Simulates doing a thousand polls at a 1.5k sample size from a population of 256 million. Most any poll was off was by 4.5%, while average variance was just 1.19%.

how-effective-sampling-is.py

import random

american_adult_population = int(245e6)
 # Warning: This array is HUGE!
 # The array is just the numbers from 0 to 245,000,000
usa_array = range(american_adult_population)
actual_average = float(american_adult_population / 2)

sample_amount = 1500

def make_sample(population, k=sample_amount):
    """
    Given a population (and an optional sample amount (k)), return a tuple of:
        - the average of the same
        - the absolute difference between the sample average and real average
        - the percent difference between the sample average and real average
    """
    sampling = random.sample(usa_array, k)
    avg = average(sampling)
    return (
        int(avg),
        abs(avg - actual_average),
        round(abs((avg - actual_average)/actual_average) * 100, 4),
    )

def average(nums):
    total = 0
    for num in nums:
        total += num
    return float(total) / len(nums)


samples = [make_sample(usa_array) for _ in range(1000)]

if __name__ == "__main__":
    percents_sample_is_off_by = [p for _, _, p in samples]
    print (
        max(percents_sample_is_off_by),
        average(percents_sample_is_off_by)
    )