Lecture 8

Distributions

Normal distributions and the sampling distribution

Dr Lincoln Colling

14 Nov 2022

Psychology as a Science

Plan for today

Today we’ll learn about the sampling distribution

But before we can do that we need to know what distributions are, where they come from, and how to describe them

• The binomial distribution

• The normal distribution

  • Processes that produce normal distributions

  • Process that don’t produce normal distributions

  • Describing normal distributions

  • Describing departures from the normal distributions

• Distributions and samples

  • The Central Limit Theorem

• The Standard Error of the Mean

The Binomial Distributions

• The binomial distribution is one of the simplest distribution you’ll come across

• To see where it comes from, we’ll just build one!

• We can build one by flipping a coin (multiple times) and counting up the number of heads that we get

viewof sequences = {
  const div = document.createElement("div");
  div.value = new vega.View(parsedSpec).initialize(div).run();
  return div;
}

viewof coin_flip_dist = Plot.plot({
  y: {
    label: "Sequences",
    grid: true
  },
  x: {
    label: "Number of heads"
  },
  marks: [Plot.barY(n_heads, { x: "x", y: "y" }), Plot.ruleY([0])]
})

viewof coins = htl.html`<input style="width:300px" type="range" id="coins" min="1" max="7" value="1" class="form-range">`

coins_label = htl.html`<label for="coins" class= "form-label" width="100%">Number of coin flips: ${
  coins - 1
}</label>`

• In Figure 1 we can see the possible sequences of events that can happen if we flip a coin (⚈ = heads and ⚆ = tails) Figure 2 look very interesting at the moment.

• In Figure 2 we just count up the number of sequences that lead to 0 heads, 1 head, 2 heads, etc

• As we flip more coins the distribution of number of heads takes on a characteristic shape

• This is the binomial distribution

The binomial distribution