Normal Probability Distribution

Continuous Random Variables

Continuous random variables are variables that can take any real value within a given interval. Unlike discrete random variables, their possible values cannot be listed individually, as there are infinitely many within any range.

• Definition: A continuous random variable is one whose set of possible values forms an interval or collection of intervals on the real number line.

• Examples: Height, weight, age, time, temperature, pressure, pH.

• Probability: The probability of a continuous random variable taking any exact value is zero, i.e., . Instead, probabilities are calculated over intervals: , , .

Density Curves

Density curves are graphical representations of the probability distribution of a continuous random variable. The area under the curve represents probability, and the total area is always 1.

• Key Point: The density curve is used to model the distribution of continuous random variables.

• Total Area: The total area under the density curve equals 1.

The Normal Distribution

The normal probability distribution is a fundamental continuous distribution characterized by its mean () and standard deviation (). It is widely used in statistics due to its natural occurrence in many real-world phenomena.

• Definition: The normal distribution is a symmetric, bell-shaped distribution defined by its mean and standard deviation.

• Characteristics:

  • The mean is at the center, and mean = mode = median.

  • The curve is symmetric and bell-shaped.

  • The standard deviation determines the width of the curve.

  • Total area under the curve is 1.

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

Standard Normal Distribution and Z-Scores

The standard normal distribution is a special case of the normal distribution with mean 0 and standard deviation 1. Z-scores are used to standardize values from any normal distribution.

• Definition: The standard normal distribution has and .

• Z-score: The Z-score transforms any normal variable to the standard normal:

• Z-table: The Z-table provides probabilities for the standard normal distribution.

Empirical Rule (68-95-99.7 Rule)

The empirical rule describes the proportion of values within certain standard deviations from the mean in a normal distribution.

• 68%: Within 1 standard deviation ()

• 95%: Within 1.96 standard deviations ()

• 99.7%: Within 3 standard deviations ()

Calculating Probabilities Using the Normal Distribution

Probabilities for intervals or specific values can be calculated using the normal distribution and Z-scores.

• Transforming to Z: Use to convert any value to the standard normal.

• Example: For IQ scores with , , the percentage above 130 is found by calculating and using the Z-table.

• Interval Probability: is the area under the curve between and .

Applications of the Normal Distribution

The normal distribution is used in various statistical applications, including quality control, medical statistics, and employee appraisals.

• Control Charts: Used to monitor processes; most values fall within 3 standard deviations of the mean.

  • Upper Control Limit (UCL):

  • Lower Control Limit (LCL):

• Example: Daily sales with , ; values outside UCL or LCL are unusual.

Using Z-values to Identify Unusual Cases

Z-values allow comparison of individual cases to the population, identifying outliers or unusual values.

• Example: Diastolic blood pressure in adult males ( mm Hg, mm Hg). A patient with 125 mm Hg has , which is highly unusual.

Employee Appraisals and Forced Ranking

Normal distribution is sometimes used in employee appraisals for large groups, but is less useful for small teams.

• Application: Forced ranking systems assume employee performance follows a normal distribution.

Summary

• Normal distribution is defined by its mean and standard deviation.

• Z-table gives probabilities for the standard normal distribution.

• Any normal distribution value can be transformed to a z-value using .