Mean and Standard Deviation

Find the mean and standard deviation of a list of numbers.

Complete Code

POD for Macro Files

PGstatisticsmacros.pl

PG problem file

Explanation

DOCUMENT();

loadMacros("PGstandard.pl", "PGML.pl", 'PGstatisticsmacros.pl', "PGcourse.pl");

Preamble

The PGstatisticsmacros.pl macro provides the stats_mean, stats_sd, and stats_SX_SXX methods.

@x = map { random(1, 10) } 0 .. 7;

$mean = stats_mean(@x);
$sd   = stats_sd(@x);

Setup

Generate random numbers and then calculate the mean and standard deviation using the stats_mean and stats_sd methods from PGstatisticsmacros.pl.

BEGIN_PGML
Find the mean and standard deviation of the following list of numbers:
[@ join(', ', @x) @]

a) Mean: [_]{$mean}{8}

b) Standard Deviation: [_]{$sd}{8}
END_PGML

Statement

This is the problem statement in PGML.

($sum_x, $sum_sq) = stats_SX_SXX(@x);
$var = $sum_sq - ($sum_x**2) / 8;

BEGIN_PGML_SOLUTION
The mean is

[``\bar{x} = \frac{1}{n} \sum_{i = 1}^n x_i = \frac{[$sum_x]}{8} = [$mean]``]

For the standard deviation, first, find the variance.

[``
    \begin{aligned}
    s^2 &= \frac{1}{n - 1} \left(
        \sum_{i = 1}^n x_i^2 - \frac{1}{n}\left(\sum_{i = 1}^n x_i\right)^2
    \right) \\
    &= \frac{1}{7} \left( [$sum_sq] - \frac{1}{8} ([$sum_x])^2\right) \\
    &= \frac{[$var]}{7}
    \end{aligned}
``]

Taking the square root gives [`s \approx [$sd]`]
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

The stats_SX_SXX method from PGstatisticsmacros.pl returns the sum and sum of squares which are used in the solution.

Note that when a long list of equalities are strung together in an equation it is a good idea to use the aligned environment to break the equalities onto separate lines. If left on a single line, the equation may extend outside of the solution container and look quite bad. Particularly on narrow screens.