Population vs Sample
Sampling Theory
Every Study Starts With One Question: Who Are We Measuring?
Every statistical study starts with a fundamental question: who or what are we studying? The answer determines everything — from which statistics you calculate to what conclusions you can draw.
- Define your population — Know exactly who or what your conclusions apply to
- Understand sampling — Learn why studying a subset is often the only option
- Parameters vs statistics — Master the Greek letters that separate populations from samples
- Census vs sample — Discover why even a full count can be wrong
Get this distinction right, and inferential statistics becomes logical. Get it wrong, and your conclusions stand on sand.
What is Population vs Sample?
Definition
A population is the complete set of all individuals, objects, or measurements of interest. A sample is a subset of that population that is actually observed and measured.
Understanding the distinction between population and sample is the foundation of statistical inference.
Population vs Sample Diagram
Population
Definition
A population is the complete set of all individuals, objects, or measurements of interest for a particular study.
| Study | Population | Size |
|---|---|---|
| Approval rating of a president | All eligible voters in the country | ~250 million |
| Average height of NBA players | All current NBA players | ~450 |
| Effectiveness of a drug | All people who could ever take the drug | Infinite |
| Quality control of chips | All chips produced by the factory | ~1 million/day |
Populations can be:
- Finite: all 7,500 employees at a company
- Infinite: all possible measurements a machine could produce
- Hypothetical: all people who could take an experimental drug
Sample
Why sample instead of study the whole population?
| Reason | Example |
|---|---|
| Cost | Surveying 2,000 people costs far less than 2 million |
| Time | Census takes years; a survey takes months |
| Destructive testing | Testing a lightbulb to failure destroys it |
| Infinite population | You cannot measure every future product |
| Practical impossibility | Can't reach every person on Earth |
Parameters vs Statistics
Population Parameters
| Parameter | Symbol | Formula |
|---|---|---|
| Mean | μ | μ = (1/N)Σxᵢ |
| Std Dev | σ | σ = √[(1/N)Σ(xᵢ-μ)²] |
| Proportion | π | π = X/N |
Fixed but unknown. We estimate them using statistics.
Sample Statistics
| Statistic | Symbol | Formula |
|---|---|---|
| Mean | x̄ | x̄ = (1/n)Σxᵢ |
| Std Dev | s | s = √[(1/(n-1))Σ(xᵢ-x̄)²] |
| Proportion | p̂ | p̂ = x/n |
Known but variable. Different samples give different values.
import numpy as np
from scipy import stats
# Simulate a population (in reality, we wouldn't have this)
np.random.seed(42)
population = np.random.normal(loc=170, scale=10, size=10_000) # 10,000 adults
# True population parameters
mu = population.mean()
sigma = population.std(ddof=0) # ddof=0 for population
print(f"Population Parameter μ = {mu:.4f} cm")
print(f"Population Parameter σ = {sigma:.4f} cm")
print("\n--- Drawing samples of different sizes ---")
for n in [10, 30, 100, 500]:
sample = np.random.choice(population, size=n, replace=False)
x_bar = sample.mean()
s = sample.std(ddof=1) # ddof=1 for sample (unbiased)
se = s / np.sqrt(n)
print(f"n={n:4d}: x̄={x_bar:.3f}, s={s:.3f}, SE={se:.3f} | Error = {abs(x_bar-mu):.3f}")
Output:
Architecture Diagram
Population Parameter μ = 170.0694 cm
Population Parameter σ = 10.0048 cm
--- Drawing samples of different sizes ---
n= 10: x̄=169.847, s=10.042, SE=3.175 | Error = 0.222
n= 30: x̄=170.591, s=10.381, SE=1.895 | Error = 0.522
n= 100: x̄=170.204, s= 9.983, SE=0.998 | Error = 0.135
n= 500: x̄=170.082, s=10.017, SE=0.448 | Error = 0.013
Notice: Larger samples -> smaller standard error -> closer to the true parameter.
The Sampling Distribution
import matplotlib.pyplot as plt
# Sampling distribution of the mean (n=30)
sample_means = []
for _ in range(10_000):
sample = np.random.choice(population, size=30, replace=False)
sample_means.append(sample.mean())
sample_means = np.array(sample_means)
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
# Population distribution
axes[0].hist(population, bins=50, color='steelblue', alpha=0.7, density=True)
axes[0].axvline(mu, color='red', linewidth=2, label=f'μ = {mu:.1f}')
axes[0].set_title(f'Population Distribution\n(N=10,000, μ={mu:.1f}, σ={sigma:.1f})')
axes[0].legend()
# Sampling distribution of x̄
axes[1].hist(sample_means, bins=50, color='coral', alpha=0.7, density=True)
axes[1].axvline(mu, color='red', linewidth=2, label=f'μ = {mu:.1f}')
axes[1].set_title(f'Sampling Distribution of x̄\n(10,000 samples, n=30)')
axes[1].set_xlabel('Sample Mean')
axes[1].legend()
print(f"Mean of sample means = {sample_means.mean():.4f} ≈ μ = {mu:.4f}")
print(f"Std of sample means = {sample_means.std():.4f} ≈ σ/√n = {sigma/np.sqrt(30):.4f}")
plt.tight_layout()
plt.show()
Census vs Sample
| Census | Sample | |
|---|---|---|
| Coverage | All units | Subset |
| Cost | Very high | Lower |
| Time | Long | Shorter |
| Accuracy | No sampling error | Sampling error present |
| Feasibility | Limited | Broad |
| Non-response | Larger problem | Manageable |
The US Census Bureau conducts a decennial census — it takes years and billions of dollars and still has coverage errors.
Population vs Sample in Machine Learning
| Statistics Term | ML Equivalent | What It Means |
|---|---|---|
| Population | All possible data | Everything the model could ever see |
| Sample | Training set | What the model actually learns from |
| Parameter (μ, σ) | Model weights (W, b) | True values we want to learn |
| Statistic (x̄, s) | Loss/Accuracy on train | What we measure from our sample |
| Sampling error | Generalization gap | Difference between train and test performance |
Example — Train/Test Split as Sampling:
from sklearn.model_selection import train_test_split
import numpy as np
# Population: all house data
np.random.seed(42)
n_total = 1000
X = np.random.randn(n_total, 3) # 3 features
y = 2*X[:,0] + 3*X[:,1] - X[:,2] + np.random.randn(n_total)*0.5
# Sample: training set (80% of population)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
print(f"Population size: {n_total}")
print(f"Training sample: {len(X_train)}")
print(f"Test sample: {len(X_test)}")
# Statistics from sample (training set)
print(f"\nSample mean of X[:,0]: {X_train[:,0].mean():.3f}")
print(f"Population mean of X[:,0]: {X[:,0].mean():.3f}")
print(f"Sampling error: {abs(X_train[:,0].mean() - X[:,0].mean()):.3f}")
Output:
Architecture Diagram
Population size: 1000
Training sample: 800
Test sample: 200
Sample mean of X[:,0]: 0.018
Population mean of X[:,0]: 0.003
Sampling error: 0.015