Spread by computer

You will write four functions that measure the spread of data: the mean deviation, the variance, the standard deviation and the variance of grouped data. The checker tests them on short lists, on a table with class marks, and on the difference between a whole group and a sample.

The idea. The mean deviation about a centre is the mean of the distances from it. The variance is the mean of the squared distances from the mean. For a sample, the sum of squared distances is divided by $n - 1$ instead of $n$. The standard deviation is the square root of the variance. For grouped data the class marks stand for the values.

Your tasks.

1. Finish mean_deviation(data, about).
2. Finish variance(data, sample=False). Divide by n - 1 when sample is True.
3. Finish std_dev(data, sample=False).
4. Finish grouped_variance(marks, freqs) with the shortcut: the mean of the squares minus the square of the mean.