What are the two important parameters needed to define a normal distribution?

The two important parameters are the mean ($\mu$) and the standard deviation ($\sigma$).

What is the formula used to calculate the Z-score?

The formula is $Z = \frac{X - \mu}{\sigma}$, where $X$ is the random variable, $\mu$ is the mean, and $\sigma$ is the standard deviation.

According to the Empirical Rule, what percentage of values lie within one standard deviation of the mean?

According to the Empirical Rule, 68% of the X values lie within one standard deviation of the mean.

What area does a value read directly from a standard Z-table represent?

The value read from a standard Z-table represents the area under the curve to the left of that specific Z-score.

If a Z-score is positive, where does the corresponding X value lie relative to the mean?

If the Z-score is positive, the corresponding X value lies above (to the right of) the mean.

How is the probability calculated for an X value falling between two points using Z-tables?

The probability is calculated by finding the difference between the area corresponding to the upper X-value (less than $X_2$) and the area corresponding to the lower X-value (less than $X_1$): $P(X_1 < X < X_2) = P(X < X_2) - P(X < X_1)$.

Standard Normal Distribution Tables, Z Scores, Probability & Empirical Rule - Stats

Normal Distribution Fundamentals
📌 The normal distribution is characterized by its bell curve shape and two key parameters: the mean ( $\mu$ ) and the standard deviation ( $\sigma$ ).
⚙️ The formula for calculating the Z-score (standardizing a variable X) is $Z = \frac{X - \mu}{\sigma}$ .
🧮 The formula to find $X$ given a Z-score is $X = \mu + Z \times \sigma$ .
📝 The probability density function (PDF) for the normal distribution involves $e$ and is given by $f(X) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{X - \mu}{\sigma}\right)^2}$ .

The Empirical Rule (68-95-99.7 Rule)
📊 68% of X values lie within one standard deviation ( $\sigma$ ) of the mean ( $\mu$ ).
🎯 95% of X values lie within two standard deviations of the mean.
🌟 99.7% of X values lie within three standard deviations of the mean.
📈 Specific percentages between standard deviations: $\pm 1\sigma$ is 34% on each side; the area between $1\sigma$ and $2\sigma$ is 13.5% on each side; and the area between $2\sigma$ and $3\sigma$ is 2.35% on each side.

Z-Score Interpretation and Table Usage
➡️ Positive Z-values correspond to X-values above the mean, while negative Z-values correspond to X-values below the mean.
🔢 When the Z-score is not a whole number (e.g., $Z=1.56$), the Z-table must be used to find the cumulative area (probability) to the left of that Z-score.
📉 For an area greater than the Z-score (e.g., $P(X > x)$), use the formula: $1 - P(X \le x)$ or $1 - (\text{Area from Z-table})$ .

Application Examples (IQ Scores and Defective Tires)
🧠 For IQ scores ( $\mu=100, \sigma=15$ ): A score of $X=80$ has a Z-score of $Z \approx -1.33$ , yielding a probability of approximately 9.18% of being less than 80.
🔨 For defective tires ($n=500, p=0.02, q=0.98$): The mean ( $\mu$ ) is $n \times p = 500 \times 0.02 = \mathbf{10}$ defective tires, and the standard deviation ( $\sigma$ ) is $\sqrt{n p q} \approx \mathbf{3.13}$ .
📉 The probability of having between 7 and 14 defective tires requires calculating Z-scores for both values and finding the difference between their cumulative probabilities ($P(X < 14) - P(X < 7)$).

Key Points & Insights
➡️ Memorize the relationship between standard deviations and percentages via the Empirical Rule (68%, 95%, 99.7%).
➡️ When calculating X from Z, remember the formula: $X = \mu + Z \times \sigma$ .
➡️ To find the probability associated with an interval $(X_1, X_2)$ , calculate the Z-scores for both points and find the difference in their cumulative probabilities: $P(X_1 < X < X_2) = P(Z < Z_2) - P(Z < Z_1)$ .
➡️ When looking up percentiles (areas) in the Z-table that fall between two Z-values (e.g., 0.65), average the corresponding Z-scores to find the best estimate for that percentile.

📸 Video summarized with SummaryTube.com on Jan 05, 2026, 14:48 UTC

Standard Normal Distribution Tables, Z Scores, Probability & Empirical Rule - Stats

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