Lecture 22: The Normal Distribution – Areas, Probabilities, and Percentiles

Normal Probability Distributions

Introduction to the Standard Normal Distribution

The normal distribution is a fundamental continuous probability distribution in statistics, characterized by its bell-shaped curve. The standard normal distribution is a special case with mean 0 and standard deviation 1. Probabilities and areas under the curve are often computed using z-scores and standard normal tables (Z-tables).

• Z-score: The number of standard deviations a value is from the mean. Calculated as $z = \frac{x - \mu}{\sigma}$.

• Standard Normal Table (Z-table): Provides the area (probability) to the left of a given z-value under the standard normal curve.

Computing Areas Under the Normal Curve

Area to the Left of a Z-value

To find the area under the standard normal curve to the left of a specific z-value:

• Use the Z-table to look up the area corresponding to the z-value.

• Example: Area to the left of $z = -0.38$ is 0.3520.

Area to the Right of a Z-value

To find the area to the right of a specific z-value:

• Subtract the area to the left from 1: $P(Z > z) = 1 - P(Z < z)$.

• Example: Area to the right of $z = 1.25$ is $1 - 0.8944 = 0.1056$.

Area Between Two Z-values

To find the area between two z-values $z_1$ and $z_2$:

• Subtract the area to the left of $z_1$ from the area to the left of $z_2$:

$P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$

• Example: Area between $z = -1.02$ and $z = 2.94$ is $0.9984 - 0.1539 = 0.8445$.

Generalization to Any Normal Random Variable

• For a normal random variable $X \sim N(\mu, \sigma^2)$, convert $x$ to a z-score: $z = \frac{x - \mu}{\sigma}$.

• Use the Z-table to find the area to the left, right, or between values as above.

Finding Cutoff Points (Percentiles and Probabilities)

Finding a Value Corresponding to a Given Probability

To find the value $x$ such that $P(X < x) = p$ (the $p$-th percentile):

1. Draw the normal curve and shade the area corresponding to $p$.

2. Use the Z-table to find the z-score corresponding to the shaded area.

3. Convert the z-score to the original value: $x = \mu + z\sigma$.

• Example: For body temperatures with $\mu = 98.2^\circ F$, $\sigma = 0.73^\circ F$, the cutoff for the lowest 3% is $x = 98.2 + (-1.88)(0.73) = 96.8^\circ F$.

Properties of Continuous Random Variables

• For any continuous random variable, the probability of observing any specific value is 0.

• Thus, $P(a < X < b) = P(a \leq X < b) = P(a < X \leq b) = P(a \leq X \leq b)$.

Computing Normal Probabilities: Step-by-Step

Step 1: Convert to Z-scores

• For $X \sim N(\mu, \sigma^2)$, convert $a$ and $b$ to z-scores: $z_a = \frac{a - \mu}{\sigma}$, $z_b = \frac{b - \mu}{\sigma}$.

Step 2: Use the Z-table

• Find the probabilities for the corresponding z-scores using the Z-table.

Step 3: Interpret the Table

• The Z-table typically gives $P(Z < z)$ for $-3.40 \leq z \leq 3.40$.

• For z-values not exactly in the table, round to two decimal places.

• Ones and tenths are in the first column; hundredths are in the first row.

Types of Normal Probability Calculations

• Left-tail: $P(Z < z)$ – direct from the table.

• Right-tail: $P(Z > z) = 1 - P(Z < z)$ or $P(Z > z) = P(Z < -z)$ (by symmetry).

• Between: $P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$.

• Outside: $P(|Z| > z) = 2P(Z < -z)$.

Example: Normal Probability Calculations

• Let $X \sim N(10, 25)$ (mean 10, SD 5). Find:

  • (a) $P(X \leq 12)$: $z = \frac{12-10}{5} = 0.40$, $P(Z \leq 0.40) = 0.6554$

  • (b) $P(X > 12)$: $P(Z > 0.40) = 1 - 0.6554 = 0.3446$

  • (c) $P(12 \leq X \leq 15)$: $z_1 = 0.40$, $z_2 = 1.00$, $P(0.40 \leq Z \leq 1.00) = 0.8413 - 0.6554 = 0.1859$

  • (d) $P(|X| \leq 15)$: $z = 1.00$, $P(-1.00 \leq Z \leq 1.00) = 0.8413 - 0.1587 = 0.6826$

Percentiles of the Normal Distribution

Definition and Calculation

• The p-th percentile $x_p$ of a continuous random variable $X$ is the value such that $P(X \leq x_p) = p$.

• For the standard normal, find $z_p$ such that $P(Z \leq z_p) = p$.

• For a general normal, $x_p = \mu + z_p \sigma$.

Finding Percentiles Using the Z-table

• If $p$ is not in the table, use the closest value.

• If two values are equally close, average the corresponding z-scores.

• Example: For $p = 0.95$, the closest table values are 0.9495 ($z_1 = 1.64$) and 0.9505 ($z_2 = 1.65$), so $z_{0.95} = 1.645$.

Worked Example: Percentiles for $X \sim N(40, 36)$

• Mean $\mu = 40$, standard deviation $\sigma = 6$.

• (a) 95th percentile: $z_{0.95} = 1.645$, $x_{0.95} = 40 + 6 \times 1.645 = 49.87$

• (b) 5th percentile: $z_{0.05} = -1.645$, $x_{0.05} = 40 + 6 \times (-1.645) = 30.13$

• (c) Value such that $P(X \geq x_0) = 0.7995$: $P(X < x_0) = 0.2005$, $z_{0.2005} = -0.84$, $x_0 = 40 + 6 \times (-0.84) = 34.96$

Summary Table: Types of Normal Probability Calculations

Type	Formula	How to Use Table
Left-tail	$P(Z < z)$	Direct from Z-table
Right-tail	$P(Z > z) = 1 - P(Z < z)$	1 minus table value
Between	$P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$	Subtract table values
Outside	$P(|Z| > z) = 2P(Z < -z)$	Double left-tail for negative z

Key Formulas

• Z-score: $z = \frac{x - \mu}{\sigma}$

• Value from percentile: $x = \mu + z\sigma$

• Area between: $P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$

Additional info:

• Table V refers to the standard normal (Z) table, which is a staple in statistics textbooks and exams.

• These methods are foundational for hypothesis testing, confidence intervals, and many inferential statistics procedures.