
  returns: float, the standard deviation of the input, 
  tot = 0.0 
  meanX = mean(X) 
  for x in X: 
    tot += (x - meanX) ** 2 
  return (tot/len(X))**0.5

The operator ** is the power, so **0.5 means doing the square root.
The function mean() was previously defined.

2 thoughts on “deviation measures (the standard deviation and the variance)

  1. Pingback: Quartiles and summary statistics in Python | Look back in respect

  2. Pingback: The coefficient of variation | Look back in respect

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