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Probability and Samples: The Distribution of Sample Means - Coggle Diagram
Probability and Samples: The Distribution of Sample Means
CENTRAL CONCEPT: Distribution of Sample Means
Definition: Collection of all possible random sample means of a particular size (n) drawn from a population
Key Formula: The distribution is characterized by:
Mean: μₘ = μ (equals population mean)
Standard deviation: σₘ = σ/√n (standard error)
CENTRAL LIMIT THEOREM (CLT)
Definition: Mathematical proposition stating that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the population's shape
Characteristics:
Works for ANY population distribution (skewed, bimodal, uniform, etc.)
Requires sufficiently large sample size (typically n ≥ 30)
Cornerstone of inferential statistics
Allows use of normal distribution for hypothesis testing
Formula Applications:
μₘ = μ (mean of sampling distribution equals population mean)
σₘ = σ/√n (standard error decreases with larger n)
As n → ∞, the distribution becomes perfectly normal
Example:
Population: Student reaction times (heavily right-skewed)
μ = 250 ms, σ = 50 ms
With n = 5: Distribution of sample means is still somewhat skewed
With n = 30: Distribution of sample means is approximately normal
With n = 100: Distribution of sample means is very close to normal
Why it matters
Enables statistical inference even when population distribution is unknown
Justifies use of z-scores and t-tests
Makes prediction and probability calculations possible
STANDARD ERROR OF M
Definition: Measure of the average distance expected between a sample mean and the population mean; the standard deviation of the sampling distribution.
Formula:
σₘ = σ/√n
σₘ = standard error of M
σ = population standard deviation
n = sample size
CharacteristicsDecreases as sample size increases
Measures precision of the sample mean
Smaller standard error = more precise estimate
Used to construct confidence intervals
Example:
Population: IQ scores with σ = 15
Sample size n = 25: σₘ = 15/√25 = 15/5 = 3
Sample size n = 100: σₘ = 15/√100 = 15/10 = 1.5
Sample size n = 400: σₘ = 15/√400 = 15/20 = 0.75
Interpretation: With n = 25, sample means typically differ from μ by about 3 points. With n = 400, they typically differ by only 0.75 points.
SAMPLING ERROR
Definition: Natural discrepancy between a statistic and its corresponding population parameter
Characteristics:
Inevitable and expected
Results from examining only part of the population
Not a "mistake" but inherent variability
Decreases as sample size increases
Sampling Error = M - μ (difference between sample mean and population mean)
Example:
Population mean test score: μ = 75
Sample mean test score: M = 78
Sampling error = 78 - 75 = 3 points
SAMPLING DISTRIBUTION
Definition: Group of statistics obtained by selecting all possible samples of a specific size from a population
Characteristics
Theoretical distribution (rarely calculated in practice)
Forms the foundation for inferential statistics
Allows us to make probability statements about samples
Different from the population distribution
Example
From a class of 100 students, you could select all possible samples of n = 25
Calculate the mean for each sample
Plot these means to create the sampling distribution
EXPECTED VALUE OF M
Definition: The mean of the distribution of sample means, which always equals the population mean.
Formula:
E(M) = μₘ = μ
Unbiased estimator (no systematic over- or under-estimation)
True regardless of sample size
Center of the sampling distribution
Example:
Population mean height: μ = 68 inches
If you take infinite samples of n = 20 and calculate each mean
The average of all those sample means = 68 inches
Expected value: E(M) = 68 inches
LAW OF LARGE NUMBERS
Definition: Rule stating that as sample size increases, the sample mean becomes increasingly likely to be close to the population mean
Formula Relationship:
As n ↑, then σₘ = σ/√n ↓
As n → ∞, then P(|M - μ| < ε) → 1 (for any small ε)
Characteristics:Theoretical foundation for why "bigger samples are better"
Explains why averages stabilize with more data
Guarantees convergence to true value (in probability)
Different from CLT (focuses on accuracy, not distribution shape)
Example 1: Coin Flips
True probability of heads: p = 0.50
After 10 flips: might get 70% heads (M = 0.70)
After 100 flips: might get 54% heads (M = 0.54)
After 1,000 flips: might get 51% heads (M = 0.51)
After 10,000 flips: likely 50.1% heads (M = 0.501)
Example 2: Education
Testing 5 students: sample mean might be 82 (true μ = 75)
Testing 30 students: sample mean might be 77 (true μ = 75)
Testing 200 students: sample mean likely 75.5 (true μ = 75)
Practical Application
Polling: Larger samples provide more accurate predictions
Quality control: More inspections = better estimate of defect rate
Clinical trials: Larger studies provide more reliable treatment effect estimates
Sampling Error <-> Standard Error
Standard error predicts the typical size of sampling error
σₘ tells us how much sampling error to expect on average
CLT <-> Expected Value
CLT guarantees that sampling distribution centers at μ
Expected value E(M) = μ is a consequence of CLT
CLT ↔ Standard Error:
Law of Large Numbers <-> Standard Error:
Inferential Statistics
These principles enable hypothesis testing
Allow calculation of confidence intervals
Make statistical prediction possible
Justify generalizing from samples to populations
LLN explains WHY larger samples are more accurate
Standard error formula (σ/√n) QUANTIFIES the improvement
CLT describes the shape (normal) of the distribution
Standard error describes the spread of that distribution