Lecture 5: Normal Curves

Fundamental Characteristics of the Normal Probability Distribution

• Continuous Data Model: The normal probability distribution is a continuous probability distribution defined for continuous random variables.

• Bell-Shaped Curve: The graph of the normal probability density function is a symmetric, bell-shaped curve, frequently referred to as the Gaussian curve.

• Domain / Range of Values: The random variable $X$ can take on any real value across the infinite interval -\negthinspace \tag*{} \text{to} +\negthinspace \tag*{}:   $-\rightarrow < X < +\rightarrow$

• Symmetry: The distribution is perfectly symmetrical about its mean $\mu$.

• Equivalence of Central Tendencies: Due to perfect symmetry:   $\text{Mean} = \text{Median} = \text{Mode}$

• Total Area Under the Curve: The total probability under the density curve (representing the total area) equals exactly $1$.

• Approximation to Discrete Binomial Probabilities: Under specific conditions, the normal distribution can be used as an approximation for discrete binomial probabilities. This normal approximation is appropriate when both of the following criteria are satisfied:   $n p > 5$   $n (1 - p) > 5$

Mathematical Formulation and Parameters

• Probability Density Function (PDF): The mathematical equation defining the normal probability density function for a random variable $X$ is:   $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$   where $-\rightarrow < X < +\rightarrow$.

• Mathematical Constants:

  • $\pi \approx 3.14$

  • $e \approx 2.718$ (Euler's constant)

• Parameters of the Distribution:

  • $\mu$: The mean of the distribution, specifying the central location.

  • $\sigma$: The standard deviation of the distribution, specifying the spread or dispersion.

• Distribution Notation: A normally distributed variable $X$ with mean $\mu$ and standard deviation $\sigma$ is denoted as:   $X \sim N(\mu, \sigma)$

Structural Variations in Normal Distributions

• Effect of Varying Means ($\mu_1 < \mu_2 < \mu_3$):

  • When standard deviations are held equal ($\sigma_1 = \sigma_2 = \sigma_3$), changing the mean $\mu$ shifts the location of the curve along the horizontal axis ($X$-axis) without changing its shape, height, or spread.

• Effect of Varying Standard Deviations ($\sigma_1 < \sigma_2 < \sigma_3$):

  • When the mean is held constant ($\mu_1 = \mu_2 = \mu_3 = \mu$), changing the standard deviation $\sigma$ alters the spread and peak of the curve. Smaller standard deviations ($\sigma_1$) result in narrow, highly peaked curves, while larger standard deviations ($\sigma_3$) produce flatter, wider curves.

Empirical Rule and Areas Under the Normal Curve

• Total Enclosed Area: The entire area under the normal curve represents a total probability of $1$.

• Interval Probabilities Relative to Standard Deviation ($\sigma$):

  • One Standard Deviation Interval ($\mu \pm 1\sigma$ ):

  • Approximately $0.68$ (68%) of the total probability lies within $\mu \pm 1\sigma$

  • The remaining $0.32$ probability is divided equally between the two outer tails, leaving $0.16$ in each tail ($P(X < \mu - 1\sigma) = 0.16$ and $P(X > \mu + 1\sigma) = 0.16$).

• Two Standard Deviations Interval ($\mu \pm 2\sigma$):

  • Approximately $0.95$ (95%) of the total probability lies within $\mu \pm 2\sigma$

  • The remaining $0.05$ probability is divided equally between the two outer tails, leaving $0.025$ in each tail ($P(X < \mu - 2\sigma) = 0.025$ and $P(X > \mu + 2\sigma) = 0.025$).

• Three Standard Deviations Interval ($\mu \pm 3\sigma$):

  • Approximately $0.997$ (99.7%) of the total probability lies within $\mu \pm 3\sigma$

  • The remaining $0.003$ probability is divided equally between the two outer tails, leaving $0.0015$ in each tail ($P(X < \mu - 3\sigma) = 0.0015$ and $P(X > \mu + 3\sigma) = 0.0015$).

The Standard Normal Probability Distribution (Z)

• Definition: The standard normal probability distribution is a special case of the normal distribution where the mean is normalized to $0$ and the standard deviation is normalized to $1$.

• Standard Parameters:

  • Mean: $\mu = 0$

  • Standard deviation: $\sigma = 1$

  • Notation: $Z \sim N(0, 1)$

• Standard Normal Probability Density Function:   $f(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}$   where $-\rightarrow < Z < +\rightarrow$.

• The $Z$-Transformation Formula: Any normally distributed variable $X \sim N(\mu, \sigma)$ can be converted to the standard normal random variable $Z$ using standardisation:   $Z = \frac{X - \mu}{\sigma}$

• Transformation Example (Blood Pressure):

  • Given a blood pressure distribution $X$ with mean $\mu = 110\,\text{mmHg}$ and standard deviation $\sigma = 10\,\text{mmHg}$:

  • For $x = 115\,\text{mmHg}$: $Z = \frac{115 - 110}{10} = 0.5$

  • For $x = 100\,\text{mmHg}$: $Z = \frac{100 - 110}{10} = -1.0$

  • For $x = 145\,\text{mmHg}$: $Z = \frac{145 - 110}{10} = 3.5$

  • For $x = 110\,\text{mmHg}$: $Z = \frac{110 - 110}{10} = 0$

Calculating Probabilities for standard Normal Distribution

• Integration Definition: The probability that $Z$ takes on values between $z_0$ and $z_1$ corresponds to the area under the PDF curve between those points:   $P(z_0 \le Z \le z_1) = \int_{z_0}^{z_1} \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} dz$

  • Explicit mathematical integration is avoided through the use of standardized tables, computer software, and web applets.

• Symmetry Relationships (for $z_0 \ge 0$):

  • Tail Equivalence: Due to curve symmetry across $Z = 0$:     $P(Z \le -z_0) = P(Z \ge z_0)$

  • Half-Curve Area: Each half of the curve split at the mean $0$ contains half the area ($0.5$):     $P(Z \le 0) = P(Z \ge 0) = 0.5$

  • Intervals from the Mean ($0$ to $z_0$):     $P(0 \le Z \le z_0) = 0.5 - P(Z \ge z_0)$     $P(-z_0 \le Z \le 0) = 0.5 - P(Z \le -z_0)$

  • Cumulative Area ($Z \le z_0$):     $P(Z \le z_0) = 0.5 + P(0 \le Z \le z_0) = 0.5 + (0.5 - P(Z \ge z_0)) = 1 - P(Z \ge z_0)$

  • Symmetric Interval ($-z_0 \le Z \le z_0$):     $P(-z_0 \le Z \le z_0) = 1 - 2 P(Z \ge z_0)$

  • Asymmetric Interval ($z_0 \neq z_1$):     $P(z_0 \le Z \le z_1) = 1 - P(Z \ge z_1) - P(Z \le z_0) = 1 - P(Z \ge z_1) - P(Z \ge -z_0)$

  • Example Calculation: For $P(-1 \le Z \le 2)$, the calculation becomes:       $P(-1 \le Z \le 2) = 1 - P(Z \ge 2) - P(Z \le -1) = 1 - P(Z \ge 2) - P(Z \ge 1)$

Structure of Standard Normal Probability Tables

• Types of Table Values: Standard normal statistical tables list probabilities for specific values of $z$ across three primary formatting conventions:

  1. Two-Sided (Two-Tailed) Value: $P(Z \ge z) + P(Z \le -z)$

  2. One-Sided (One-Tailed) Value: $P(Z \ge z)$

  3. Cumulative Probability: $P(Z \le z)$

• Tabulated Standard Values:

  • For $z = 0.00$:

  • Two-sided probability: $1.00$

  • One-sided probability: $0.5$

  • Cumulative probability: $0.5$

  • For $z = 1.645$:

  • Two-sided probability: $0.10$

  • One-sided probability: $0.05$

  • Cumulative probability: $0.95$

  • For $z = 1.96$:

  • Two-sided probability: $0.0500$

  • One-sided probability: $0.0250$

  • Cumulative probability: $0.9750$

• Diagnostic Interpretation of Patterns:

  • Identical Distributions ($Y = X$ Line): A straight line along $Y = X$ where $x_2(p) = x_1(p)$ indicates identical distributions.

  • Shifted Means (Parallel Line to $Y = X$): A straight line parallel to $Y = X$ indicates that only the means differ, while variances and shapes are identical.

  • Differing Means and Variances (Non-Parallel Straight Line): A straight line that is not parallel to $Y = X$ indicates differing means and differing variances.

  • Differing Distribution Shapes (Curved Line): A curved plot indicates that the underlying shapes of the two distributions differ (e.g., one is skewed while the other is symmetric).

Case Study: Carotid Endarterectomy (CE) Total Charges

• Objective: Evaluate the distribution of total charges for Carotid Endarterectomy (CE) procedures and test whether a $\log_{10}$ transformation satisfies the assumption of normality.

• Stata Analysis Commands:

  • Plotting raw charge histogram with normal density overlay:     .hist totchg, normal kdensity title(Histogram of CE Total Charges)

  • Creating log-transformed variable:     .gen ltc = log10(totchg)

  • Plotting log-transformed charge histogram:     .hist ltc, normal kdensity title(Histogram of log10 CE Total Charges)

  • Generating Q-Q plots against standard normal distribution:     .qnorm totchg, title(Q-Q Plot of CE Total Charges) grid     .qnorm ltc, title(Q-Q Plot of CE Log10 Total Charges) grid

• Empirical Visual Results:

  • Raw Total Charges (totchg):

  • Histogram: Shows severe positive (right) skewness with high density near lower charges and a long tail extending beyond $\$100000$.

- *Q-Q Plot:* Displays sharp upward curvature at higher quantiles relative to the straight normal reference line, confirming heavy right-tail skewness.

• Log10 Transformed Charges (ltc):

  • Histogram: Displays a substantially more symmetric bell-shaped distribution centered near $3.7$.

- *Q-Q Plot:* Follows the linear reference line much more closely across central percentiles, though mild departures remain at extreme tails.

• Analytical Conclusions:

  • Raw Carotid Endarterectomy (CE) total charges are strongly skewed toward high values.

  • The $\log_{10}$ transformation creates a substantially more symmetric distribution.

  • However, Q-Q plot diagnostic evaluation indicates CE charges remain somewhat skewed to high values even on the $\log_{10}$ scale, illustrating the importance of visual diagnostic tools prior to applying normal distribution theory.