Lecture 13
The Normal Distribution

ABD 3e Chapter 10

Chris Merkord

Bell-shaped curves = normal distribution

  • Many numerical variables have bell-shaped frequency distributions.

  • Normal distribution = theoretical probability distribution describing many bell curves

    • Continuous numerical variables

    • Symmetric, unimodal

    • Lower probability further from mean

Example: Birth weights of the 4,017,264 singleton births recorded by birth certificate in the United States in 1991. Whitlock & Schluter 3e

Example: Birth weights of the 4,017,264 singleton births recorded by birth certificate in the United States in 1991. Whitlock & Schluter 3e

The Normal Distribution

The normal distribution

  • The normal distribution is a continuous probability distribution describing a bell shaped curve.

  • Has two parameters to describe it location and spread:

    • Mean (location)

    • Standard deviation (spread)

The normal distribution for a variable Y with mean and variance equal to that in the baby birth weight data.

The normal distribution for a variable Y with mean and variance equal to that in the baby birth weight data.

The normal distribution approximates frequency distributions in nature

Examples

  • Human body temperature, in degrees Fahrenheit (Shoemaker 1996)

  • University undergraduate brain size (measured in number of megapixels on an MRI scan) (Willerman et al. 2991)

  • The number of bristles on the fourth and fifth segments of the abdomens of fruit flies (Falconer and macKay 1995)

The black lines show normal distributions with the same mean and standard deviation as measured in the data.

Example normal distributions. Whitlock & Schluter 3e

Example normal distributions. Whitlock & Schluter 3e

The formula for the normal distribution

  • Rarely used by hand (do not memorize)

  • Mean can be any value

  • Standard deviation can be any positive value

  • Thus β€œnormal distribution” is really an infinite number of distributions, each with its own

    1. Mean

    2. Standard deviation

\[ f(Y) = \frac{1}{\sqrt{2\pi\sigma^2}} \, e^{-\frac{(Y-\mu)^2}{2\sigma^2}} \]

Properties of the normal distribution

  • It is a continuous distribution:
    • probability measured by the area under the curve (AUC) not height of the curve
  • Symmetrical
  • Unimodal
  • Mode = mean = median

About 2/3 of AUC is < 1 sd from mean

About 95% of AUC is < 2 sd from mean

The standard normal distribution

  • The standard normal distribution is a normal distribution with:

    • mean = zero

    • standard deviation = one

  • Uses the symbol \(Z\) to indicate a variable having a standard normal distribution

Statistical Table B: The standard normal ( \(Z\) ) distribution

  • Gives probabilities under the right tail of the standard normal distribution.

  • An example of a probability under the standard normal curve: the probability of sampling a value greater than or equal to the value \(1.96\) is \(0.025\), or \(2.5\%\).

Using Statistical table B

  • To find the area under the standard normal distribution to the right of \(Z=1.96\) :

    1. Find the first two digits \(1.9\) on the left

    2. Find the third digit \(6\) on the top

    3. The intersection gives the area under the curve

  • R equivalent:

    1 - pnorm(q = 1.96)

To find a AUC for a range, use subtraction

Example: What proportion of the population have values between X and Y?

To solve: Subtract 𝑍-score for Y from 𝑍-score for X

Z-Scores (standard normal deviates)

Using the standard normal to describe any normal distribution

  • There are an infinite number of normal distributions but they are all similar in shape
  • This allows us to use a simple transformation to obtain probabilities under any normal distribution
  • A standard normal deviate, or 𝒁-score, tells us how many standard deviations a particular value is from the mean

\[ Z = \frac{Y-\mu}{\sigma} \]

Where:

  • \(Z\) is a standard normal value

  • \(Y\) is any particular value

  • \(\mu\) is the population mean

  • \(\sigma\) is the population standard deviation

Practical application of 𝑍 scores

  • In many (most?) kinds of modeling, raw values of predictor variables should be converted to 𝑍 scores.

  • Two reasons:

    1. Effects size of variables can be compared directly (e.g. when doing multiple regression)

    2. Facilitates model convergence (computers have an easier time optimizing likelihoods to estimate model parameters). In terms that matter to you: the computer goes faster and is less likely to throw an error.

The normal distribution of sample means

  • \(\bar{Y}\) is the mean of a single sample.

  • If a variable \(Y\) has a normal distribution in a population, then the distribution of sample means is also normal

  • The standard deviation of the sampling distribution for \(\bar{Y}\) is known as the standard error of the mean

\[ \sigma_{\bar{Y}}=\frac{\sigma}{\sqrt{n}} \]

As sample size \(n\) increases, \(\sigma_{\bar{Y}}\) decreases

Example: The distribution of sample means based on sample sizes of

  • 𝑛=10 (red)

  • 𝑛=100 (blue)

  • 𝑛=1000 (black)

Note the change in scale from previous figures.

Calculating probabilities of sample means

  • You can calculate the probability of obtaining a sample with a mean in a given range:

    • To do this, calculate the 𝑍 score for a given sample mean \(\bar{Y}\)

    • Then calculate the 𝑍 score for another sample mean \(\bar{Y}\)

    • Subtract one 𝑍 score from the other to get the AUC between the two 𝑍 scores

Practice Problem #1

Question

The natural log of growth (change in radius per year in mm) of Engelmann spruce is approximately normally distributed with mean of 0.037 log units and standard deviation 0.385.

Following these steps, determine the probability that a tree has a bad year, defined as having growth less than βˆ’0.050 log units in a year.

STEP 1: Sketch a normal distribution with \(\mu=0.037\), \(\operatorname{sd}=0.385\)

  1. Draw axes
  1. Mean goes in middle
  1. Calculate width:

\[ \mu \pm 3\sigma \\0.037 \pm 3 \times 0.385 \\0.037 \pm 1.155 \\-1.118 < \mu < 1.192 \]

  1. Label x-axis ticks
  1. Label x-axis
  1. Draw bell-shape and label y-axis

STEP 2: Mark the values that we are trying to determine

(i.e., those values less than βˆ’0.05)

… From previous step

  1. Mark the value (-0.05)

  1. Fill in the area (< -0.05)

STEP 3: Calculate the standard normal deviate (𝑍)

Calculate the standard normal deviate (𝑍) associated with the value we are interested in here, βˆ’0.05

  • Mean \(\mu=0.037\) log units

  • Standard deviation \(\sigma=0.385\) log units

  • Value of interest \(Y=-0.05\) log units

\[ Z = \frac{Y-\mu}{\sigma} \]

\[ \frac{-0.05-0.037}{0.385} \]

\[ -0.226 \]

STEP 4: Find probability of value < the Z-score

We are interested in the probability of getting a value less than -0.226.

Goal: Find the area to the left of the red line

Problem: the Z table only shows areas to right of a given value

Solution: Find an equivalent area to right of a given value

STEP 5: Find the P-value in the statistical table

What is the probability that a random draw from a standard normal distribution will be greater than 0.226?

  • Find area to right of \(Z=0.226\)

  • Use statistical table B

  • Area to right of \(Z=0.226\) is about \(0.40905\)

  • Probability of randomly drawing a value greater than \(Z=0.226\) is \(P=0.40905\)

  • or about \(41\%\)

STEP 6: Calculate original area of interest

What is the probability that a random draw from a standard normal distribution will be less than βˆ’0.226?

  • Remember, the area we were interested in was not the area to the right of +0.226 but the area to the left of -0.226

  • What is that area?

  • Same as area to the right of +0.226:

    \(0.40905\)

STEP 7: Answer the question

What is the probability that a tree has a bad growth year, that is, less than βˆ’0.05 log units?

  • Probability of \(Y \le -0.05 \text{ log units}\) ?

  • Same as probability of \(Z \le -0.226\)

  • \(P=0.40904\)