eyeprolog

Statistics summary

What are the mean, variance, and standard deviation of a small sample?

Descriptive statistics are usually a single library call. Here they are a few recursive rules, so each number in the result comes with a derivation you can read: the sum, the mean, every squared deviation, and the final square root.

Run it in the playground · Source

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The question

For the textbook sample 2, 4, 4, 4, 5, 5, 7, 9, compute:

• the count of values;
• the mean, the sum divided by the count;
• the population variance, the average squared distance from the mean;
• the population standard deviation, the square root of the variance.

The expected answers are 8, 5, 4, and 2, which makes the sample a favorite for checking statistics code.

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The data and the folds

sample(scores, [2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0]).

sum([], 0.0).
sum([X|Xs], Total) :-
  sum(Xs, Rest),
  (Total is X + Rest).

squared_error_sum([], _mean, 0.0).
squared_error_sum([X|Xs], Mean, Total) :-
  (Delta is X - Mean),
  (Squared is Delta ** 2.0),
  squared_error_sum(Xs, Mean, Rest),
  (Total is Squared + Rest).

Both are folds: walk the list, combine each element with the result for the rest.

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The statistics

population_variance(Name, Variance) :-
  sample(Name, Values),
  mean(Name, Mean),
  squared_error_sum(Values, Mean, Sumsquarederrors),
  length(Values, Count),
  (Variance is Sumsquarederrors / Count).

population_stddev(Name, Stddev) :-
  population_variance(Name, Variance),
  (Stddev is Variance ** 0.5).

“Population” means dividing by n, not n − 1: the sample is treated as the whole population.

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The answer

eyeprolog examples/statistics-summary.pl

mean(scores, 5.0).
count(scores, 8).
populationVariance(scores, 4.0).
populationStddev(scores, 2.0).

The four expected values, computed in floating point.

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The reason

eyeprolog --proof examples/statistics-summary.pl

step(population_variance(scores, 4.0),
     rule(7),
     ['Name' = scores,
      'Variance' = 4.0,
      'Values' = [2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0],
      'Mean' = 5.0,
      'Sumsquarederrors' = 32.0,
      'Count' = 8],
     [sample(scores, [2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0]),
      mean(scores, 5.0),
      squared_error_sum([2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0], 5.0, 32.0),
      length([2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0], 8),
      4.0 is 32.0 / 8]).

The variance is 4 because the squared deviations add up to 32 and there are 8 values. Each intermediate result has its own step further down the proof.

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The check

condition('C1', resolution, ok, 25).
condition('C5', re_decision, ok, 29).
steps(54).
verified(25).
recomputed(29).
trusted(0).
verdict(checked).

The checker replays the 25 rule steps and independently recomputes the 29 arithmetic steps, from each is/2 addition to the final ** 0.5. Nothing is trusted.

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Try it

• Add a ninth value 5.0 to the sample. The mean stays 5.0; predict whether the variance goes up or down. (It drops to about 3.56.)
• Add sample_variance/2, which divides by Count - 1, and compare it with the population variance.
• Replace the floats with integers such as 2 and 4. Which results stay integers, and which become floats?

Learn more in The Art of EyeProlog, Chapters 6 (arithmetic) and 29 (numerical models).

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