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Sampling and Empirical Distributions

An important part of data science consists of making conclusions based on the data in random samples. In order to correctly interpret their results, data scientists have to first understand exactly what random samples are.

In this chapter we will take a more careful look at sampling, with special attention to the properties of large random samples.

Let’s start by drawing some samples. Our examples are based on the top_movies.csv data set.

top1 = Table.read_table(path_data + 'top_movies.csv')
top2 = top1.with_column('Row Index', np.arange(top1.num_rows))
top = top2.move_to_start('Row Index')

top.set_format(make_array(3, 4), NumberFormatter)
Row Index Title Studio Gross Gross (Adjusted) Year
0 Star Wars: The Force Awakens Buena Vista (Disney) 906,723,418 906,723,400 2015
1 Avatar Fox 760,507,625 846,120,800 2009
2 Titanic Paramount 658,672,302 1,178,627,900 1997
3 Jurassic World Universal 652,270,625 687,728,000 2015
4 Marvel's The Avengers Buena Vista (Disney) 623,357,910 668,866,600 2012
5 The Dark Knight Warner Bros. 534,858,444 647,761,600 2008
6 Star Wars: Episode I - The Phantom Menace Fox 474,544,677 785,715,000 1999
7 Star Wars Fox 460,998,007 1,549,640,500 1977
8 Avengers: Age of Ultron Buena Vista (Disney) 459,005,868 465,684,200 2015
9 The Dark Knight Rises Warner Bros. 448,139,099 500,961,700 2012

... (190 rows omitted)

Sampling Rows of a Table

Each row of a data table represents an individual; in top, each individual is a movie. Sampling individuals can thus be achieved by sampling the rows of a table.

The contents of a row are the values of different variables measured on the same individual. So the contents of the sampled rows form samples of values of each of the variables.

Deterministic Samples

When you simply specify which elements of a set you want to choose, without any chances involved, you create a deterministic sample.

You have done this many times, for example by using take:

top.take(make_array(3, 18, 100))
Row Index Title Studio Gross Gross (Adjusted) Year
3 Jurassic World Universal 652,270,625 687,728,000 2015
18 Spider-Man Sony 403,706,375 604,517,300 2002
100 Gone with the Wind MGM 198,676,459 1,757,788,200 1939

You have also used where:

top.where('Title', are.containing('Harry Potter'))
Row Index Title Studio Gross Gross (Adjusted) Year
22 Harry Potter and the Deathly Hallows Part 2 Warner Bros. 381,011,219 417,512,200 2011
43 Harry Potter and the Sorcerer's Stone Warner Bros. 317,575,550 486,442,900 2001
54 Harry Potter and the Half-Blood Prince Warner Bros. 301,959,197 352,098,800 2009
59 Harry Potter and the Order of the Phoenix Warner Bros. 292,004,738 369,250,200 2007
62 Harry Potter and the Goblet of Fire Warner Bros. 290,013,036 393,024,800 2005
69 Harry Potter and the Chamber of Secrets Warner Bros. 261,988,482 390,768,100 2002
76 Harry Potter and the Prisoner of Azkaban Warner Bros. 249,541,069 349,598,600 2004

While these are samples, they are not random samples. They don’t involve chance.

Probability Samples

For describing random samples, some terminology will be helpful.

A population is the set of all elements from whom a sample will be drawn.

A probability sample is one for which it is possible to calculate, before the sample is drawn, the chance with which any subset of elements will enter the sample.

In a probability sample, all elements need not have the same chance of being chosen.

A Random Sampling Scheme

For example, suppose you choose two people from a population that consists of three people A, B, and C, according to the following scheme:

  • Person A is chosen with probability 1.
  • One of Persons B or C is chosen according to the toss of a coin: if the coin lands heads, you choose B, and if it lands tails you choose C.

This is a probability sample of size 2. Here are the chances of entry for all non-empty subsets:

A: 1 
B: 1/2
C: 1/2
AB: 1/2
AC: 1/2
BC: 0
ABC: 0

Person A has a higher chance of being selected than Persons B or C; indeed, Person A is certain to be selected. Since these differences are known and quantified, they can be taken into account when working with the sample.

A Systematic Sample

Imagine all the elements of the population listed in a sequence. One method of sampling starts by choosing a random position early in the list, and then evenly spaced positions after that. The sample consists of the elements in those positions. Such a sample is called a systematic sample.

Here we will choose a systematic sample of the rows of top. We will start by picking one of the first 10 rows at random, and then we will pick every 10th row after that.

"""Choose a random start among rows 0 through 9;
then take every 10th row."""

start = np.random.choice(np.arange(10))
top.take(np.arange(start, top.num_rows, 10))
Row Index Title Studio Gross Gross (Adjusted) Year
2 Titanic Paramount 658,672,302 1,178,627,900 1997
12 The Hunger Games: Catching Fire Lionsgate 424,668,047 444,697,400 2013
22 Harry Potter and the Deathly Hallows Part 2 Warner Bros. 381,011,219 417,512,200 2011
32 American Sniper Warner Bros. 350,126,372 374,796,000 2014
42 Iron Man Paramount 318,412,101 385,808,100 2008
52 Skyfall Sony 304,360,277 329,225,400 2012
62 Harry Potter and the Goblet of Fire Warner Bros. 290,013,036 393,024,800 2005
72 Jaws Universal 260,000,000 1,114,285,700 1975
82 Twister Warner Bros. 241,721,524 475,786,700 1996
92 Ghost Paramount 217,631,306 447,747,400 1990

... (10 rows omitted)

Run the cell a few times to see how the output varies.

This systematic sample is a probability sample. In this scheme, all rows have chance $1/10$ of being chosen. For example, Row 23 is chosen if and only if Row 3 is chosen, and the chance of that is $1/10$.

But not all subsets have the same chance of being chosen. Because the selected rows are evenly spaced, most subsets of rows have no chance of being chosen. The only subsets that are possible are those that consist of rows all separated by multiples of 10. Any of those subsets is selected with chance 1/10. Other subsets, like the subset containing the first 11 rows of the table, are selected with chance 0.

Random Samples Drawn With or Without Replacement

In this course, we will mostly deal with the two most straightforward methods of sampling.

The first is random sampling with replacement, which (as we have seen earlier) is the default behavior of np.random.choice when it samples from an array.

The other, called a “simple random sample”, is a sample drawn at random without replacement. Sampled individuals are not replaced in the population before the next individual is drawn. This is the kind of sampling that happens when you deal a hand from a deck of cards, for example.

In this chapter, we will use simulation to study the behavior of large samples drawn at random with or without replacement.

Drawing a random sample requires care and precision. It is not haphazard, even though that is a colloquial meaning of the word “random”. If you stand at a street corner and take as your sample the first ten people who pass by, you might think you’re sampling at random because you didn’t choose who walked by. But it’s not a random sample – it’s a sample of convenience. You didn’t know ahead of time the probability of each person entering the sample; perhaps you hadn’t even specified exactly who was in the population.