Q15: Normal Dist

Normal Distribution

A bell-shaped graph where most values are near the average and fewer values are far away.

Bell Shape

The shape of a normal distribution with a high center and lower sides.

Symmetry

A shape where the left and right sides are mirror images.

Mean (μ)

The average value and the center of a normal distribution.

Standard Deviation (σ)

A number that tells you how spread out the values are from the mean.

Variance (σ²)

The standard deviation multiplied by itself.

Normal Distribution Notation X ~ N(μ σ²)

A way to write a normal distribution where μ is the mean and σ² is the variance.

Median

The middle value when all values are put in order.

Mode

The value that appears most often.

Mean Median and Mode in a Normal Distribution

They are all equal and located at the center.

Total Area Under the Curve

The full area under a normal curve equals 1 or 100 percent.

Probability

The chance that something will happen.

Probability Density Function (PDF)

A function that describes the shape of a distribution and how probability is spread across values.

Area Under the Curve

The area that represents the probability of values falling within a range.

Probability of a Single Exact Value

For a continuous distribution the probability of one exact value is zero.

68–95–99.7 Rule

A rule saying about 68 percent of values are within 1 standard deviation 95 percent within 2 and 99.7 percent within 3.

One Standard Deviation from the Mean

The range from μ − σ to μ + σ containing about 68 percent of values.

Two Standard Deviations from the Mean

The range from μ − 2σ to μ + 2σ containing about 95 percent of values.

Three Standard Deviations from the Mean

The range from μ − 3σ to μ + 3σ containing about 99.7 percent of values.

Z-Score

A number that tells you how many standard deviations a value is above or below the mean.

Z-Score Formula

z = (x − μ) / σ.

Positive Z-Score

A Z-score above zero meaning the value is above the mean.

Negative Z-Score

A Z-score below zero meaning the value is below the mean.

Standard Normal Distribution

A normal distribution with mean 0 and standard deviation 1.

Z = 0

A Z-score of zero meaning the value equals the mean.

Z = 1

A value one standard deviation above the mean.

Z = −1

A value one standard deviation below the mean.

pnorm() in R

An R function that calculates the probability to the left of a value by default.

Lower-Tail Probability

The chance that a value is less than a chosen number.

Upper-Tail Probability

The chance that a value is greater than a chosen number.

Finding an Upper-Tail Probability in R

Use 1 - pnorm(value mean = mean sd = standard deviation).

Probability Between Two Values in R

Subtract the pnorm() result for the lower value from the result for the upper value.

rnorm() in R

An R function that generates random values from a normal distribution.

n in rnorm()

The number of random values to generate.

mean in rnorm()

The mean used to generate the random values.

sd in rnorm()

The standard deviation used to generate the random values.

Central Limit Theorem (CLT)

The rule that the distribution of sample means becomes roughly normal as sample size grows under suitable conditions.

Sampling Distribution

The distribution of a statistic such as the sample mean across many possible samples.

Sample Mean (x̄)

The average value calculated from a sample.

Standard Error (SE)

A measure of how much sample means vary across different samples.

Standard Error Formula

SE = σ / √n.

Sample Size (n)

The number of observations in a sample.

Effect of Increasing Sample Size

As sample size grows the standard error of the sample mean gets smaller.

CLT and Original Data

The CLT does not say the original data become normal

it describes the distribution of sample means.

Independent Observations

Observations that do not depend on each other.

Quantile-Quantile Plot (QQ Plot)

A graph that compares your data with the values expected from a normal distribution.

QQ Plot Showing Normality

Points that roughly follow a straight line suggest the data fit a normal distribution reasonably well.

QQ Plot Showing Non-Normality

Curves or large bends in the points suggest the data may not follow a normal distribution.

Outlier

A value that is unusually far from most other values.

Data Transformation

A mathematical change to data that may make patterns easier to model.

Log Transformation

A change that uses logarithms to shrink large values more than small values.

When to Use a Log Transformation

It may help with strongly right-skewed data or data whose spread grows with their average.

Limits of a Log Transformation

A log transformation does not always make data normal and usually requires positive values.

Right-Skewed Data

Data with a long tail toward larger values.

Normality

How closely data follow the shape of a normal distribution.

Normal Enough

Close enough to normal for the planned statistical method to work well.

Normality in Linear Models

The normality assumption often applies to the model errors or residuals rather than the raw data.

Changing the Mean

Changing the mean moves the normal curve left or right.

Changing the Standard Deviation

Changing the standard deviation changes how wide or narrow the normal curve is.

PDF vs pnorm()

A PDF describes the distribution shape while pnorm() calculates cumulative probability.

SD vs SE

SD measures spread among individual values while SE measures spread among sample means.

Why the Normal Distribution Matters

It helps calculate probabilities and supports many methods used to make conclusions from sample data