Discovering Statistics: An Interactive Primer

Building Intuition for Basic Concepts

Alasdair Warwick

2025-01-02

Contents

• ⁠Single sample: probability basics - normal distribution and binomial (coins)
• ⁠⁠2 samples: normal and chi square
• Adjusting for confounders - (stratifying) & linear regression. Logistic regression

Normal distribution

Try this

1. Start with default settings (mean=0, SD=1, n=30)
2. Reduce sample size to n=5 and observe how t-distribution tails become heavier
3. Increase sample size to n=100 and see how t-distribution converges to normal
4. Adjust mean and standard deviation to see how both distributions shift and scale
5. Compare the width of confidence intervals (dashed lines) between distributions

Notes

• The normal distribution (blue) is symmetric and bell-shaped
• The t-distribution (red) has heavier tails than the normal distribution
• As sample size increases, the t-distribution approaches the normal distribution
• The choice between them depends on whether the population standard deviation is known:
  • Normal distribution: Used when population standard deviation is known (rare in practice)
  • t-distribution: Used when estimating standard deviation from sample data (most common)
  • t-distribution accounts for added uncertainty in standard deviation estimate

Equations

The mean (\(\mu\)) and standard deviation (\(\sigma\)) are calculated as follows:

• Mean (\(\mu\)): \[ \mu = \frac{\sum_{i=1}^{n} x_i}{n} \]

• Standard Deviation (σ): \[ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}} \]

Where:

• \(x_i\) represents each individual data point
• \(n\) is the number of data points

Coin Flip Simulation

This simulation demonstrates the binomial distribution through repeated coin flips.

Try this

• Flip a coin 10 times
• Click ‘Play’ to repeat the experiment (select the ‘Multi-flip’ check box to perform 10 coin flips each time)

Binomial distribution

Binomial distribution

The binomial distribution is given by the formula:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Where:

• \(P(X = k)\) is the probability of getting exactly \(k\) successes in \(n\) trials.
• \(\binom{n}{k}\) is the binomial coefficient, which represents the number of ways to choose \(k\) successes from \(n\) trials.
• \(p\) is the probability of success on a single trial.
• \((1-p)\) is the probability of failure on a single trial.

Sampling variation

Comparing 2 samples

Next slides

To do

print("Hi")

Hi