The Normal Distribution: Properties, Z-Scores, and Probability Calculations

The Normal Distribution

Introduction to Continuous Probability Distributions

Continuous probability distributions describe the probabilities of outcomes for variables that can take on any value within a given range. Unlike discrete distributions, which focus on specific values, continuous distributions are concerned with intervals of values.

• Continuous variables represent measurements (e.g., height, weight, time).

• Possible values span from $-\infty$ to $+\infty$ and can be any real number.

• Probability is assigned to intervals of values, not to individual points.

• The area under the curve of the probability density function (PDF) represents the probability for a given interval.

• The total area under the curve equals 1, representing all possible outcomes.

Types of Continuous Distributions

Common continuous distributions include:

• Uniform Distribution: All intervals of the same length have equal probability.

• Normal Distribution: Bell-shaped, symmetric curve; most data centered around the mean.

• Exponential Distribution: Often used to model time until an event occurs.

Comparison Table of Continuous Distributions

Distribution	Shape	Key Feature
Uniform	Rectangular	Equal probability for all intervals
Normal	Bell-shaped, symmetric	Most data near mean
Exponential	Right-skewed	Models waiting times

Properties of the Normal Distribution

Defining Characteristics

The normal distribution is the most widely used continuous probability distribution in statistics. It is characterized by its bell-shaped, symmetric curve.

• Mean ($\mu$): The center of the distribution; location of the peak.

• Standard deviation ($\sigma$): Measures the spread or variability; determines the width of the curve.

• Most data is concentrated around the mean, with probabilities decreasing as values move further from the mean.

Mathematical Formula

The probability density function (PDF) of the normal distribution is:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2}$

Effects of Changing Mean and Standard Deviation

• Changing the mean ($\mu$) shifts the curve left or right along the number line.

• Changing the standard deviation ($\sigma$) alters the spread: smaller $\sigma$ makes the curve narrower and taller; larger $\sigma$ makes it wider and shorter.

Illustrative Table: Effects of Parameters

Parameter	Effect
Increase $\mu$	Curve shifts right
Decrease $\mu$	Curve shifts left
Increase $\sigma$	Curve becomes wider, less peaked
Decrease $\sigma$	Curve becomes narrower, more peaked

Z-Scores and Standardization

Definition of Z-Score

A z-score indicates how many standard deviations a value is from the mean. It allows comparison across different normal distributions by standardizing values.

• Negative z-scores: Values below the mean.

• Positive z-scores: Values above the mean.

Formula for Z-Score

$z = \frac{x - \mu}{\sigma}$

Interpreting Z-Scores

• If $z = 0$, the value is at the mean.

• If $z > 0$, the value is above the mean.

• If $z < 0$, the value is below the mean.

Finding Probabilities Using the Normal Distribution

Area Under the Curve

Probabilities for intervals are found by calculating the area under the normal curve for the specified range.

• The total area under the curve is 1.

• Area to the left of the mean is 0.5; area to the right is also 0.5.

• Probabilities for specific z-scores can be found using the standard normal table (Table A).

Using the Standard Normal Table

• Find the z-score for the value of interest.

• Look up the z-score in the table to find the area to the left ($P(z < a)$).

• For area to the right, subtract the table value from 1: $P(z > a) = 1 - P(z < a)$.

• For area between two z-scores: $P(a < z < b) = P(z < b) - P(z < a)$.

Example Table: Standard Normal Probabilities

Z-Score	Area to Left	Area to Right
-1.00	0.1587	0.8413
0.00	0.5000	0.5000
1.00	0.8413	0.1587

Finding Probabilities for X-Values

Steps to Calculate Probability for a Given X

1. Calculate the z-score using $z = \frac{x - \mu}{\sigma}$.

2. Use the standard normal table to find the area (probability) corresponding to the z-score.

3. Interpret the area as the probability that $X$ falls within the specified interval.

Example

Suppose the amount of drink dispensed by a machine is normally distributed with $\mu = 10.25$ ounces and $\sigma = 0.50$ ounces. What is the probability of getting less than 10 ounces?

• Calculate z-score: $z = \frac{10 - 10.25}{0.50} = -0.5$

• Look up $z = -0.5$ in the standard normal table: $P(z < -0.5) \approx 0.3085$

• Probability is 0.3085.

Finding X-Values Given Probabilities

Steps to Find X for a Given Probability

1. Find the z-score corresponding to the given probability using the standard normal table.

2. Convert the z-score to an x-value using $x = \mu + z\sigma$.

Example

Suppose exam scores are normally distributed with $\mu = 62$ and $\sigma = 6.3$. What score corresponds to the lowest 10% of the class?

• Find z-score for 10%: $P(z < a) = 0.10$; from the table, $z \approx -1.28$

• Calculate x: $x = 62 + (-1.28)(6.3) = 53.936$

Summary Table: Key Formulas

Purpose	Formula
Z-score	$z = \frac{x - \mu}{\sigma}$
X-value from z-score	$x = \mu + z\sigma$
Area to left of z	Use standard normal table
Area to right of z	$1 - P(z < a)$
Area between two z-scores	$P(z < b) - P(z < a)$

Key Points to Remember

• The normal distribution is defined by its mean and standard deviation.

• Z-scores standardize values for comparison and probability calculation.

• Probabilities are found using the area under the curve, often with the help of the standard normal table.

• Always check whether you need the area to the left, right, or between values.

Additional info: Some examples and table values were inferred and expanded for clarity and completeness.