Normal Distribution: Concepts, Applications, and Evaluating Normality

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 ($oldsymbol{ ext{μ}}$).

• Mean, median, and mode are equal.

• Defined by two parameters: mean ($oldsymbol{ ext{μ}}$) and standard deviation ($oldsymbol{ ext{σ}}$).

• 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:

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

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:

$Z = \frac{X - \mu}{\sigma}$

• Values above the mean have positive Z-values.

• Values below the mean have negative Z-values.

Example

If $X$ is distributed normally with mean 100 and standard deviation 50, the Z value for $X = 200$ is:

$Z = \frac{200 - 100}{50} = 2.0$

Probability and Area Under the Curve

Concept

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

• For any normal distribution:

$P(-\infty < X < \infty) = 1.0$

• Area to the left of the mean: 0.5

• Area to the right of the mean: 0.5

Finding Probabilities

• To find $P(a < X < b)$, calculate the area under the curve between $a$ and $b$.

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

Example

Suppose $X$ is normal with mean 18.0 and standard deviation 5.0. To find $P(X < 18.6)$:

• Calculate $Z = \frac{18.6 - 18.0}{5.0} = 0.12$

• Look up $Z = 0.12$ in the standard normal table to find the probability.

Upper and Lower Tail Probabilities

• Upper tail: $P(X > a)$

• Lower tail: $P(X < a)$

• For probabilities between two values, $P(a < X < b)$, calculate Z for both and subtract the lower from the upper cumulative probability.

Example

Suppose $X$ is normal with mean 18.0 and standard deviation 5.0. Find $P(18 < X < 18.6)$:

• $Z_1 = \frac{18 - 18}{5} = 0$

• $Z_2 = \frac{18.6 - 18}{5} = 0.12$

• $P(18 < X < 18.6) = P(0 < Z < 0.12)$

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:

$X = \mu + Z\sigma$

Example

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

• Find $Z$ for 0.20 in the lower tail: $Z = -0.84$

• Calculate $X = 18.0 + (-0.84) \times 5.0 = 13.8$

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 $ext{μ}=50$ and $ext{σ}=4$, find probabilities for specific values and ranges.

• Given spending data with $ext{μ}=250$ and $ext{σ}=20$, 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

Fill Amount (liters)	Relative Frequency
1.040	0.010
1.045	0.020
1.050	0.040
1.055	0.050
1.060	0.030
1.065	0.020
1.070	0.010

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

Standardized Normal Probability Table (Excerpt)

Z	0.00	0.01	0.02
0.8	0.7881	0.7910	0.7939
0.9	0.8159	0.8186	0.8212
1.0	0.8413	0.8438	0.8461

To find $P(Z < 0.84)$, 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.