Normal Distribution

Introduction

The normal distribution is a fundamental concept in statistics for business, describing how data values are distributed in many real-world scenarios. It is characterized by its bell-shaped, symmetrical curve and is widely used for probability calculations and statistical inference.

Continuous Probability Distributions

Definition and Examples

• Continuous variable: A variable that can assume any value on a continuum (uncountable number of values).

• Examples include:

  • Thickness of an item

  • Time required to complete a task

  • Temperature of a solution

  • Height in inches

Shapes of Continuous Distributions

• Normal Distribution: Symmetrical, bell-shaped, ranges from negative to positive infinity.

• Uniform Distribution: Symmetrical, rectangular, every value between the smallest and largest is equally likely.

• Exponential Distribution: Right-skewed, mean > median, ranges from zero to positive infinity.

The Normal Distribution

Properties

• Bell-shaped and symmetrical about the mean ().

• Mean, median, and mode are equal.

• Defined by two parameters: mean () and standard deviation ().

• Empirical Rule:

  • Approximately 68% of data falls within 1 standard deviation of the mean.

  • Approximately 95% within 2 standard deviations.

  • Approximately 99.7% within 3 standard deviations.

Probability Density Function

The formula for the normal probability density function is:

Effect of Parameters

• Changing μ shifts the distribution left or right.

• Changing σ alters the spread (width) of the curve.

• Distributions with the same mean but different standard deviations have different spreads.

The Standardized Normal Distribution

Definition

• Also known as the Z distribution.

• Mean is 0, standard deviation is 1.

• Standardization formula:

• Values above the mean have positive Z-values.

• Values below the mean have negative Z-values.

Example

If is distributed normally with mean 100 and standard deviation 50, the Z value for is:

Probability and Area Under the Curve

Concept

• Probability is measured by the area under the curve between two values.

• For any normal distribution:

• Area to the left of the mean: 0.5

• Area to the right of the mean: 0.5

Finding Probabilities

• To find , calculate the area under the curve between and .

• Use the Cumulative Standardized Normal Table to find probabilities for Z values.

Example

Suppose is normal with mean 18.0 and standard deviation 5.0. To find :

• Calculate

• Look up in the standard normal table to find the probability.

Upper and Lower Tail Probabilities

• Upper tail:

• Lower tail:

• For probabilities between two values, , calculate Z for both and subtract the lower from the upper cumulative probability.

Example

Suppose is normal with mean 18.0 and standard deviation 5.0. Find :

Finding the X Value for a Known Probability

Steps

1. Find the Z value for the known probability (using the standard normal table).

2. Convert to X units using the formula:

Example

Suppose is normal with mean 18.0 and standard deviation 5.0. Find such that 20% of download times are less than :

• Find for 0.20 in the lower tail:

• Calculate

Using Technology for Normal Probabilities

Excel, JMP, and Minitab Templates

• Statistical software can compute cumulative normal probabilities efficiently.

• Input mean and standard deviation, specify the value or range, and obtain the probability.

Exercises: Working with the Normal Distribution

Sample Problems

• Given a normal distribution with and , find probabilities for specific values and ranges.

• Given spending data with and , calculate probabilities and percentiles for online shopping behavior.

Evaluating Normality

Theoretical Properties

• Normal distribution is bell-shaped and symmetrical.

• Mean equals median.

• Empirical rule applies.

• Interquartile range is approximately 1.33 standard deviations.

Assessing Data Normality

• Construct charts or graphs (histogram, boxplot, stem-and-leaf).

• Compute descriptive summary measures (mean, median, mode, interquartile range, range).

• Observe the distribution:

  • ~2/3 of data within 1 standard deviation

  • ~80% within 1.28 standard deviations

  • ~95% within 2 standard deviations

• Evaluate normal probability plot:

  • Linear plot indicates normality

  • Nonlinear plot indicates deviation (left-skewed, right-skewed)

Normal Probability Plot Interpretation

• Plot of observed data values (X) against standardized normal quantile values (Z).

• Approximately linear plot suggests data is normally distributed.

• Nonlinear plots indicate skewness or deviation from normality.

Constructing a Normal Probability Plot

• Arrange data into ordered array.

• Find corresponding standardized normal quantile values (Z).

• Plot pairs of (X, Z) and evaluate for linearity.

Tables

Relative Frequency Table Example

Amounts cluster around the 1.050–1.055 interval, forming a bell-shaped pattern.

Standardized Normal Probability Table (Excerpt)

To find , use the value 0.7995 from the table.

Chapter Summary

• Computing probabilities from the normal distribution.

• Using the normal distribution to solve business problems.

• Using the normal probability plot to determine whether a set of data is approximately normally distributed.

Additional info: Some context and examples were expanded for clarity and completeness.