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

Statistics functions mean and standard deviation are used so we load PGstatisticsmacros.pl.

for $i (0 .. 7) {
    $x[$i] = random(1, 10);
}

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

Setup

First, generate random numbers and then calculate the mean and standard deviation using the methods stats_mean and stats_sd from the macro PGstatisticsmacros.pl.

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

a) Mean = [_____]{$mean}

b) Standard Deviation = [___]{$sd}
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
For the mean, use the formula

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

For the standard deviation, first, find the variance

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

and taking the square root

[``s = [$sd]``]
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

We use the method stats_SX_SXX of the PGstatisticsmacros.pl which returns the sum and sum of squares which aids in the solution.