Normal Probability Distributions

Definition and Properties

The normal distribution is a fundamental continuous probability distribution in statistics, characterized by its symmetric, bell-shaped curve. It is widely used to model real-world phenomena where data tend to cluster around a central value with no bias left or right.

• Definition: A normal distribution is a continuous probability distribution for a random variable, x. Its graph is called the normal curve.

• Key Properties:

  • The mean, median, and mode are all equal and located at the center of the distribution.

  • The curve is bell-shaped and symmetric about the mean.

  • The total area under the curve is equal to 1, representing the total probability.

  • The curve approaches but never touches the x-axis as it extends infinitely in both directions.

  • Between μ − σ and μ + σ (where μ is the mean and σ is the standard deviation), the curve is concave downward; outside this interval, it is concave upward. The points where the curvature changes are called inflection points.

Example: Consider several normal curves with different means and standard deviations. The curve furthest to the right has the greatest mean, while the curve with the widest spread has the largest standard deviation.

Comparing Normal Curves

• Mean: The location of the curve on the x-axis indicates its mean. Curves centered further to the right have higher means.

• Standard Deviation: The spread or width of the curve reflects the standard deviation. A wider curve indicates a larger standard deviation.

Example: If a magenta curve is centered at -2 and other curves are centered at 0, the latter have the greater mean. Among curves with the same mean, the one that is most spread out (e.g., the blue curve) has the largest standard deviation.

The Standard Normal Distribution

Definition and Properties

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is denoted by the variable z and is used extensively for probability calculations and statistical inference.

• Mean (μ): 0

• Standard deviation (σ): 1

• Properties:

  • Bell-shaped and symmetric about 0.

  • Inflection points at z = -1 and z = 1.

  • Total area under the curve is 1.

Cumulative Areas and Z-Scores

• Cumulative area is the probability that a standard normal variable is less than or equal to a given z-score.

• For z close to -3.49, the cumulative area is near 0.

• For z = 0, the cumulative area is 0.5 (half the distribution).

• For z close to 3.49, the cumulative area is near 1.

• The cumulative area increases as z increases.

Finding Areas Under the Standard Normal Curve

Using the Standard Normal Table

The standard normal table (z-table) provides cumulative probabilities for z-scores. These probabilities represent the area under the curve to the left of a given z-value.

• To find the cumulative probability for a z-score:

  • Locate the first two digits of the z-score in the leftmost column.

  • Find the second decimal place in the top row.

  • The intersection gives the cumulative probability.

Example 1: Find the cumulative area for z = 1.15.

• First column: 1.1

• Top row: 0.05

• Intersection: 0.8749

• Interpretation: The probability that z < 1.15 is 0.8749.

Example 2: Find the cumulative area for z = 1.23.

• First column: 1.2

• Top row: 0.03

• Intersection: 0.8907

• Interpretation: The probability that z < 1.23 is 0.8907.

Finding Areas to the Right of a Z-Score

• To find the area to the right of a z-score, subtract the cumulative probability from 1:

Example: Find the area to the right of z = -2.16.

• Cumulative probability for z = -2.16: 0.0154

• Area to the right: 1 - 0.0154 = 0.9846

Finding Areas Between Two Z-Scores

• To find the area between two z-scores, subtract the cumulative probability of the smaller z from the larger z:

Example: Find the area between z = -2.165 and z = -1.35.

• Cumulative probability for z = -1.35: 0.08851

• Cumulative probability for z = -2.165: 0.01519

• Area between: 0.08851 - 0.01519 = 0.07332

Summary Table: Types of Areas Under the Standard Normal Curve

Key Takeaways

• The normal distribution is central to probability and statistics, with well-defined properties and applications.

• The standard normal distribution (z-distribution) simplifies calculations by standardizing the mean and standard deviation.

• Areas under the curve correspond to probabilities and can be found using the standard normal table.

• Understanding how to interpret and calculate these areas is essential for hypothesis testing, confidence intervals, and many other statistical procedures.