Code
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import seaborn as sns
sns.set_theme(style='darkgrid', font_scale = 1.5,
rc={'figure.figsize':(7,5)})
rng = np.random.default_rng()9 Sampling
In data science, understanding characteristics of a population starts with having quality data to investigate. While it is often impossible to collect all the data describing a population, we can overcome this by properly sampling from the population. In this note, we will discuss appropriate techniques for sampling from populations.
9.1 Censuses and Surveys
In general: a census is “a complete count or survey of a population, typically recording various details of individuals.” An example is the U.S. Decennial Census which was held in April 2020. It counts every person living in all 50 states, DC, and US territories, not just citizens. Participation is required by law (it is mandated by the U.S. Constitution). Important uses include the allocation of Federal funds, congressional representation, and drawing congressional and state legislative districts. The census is composed of a survey mailed to different housing addresses in the United States.
A survey is a set of questions. An example is workers sampling individuals and households. What is asked and how it is asked can affect how the respondent answers or even whether or not they answer in the first place.
While censuses are great, it is often very difficult and expensive to survey everyone in a population. Imagine the amount of resources, money, time, and energy the U.S. spent on the 2020 Census. While this does give us more accurate information about the population, it’s often infeasible to execute. Thus, we usually survey a subset of the population instead.
A sample is (usually) a subset of the population that is often used to make inferences about the population. If our sample is a good representation of our population, then we can use it to glean useful information at a lower cost. That being said, how the sample is drawn will affect the reliability of such inferences. Two common sources of error in sampling are chance error, where random samples can vary from what is expected in any direction, and bias, which is a systematic error in one direction. Biases can be the result of many things, for example, our sampling scheme or survey methods.
Let’s define some useful vocabulary:
- Population: The group that you want to learn something about.
- Individuals in a population are not always people. Other populations include bacteria in your gut (sampled using DNA sequencing), trees of a certain species, small businesses receiving a microloan, or published results in an academic journal or field.
- Sampling Frame: The list from which the sample is drawn.
- For example, if sampling people, then the sampling frame is the set of all people that could possibly end up in your sample.
- Sample: Who you actually end up sampling. The sample is therefore a subset of your sampling frame.
While ideally, these three sets would be exactly the same, they usually aren’t in practice. For example, there may be individuals in your sampling frame (and hence, your sample) that are not in your population. And generally, sample sizes are much smaller than population sizes.
9.2 Bias: A Case Study
The following case study is adapted from Statistics by Freedman, Pisani, and Purves, W.W. Norton NY, 1978.
In 1936, President Franklin D. Roosevelt (Democratic) went up for re-election against Alf Landon (Republican). As is usual, polls were conducted in the months leading up to the election to try and predict the outcome. The Literary Digest was a magazine that had successfully predicted the outcome of 5 general elections coming into 1936. In their polling for the 1936 election, they sent out their survey to 10 million individuals whom they found from phone books, lists of magazine subscribers, and lists of country club members. Of the roughly 2.4 million people who filled out the survey, only 43% reported they would vote for Roosevelt; thus, the Digest predicted that Landon would win.
On election day, Roosevelt won in a landslide, winning 61% of the popular vote of about 45 million voters. How could the Digest have been so wrong with their polling?
It turns out that the Literary Digest sample was not representative of the population. Their sampling frame of people found in phone books, lists of magazine subscribers, and lists of country club members were more affluent and tended to vote Republican. As such, their sampling frame was inherently skewed in Landon’s favor. The Literary Digest completely overlooked the lion’s share of voters who were still suffering through the Great Depression. Furthermore, they had a dismal response rate (about 24%); who knows how the other non-respondents would have polled? The Digest folded just 18 months after this disaster.
At the same time, George Gallup, a rising statistician, also made predictions about the 1936 elections. Despite having a smaller sample size of “only” 50,000 (this is still more than necessary; more when we cover the Central Limit Theorem), his estimate that 56% of voters would choose Roosevelt was much closer to the actual result (61%). Gallup also predicted the Digest’s prediction within 1% with a sample size of only 3000 people by anticipating the Digest’s affluent sampling frame and subsampling those individuals.
So what’s the moral of the story? Samples, while convenient, are subject to chance error and bias. Election polling, in particular, can involve many sources of bias. To name a few:
- Selection bias systematically excludes (or favors) particular groups.
- Example: the Literary Digest poll excludes people not in phone books.
- How to avoid: Examine the sampling frame and the method of sampling.
- Response bias occurs because people don’t always respond truthfully. Survey designers pay special detail to the nature and wording of questions to avoid this type of bias.
- Example: Illegal immigrants might not answer truthfully when asked citizenship questions on the census survey.
- How to avoid: Examine the nature of questions and the method of surveying. Randomized response - flip a coin and answer yes if heads or answer truthfully if tails.
- Non-response bias occurs because people don’t always respond to survey requests, which can skew responses.
- Example: Only 2.4m out of 10m people responded to the Literary Digest’s poll.
- How to avoid: Keep surveys short, and be persistent.
Randomized Response
Suppose you want to ask someone a sensitive question: “Have you ever cheated on an exam?” An individual may be embarrassed or afraid to answer truthfully and might lie or not answer the question. One solution is to leverage a randomized response:
First, you can ask the individual to secretly flip a fair coin; you (the surveyor) don’t know the outcome of the coin flip.
Then, you ask them to answer “Yes” if the coin landed heads and to answer truthfully if the coin landed tails.
The surveyor doesn’t know if the “Yes” means that the person cheated or if it means that the coin landed heads. The individual’s sensitive information remains secret. However, if the response is “No”, then the surveyor knows the individual didn’t cheat. We assume the individual is comfortable revealing this information.
Generally, we can assume that the coin lands heads 50% of the time, masking the remaining 50% of the “No” answers. We can therefore double the proportion of “No” answers to estimate the true fraction of “No” answers.
Election Polls
Today, the Gallup Poll is one of the leading polls for election results. The many sources of biases – who responds to polls? Do voters tell the truth? How can we predict turnout? – still remain, but the Gallup Poll uses several tactics to mitigate them. Within their sampling frame of “civilian, non-institutionalized population” of adults in telephone households in continental U.S., they use random digit dialing to include both listed/unlisted phone numbers and to avoid selection bias. Additionally, they use a within-household selection process to randomly select households with one or more adults. If no one answers, re-call multiple times to avoid non-response bias.
9.3 Probability Samples
When sampling, it is essential to focus on the quality of the sample rather than the quantity of the sample. A huge sample size does not fix a bad sampling method. Our main goal is to gather a sample that is representative of the population it came from. In this section, we’ll explore the different types of sampling and their pros and cons.
A convenience sample is whatever you can get ahold of; this type of sampling is non-random. Note that haphazard sampling is not necessarily random sampling; there are many potential sources of bias.
In a probability sample, we provide the chance that any specified set of individuals will be in the sample (individuals in the population can have different chances of being selected; they don’t all have to be uniform), and we sample at random based off this known chance. For this reason, probability samples are also called random samples. The randomness provides a few benefits:
- Because we know the source probabilities, we can measure the errors.
- Sampling at random gives us a more representative sample of the population, which reduces bias. (Note: this is only the case when the probability distribution we’re sampling from is accurate. Random samples using “bad” or inaccurate distributions can produce biased estimates of population quantities.)
- Probability samples allow us to estimate the bias and chance error, which helps us quantify uncertainty (more in a future lecture).
The real world is usually more complicated, and we often don’t know the initial probabilities. For example, we do not generally know the probability that a given bacterium is in a microbiome sample or whether people will answer when Gallup calls landlines. That being said, still we try to model probability sampling to the best of our ability even when the sampling or measurement process is not fully under our control.
A few common random sampling schemes:
- A uniform random sample with replacement is a sample drawn uniformly at random with replacement.
- Random doesn’t always mean “uniformly at random,” but in this specific context, it does.
- Some individuals in the population might get picked more than once.
- A simple random sample (SRS) is a sample drawn uniformly at random without replacement.
- Every individual (and subset of individuals) has the same chance of being selected from the sampling frame.
- Every pair has the same chance as every other pair.
- Every triple has the same chance as every other triple.
- And so on.
- A stratified random sample, where random sampling is performed on strata (specific groups), and the groups together compose a sample.
9.3.1 Example Scheme 1: Probability Sample
Suppose we have 3 TA’s (Arman, Boyu, Charlie): I decide to sample 2 of them as follows:
- I choose A with probability 1.0
- I choose either B or C, each with a probability of 0.5.
We can list all the possible outcomes and their respective probabilities in a table:
| Outcome | Probability |
|---|---|
| {A, B} | 0.5 |
| {A, C} | 0.5 |
| {B, C} | 0 |
This is a probability sample (though not a great one). Of the 3 people in my population, I know the chance of getting each subset. Suppose I’m measuring the average distance TAs live from campus.
- This scheme does not see the entire population!
- My estimate using the single sample I take has some chance error depending on if I see AB or AC.
- This scheme is biased towards A’s response.
9.3.2 Example Scheme 2: Simple Random Sample
Consider the following sampling scheme:
- A class roster has 1100 students listed alphabetically.
- Pick one of the first 10 students on the list at random (e.g. Student 8).
- To create your sample, take that student and every 10th student listed after that (e.g. Students 8, 18, 28, 38, etc.).
Is this a probability sample?
Yes. For a sample [n, n + 10, n + 20, …, n + 1090], where 1 <= n <= 10, the probability of that sample is 1/10. Otherwise, the probability is 0.
Only 10 possible samples!Does each student have the same probability of being selected?
Yes. Each student is chosen with a probability of 1/10.Is this a simple random sample?
No. The chance of selecting (8, 18) is 1/10; the chance of selecting (8, 9) is 0.9.3.3 Demo: Barbie v. Oppenheimer
We are trying to collect a sample from Berkeley residents to predict the which one of Barbie and Oppenheimer would perform better on their opening day, July 21st.
First, let’s grab a dataset that has every single resident in Berkeley (this is a fake dataset) and which movie they actually watched on July 21st.
Let’s load in the movie.csv table. We can assume that:
is_maleis a boolean that indicates if a resident identifies as male.- There are only two movies they can watch on July 21st: Barbie and Oppenheimer.
- Every resident watches a movie (either Barbie or Oppenheimer) on July 21st.
movie = pd.read_csv("data/movie.csv")
# create a 1/0 int that indicates Barbie vote
movie['barbie'] = (movie['movie'] == 'Barbie').astype(int)
movie.head()| age | is_male | movie | barbie | |
|---|---|---|---|---|
| 0 | 35 | False | Barbie | 1 |
| 1 | 42 | True | Oppenheimer | 0 |
| 2 | 55 | False | Barbie | 1 |
| 3 | 77 | True | Oppenheimer | 0 |
| 4 | 31 | False | Barbie | 1 |
What fraction of Berkeley residents chose Barbie?
actual_barbie = np.mean(movie["barbie"])
actual_barbienp.float64(0.5302792307692308)This is the actual outcome of the competition. Based on this result, Barbie would win. How did our sample of retirees do?
9.3.3.1 Convenience Sample: Retirees
Let’s take a convenience sample of people who have retired (>= 65 years old). What proportion of them went to see Barbie instead of Oppenheimer?
convenience_sample = movie[movie['age'] >= 65] # take a convenience sample of retirees
np.mean(convenience_sample["barbie"]) # what proportion of them saw Barbie? np.float64(0.3744755089093924)Based on this result, we would have predicted that Oppenheimer would win! What happened? Is it possible that our sample is too small or noisy?
# what's the size of our sample?
len(convenience_sample)359396# what proportion of our data is in the convenience sample?
len(convenience_sample)/len(movie)0.27645846153846154Seems like our sample is rather large (roughly 360,000 people), so the error is likely not due to solely to chance.
9.3.3.2 Check for Bias
Let us aggregate all choices by age and visualize the fraction of Barbie views, split by gender.
votes_by_barbie = movie.groupby(["age","is_male"]).agg("mean", numeric_only=True).reset_index()
votes_by_barbie.head()| age | is_male | barbie | |
|---|---|---|---|
| 0 | 18 | False | 0.819594 |
| 1 | 18 | True | 0.667001 |
| 2 | 19 | False | 0.812214 |
| 3 | 19 | True | 0.661252 |
| 4 | 20 | False | 0.805281 |
Code
# A common matplotlib/seaborn pattern: create the figure and axes object, pass ax
# to seaborn for drawing into, and later fine-tune the figure via ax.
fig, ax = plt.subplots();
red_blue = ["#bf1518", "#397eb7"]
with sns.color_palette(red_blue):
sns.pointplot(data=votes_by_barbie, x = "age", y = "barbie", hue = "is_male", ax=ax)
new_ticks = [i.get_text() for i in ax.get_xticklabels()]
ax.set_xticks(range(0, len(new_ticks), 10), new_ticks[::10])
ax.set_title("Preferences by Demographics");
- We see that retirees (in Berkeley) tend to watch Oppenheimer.
- We also see that residents who identify as non-male tend to prefer Barbie.
9.3.3.3 Simple Random Sample
Suppose we took a simple random sample (SRS) of the same size as our retiree sample:
n = len(convenience_sample)
random_sample = movie.sample(n, replace = False) ## By default, replace = False
np.mean(random_sample["barbie"])np.float64(0.5301589333214616)This is very close to the actual vote of 0.5302792307692308!
It turns out that we can get similar results with a much smaller sample size, say, 800:
n = 800
random_sample = movie.sample(n, replace = False)
# Compute the sample average and the resulting relative error
sample_barbie = np.mean(random_sample["barbie"])
err = abs(sample_barbie-actual_barbie)/actual_barbie
# We can print output with Markdown formatting too...
from IPython.display import Markdown
Markdown(f"**Actual** = {actual_barbie:.4f}, **Sample** = {sample_barbie:.4f}, "
f"**Err** = {100*err:.2f}%.")Actual = 0.5303, Sample = 0.5050, Err = 4.77%.
We’ll learn how to choose this number when we (re)learn the Central Limit Theorem later in the semester.
9.3.3.4 Quantifying Chance Error
In our SRS of size 800, what would be our chance error?
Let’s simulate 1000 versions of taking the 800-sized SRS from before:
nrep = 1000 # number of simulations
n = 800 # size of our sample
poll_result = []
for i in range(0, nrep):
random_sample = movie.sample(n, replace = False)
poll_result.append(np.mean(random_sample["barbie"]))Code
fig, ax = plt.subplots()
sns.histplot(poll_result, stat='density', ax=ax)
ax.axvline(actual_barbie, color="orange", lw=4);/Users/nikhilreddy/course-notes/ds100env/lib/python3.12/site-packages/seaborn/_oldcore.py:1119: FutureWarning:
use_inf_as_na option is deprecated and will be removed in a future version. Convert inf values to NaN before operating instead.
What fraction of these simulated samples would have predicted Barbie?
poll_result = pd.Series(poll_result)
np.sum(poll_result > 0.5)/1000np.float64(0.958)You can see the curve looks roughly Gaussian/normal. Using KDE:
Code
sns.histplot(poll_result, stat='density', kde=True);/Users/nikhilreddy/course-notes/ds100env/lib/python3.12/site-packages/seaborn/_oldcore.py:1119: FutureWarning:
use_inf_as_na option is deprecated and will be removed in a future version. Convert inf values to NaN before operating instead.
9.4 Summary
Understanding the sampling process is what lets us go from describing the data to understanding the world. Without knowing / assuming something about how the data were collected, there is no connection between the sample and the population. Ultimately, the dataset doesn’t tell us about the world behind the data.