AP Statistics Flashcards: The Normal Distribution Revisited

What this deck covers

This deck focuses on The Normal Distribution Revisited, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What does a z-score represent?

Answer: Number of standard deviations from the mean. Measures distance from mean in standard deviation units.

Flashcard 2: What is the shape of a normal distribution curve?

Answer: A symmetric, bell-shaped curve. Characteristic appearance of normal probability distributions.

Flashcard 3: What is the relationship between mean, median, and mode in a normal distribution?

Answer: They are equal. Symmetry property of normal distributions.

Flashcard 4: Determine the probability of $z < -2$.

Answer: Approximately 2.28%. Lower tail probability beyond two standard deviations.

Flashcard 5: What is the mean of a standard normal distribution?

Answer: The mean is 0. Defining characteristic of the standard normal distribution.

Flashcard 6: What z-score corresponds to the 90th percentile?

Answer: $z \approx 1.28$. Critical value for 90% of data below this point.

Flashcard 7: Describe the tails of a normal distribution.

Answer: The tails are asymptotic to the x-axis. Approach but never touch the horizontal axis.

Flashcard 8: Find the z-score for the 75th percentile.

Answer: $z \approx 0.675$. Third quartile z-score in standard normal distribution.

Flashcard 9: Identify the symbol for the mean in a normal distribution.

Answer: $\mu$ (mu). Greek letter representing population mean parameter.

Flashcard 10: Calculate the probability of $z > 1.96$.

Answer: Approximately 2.5%. Upper tail beyond critical value for 95% confidence.

Flashcard 11: What is the total area under a normal distribution curve?

Answer: The total area is 1. Property of all probability distributions.

Flashcard 12: What does the term 'standard normal distribution' mean?

Answer: Normal distribution with $\mu = 0$, $\sigma = 1$. Normalized version with mean 0 and standard deviation 1.

Flashcard 13: What does the term 'standard normal distribution' mean?

Answer: Normal distribution with $\mu = 0$, $\sigma = 1$. Normalized version with mean 0 and standard deviation 1.

Flashcard 14: What is the complementary probability for $z > 1.645$?

Answer: 5%. Upper tail probability beyond 1.645 standard deviations.

Flashcard 15: What is the probability of $z < -3$?

Answer: Approximately 0.13%. Lower tail probability beyond three standard deviations.

Flashcard 16: What does a z-score represent?

Answer: Number of standard deviations from the mean. Measures distance from mean in standard deviation units.

Flashcard 17: Find the z-score for the 75th percentile.

Answer: $z \approx 0.675$. Third quartile z-score in standard normal distribution.

Flashcard 18: What is the cumulative distribution function (CDF) at $z = 0$?

Answer: 0.5. At mean, exactly half the data is below.

Flashcard 19: What is the probability of $z < 0.33$?

Answer: Approximately 62.97%. Cumulative probability up to 0.33 standard deviations.

Flashcard 20: What is the probability of $z < -1.96$ or $z > 1.96$?

Answer: Approximately 5%. Combined tail probabilities beyond critical values.

Flashcard 21: Find the x-value for $z = -1$, $\mu = 100$, $\sigma = 10$.

Answer: $x = 90$. Solving $-1 = \frac{x-100}{10}$ gives $x = 90$.

Flashcard 22: What proportion of data falls within 2 standard deviations in a normal distribution?

Answer: Approximately 95%. From the empirical rule for normal distributions.

Flashcard 23: Convert $z = 0.5$ to a percentage of data below.

Answer: Approximately 69.15%. Half standard deviation above mean corresponds to this percentile.

Flashcard 24: Determine the interquartile range (IQR) for a standard normal distribution.

Answer: IQR = $1.349$. Distance between 75th and 25th percentiles.

Flashcard 25: What percentage of data is beyond $z = 3$?

Answer: Approximately 0.13%. Combined tail areas beyond three standard deviations.

Flashcard 26: Identify the symbol for standard deviation in a normal distribution.

Answer: $\sigma$ (sigma). Greek letter representing population standard deviation parameter.

Flashcard 27: Find the z-score for the 25th percentile.

Answer: $z \approx -0.675$. First quartile z-score in standard normal distribution.

Flashcard 28: Calculate the probability for $1 < z < 2$.

Answer: Approximately 13.59%. Area between one and two standard deviations above mean.

Flashcard 29: What is the probability of $z < -3$?

Answer: Approximately 0.13%. Lower tail probability beyond three standard deviations.

Flashcard 30: What is the probability of $z < 0.33$?

Answer: Approximately 62.97%. Cumulative probability up to 0.33 standard deviations.

Flashcard 31: Find the probability of $z = 0.84$.

Answer: Approximately 79.93%. Cumulative probability up to 0.84 standard deviations.

Flashcard 32: What is the cumulative distribution function (CDF) at $z = 0$?

Answer: 0.5. At mean, exactly half the data is below.

Flashcard 33: What is the relationship between mean, median, and mode in a normal distribution?

Answer: They are equal. Symmetry property of normal distributions.

Flashcard 34: Find the probability of $z = 0.84$.

Answer: Approximately 79.93%. Cumulative probability up to 0.84 standard deviations.

Flashcard 35: What proportion of data falls within 3 standard deviations in a normal distribution?

Answer: Approximately 99.7%. From the empirical rule for normal distributions.

Flashcard 36: Calculate the probability of $z > 1.96$.

Answer: Approximately 2.5%. Upper tail beyond critical value for 95% confidence.

Flashcard 37: What is the skewness of a normal distribution?

Answer: The skewness is 0. Normal distributions are perfectly symmetric, not skewed.

Flashcard 38: What percentage of data is beyond $z = 3$?

Answer: Approximately 0.13%. Combined tail areas beyond three standard deviations.

Flashcard 39: What proportion of data falls within 1 standard deviation in a normal distribution?

Answer: Approximately 68%. From the empirical rule for normal distributions.

Flashcard 40: Find the z-score for $x = 92$, $\mu = 100$, $\sigma = 8$.

Answer: $z = -1$. Using $z = \frac{92-100}{8} = -1$.

Flashcard 41: Given $z = 2$, what percentage of data is below this z-score?

Answer: Approximately 97.5%. Two standard deviations above mean in standard normal distribution.

Flashcard 42: What proportion of data falls within 2 standard deviations in a normal distribution?

Answer: Approximately 95%. From the empirical rule for normal distributions.

Flashcard 43: What is the z-score formula?

Answer: $z = \frac{x - \mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.

Flashcard 44: Find the x-value for $z = -1$, $\mu = 100$, $\sigma = 10$.

Answer: $x = 90$. Solving $-1 = \frac{x-100}{10}$ gives $x = 90$.

Flashcard 45: Find the z-score for $x = 85$, $\mu = 80$, $\sigma = 5$.

Answer: $z = 1$. Using $z = \frac{85-80}{5} = 1$.

Flashcard 46: Describe the tails of a normal distribution.

Answer: The tails are asymptotic to the x-axis. Approach but never touch the horizontal axis.

Flashcard 47: State the empirical rule for a normal distribution.

Answer: 68%-95%-99.7% of data within 1, 2, 3 standard deviations. Also known as the 68-95-99.7 rule for normal distributions.

Flashcard 48: Calculate the probability for $1 < z < 2$.

Answer: Approximately 13.59%. Area between one and two standard deviations above mean.

Flashcard 49: What is the z-score formula?

Answer: $z = \frac{x - \mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.

Flashcard 50: Identify the symmetry property of a normal distribution.

Answer: Symmetric around the mean. Mean divides distribution into two equal halves.

Flashcard 51: Find the z-score for $x = 92$, $\mu = 100$, $\sigma = 8$.

Answer: $z = -1$. Using $z = \frac{92-100}{8} = -1$.

Flashcard 52: Find the z-score for the 25th percentile.

Answer: $z \approx -0.675$. First quartile z-score in standard normal distribution.

Flashcard 53: What is the standard deviation of a standard normal distribution?

Answer: The standard deviation is 1. Standard normal has unit variance and standard deviation.

Flashcard 54: Determine the interquartile range (IQR) for a standard normal distribution.

Answer: IQR = $1.349$. Distance between 75th and 25th percentiles.

Flashcard 55: What is the complementary probability for $z > 1.645$?

Answer: 5%. Upper tail probability beyond 1.645 standard deviations.

Flashcard 56: What z-score corresponds to the 90th percentile?

Answer: $z \approx 1.28$. Critical value for 90% of data below this point.

Flashcard 57: What proportion of data falls within 1 standard deviation in a normal distribution?

Answer: Approximately 68%. From the empirical rule for normal distributions.

Flashcard 58: Identify the symbol for standard deviation in a normal distribution.

Answer: $\sigma$ (sigma). Greek letter representing population standard deviation parameter.

Flashcard 59: What proportion of data falls within 3 standard deviations in a normal distribution?

Answer: Approximately 99.7%. From the empirical rule for normal distributions.

Flashcard 60: Determine the probability of $z < -2$.

Answer: Approximately 2.28%. Lower tail probability beyond two standard deviations.

Flashcard 61: Find the z-score for $x = 85$, $\mu = 80$, $\sigma = 5$.

Answer: $z = 1$. Using $z = \frac{85-80}{5} = 1$.

Flashcard 62: State the empirical rule for a normal distribution.

Answer: 68%-95%-99.7% of data within 1, 2, 3 standard deviations. Also known as the 68-95-99.7 rule for normal distributions.

Flashcard 63: What is the probability of $z < -1.96$ or $z > 1.96$?

Answer: Approximately 5%. Combined tail probabilities beyond critical values.

Flashcard 64: Identify the symbol for the mean in a normal distribution.

Answer: $\mu$ (mu). Greek letter representing population mean parameter.

Flashcard 65: Given $z = 2$, what percentage of data is below this z-score?

Answer: Approximately 97.5%. Two standard deviations above mean in standard normal distribution.

Flashcard 66: Convert $z = 0.5$ to a percentage of data below.

Answer: Approximately 69.15%. Half standard deviation above mean corresponds to this percentile.

Flashcard 67: Identify the symmetry property of a normal distribution.

Answer: Symmetric around the mean. Mean divides distribution into two equal halves.