AP Statistics Flashcards: Mean And Standard Deviation

What this deck covers

This deck focuses on Mean And Standard Deviation, 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: Identify the standard deviation of a Poisson random variable with rate $\text{lambda}$.

Answer: Standard deviation is $\text{sqrt(lambda)}$. Poisson standard deviation is square root of rate parameter.

Flashcard 2: What is the variance of the difference of two independent random variables $X$ and $Y$?

Answer: $Var(X-Y) = Var(X) + Var(Y)$. Variances still add even when subtracting variables.

Flashcard 3: What is the formula for the variance of a linear transformation $Y = a + bX$?

Answer: $Var(Y) = b^2 Var(X)$. Variance multiplied by $b^2$; constant $a$ has no effect.

Flashcard 4: If $X$ is a random variable with variance 25, what is $Var(2X + 3)$?

Answer: $Var(2X + 3) = 100$. Constant doesn't affect variance: $Var(2X + 3) = 4(25)$.

Flashcard 5: Calculate the variance for $P(X=1)=0.4$, $P(X=2)=0.6$. Mean is $1.6$.

Answer: Variance is $0.24$. Sum $(x_i - 1.6)^2 \times p_i$ for each value.

Flashcard 6: Find the variance of a random variable $X$ with $Var(X)=9$ after $Y=2X-3$.

Answer: $Var(Y) = 36$. Apply variance rule: $Var(Y) = 2^2 \times 9 = 36$.

Flashcard 7: What is the variance of a Poisson random variable with rate $\text{lambda}$?

Answer: $Var(X) = \text{lambda}$. For Poisson distribution, variance equals the rate parameter.

Flashcard 8: What is the variance of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$?

Answer: $Var(X) = \text{sigma}^2$. Normal distribution variance is $\sigma^2$ by definition.

Flashcard 9: What is the mean of the sum of two independent random variables $X$ and $Y$?

Answer: $E(X+Y) = E(X) + E(Y)$. Expected values always add for independent variables.

Flashcard 10: What is the mean of a Poisson random variable with rate $\text{lambda}$?

Answer: $E(X) = \text{lambda}$. For Poisson distribution, mean equals the rate parameter.

Flashcard 11: For $X$ uniform from 2 to 8, calculate the variance.

Answer: Variance is $3$. Uniform variance: $\frac{(8-2)^2}{12} = \frac{36}{12} = 3$.

Flashcard 12: Calculate the mean of a random variable $X$ with $E(X) = 5$ after transformation $Y = 3X + 2$.

Answer: $E(Y) = 17$. Apply linear transformation: $E(Y) = 3(5) + 2 = 17$.

Flashcard 13: State the formula for the variance of a discrete random variable $X$.

Answer: $Var(X) = \text{sum of } [(x_i - \bar{x})^2 \times p_i]$. Each squared deviation from mean times its probability, then sum.

Flashcard 14: Calculate the variance of $X$ with $Var(X)=16$ after $Y=-X$.

Answer: $Var(Y) = 16$. Multiplying by $-1$ doesn't change variance: $(-1)^2 = 1$.

Flashcard 15: What is the effect on the mean if $X$ is shifted by $c$, i.e., $Y = X + c$?

Answer: Mean increases by $c$, $E(Y) = E(X) + c$. Adding constant shifts mean but doesn't change spread.

Flashcard 16: What is the effect on the variance if $X$ is shifted by $c$, i.e., $Y = X + c$?

Answer: Variance remains unchanged, $Var(Y) = Var(X)$. Adding constant doesn't affect variability measures.

Flashcard 17: Calculate the variance for $P(X=1)=0.4$, $P(X=2)=0.6$. Mean is $1.6$.

Answer: Variance is $0.24$. Sum $(x_i - 1.6)^2 \times p_i$ for each value.

Flashcard 18: How do you calculate the standard deviation from variance?

Answer: Take the square root of the variance, $\text{SD}(X) = \text{sqrt}(Var(X))$. Standard deviation is always the positive square root of variance.

Flashcard 19: What is the mean of $X$ if $X$ follows a normal distribution with mean $\text{mu}$?

Answer: $E(X) = \text{mu}$. Normal distribution mean is the location parameter $\mu$.

Flashcard 20: Identify the standard deviation of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$.

Answer: Standard deviation is $\text{sigma}$. Standard deviation is square root of variance: $\sqrt{\sigma^2} = \sigma$.

Flashcard 21: What is the mean of the difference of two independent random variables $X$ and $Y$?

Answer: $E(X-Y) = E(X) - E(Y)$. Expected values subtract when finding difference.

Flashcard 22: Calculate the mean of a random variable $X$ with $E(X) = 5$ after transformation $Y = 3X + 2$.

Answer: $E(Y) = 17$. Apply linear transformation: $E(Y) = 3(5) + 2 = 17$.

Flashcard 23: Calculate the variance for $X$ with $Var(X) = 4$ after transformation $Y = 3X + 2$.

Answer: $Var(Y) = 36$. Apply variance rule: $Var(Y) = 3^2 \times 4 = 36$.

Flashcard 24: What is the formula for the variance of a linear transformation $Y = a + bX$?

Answer: $Var(Y) = b^2 Var(X)$. Variance multiplied by $b^2$; constant $a$ has no effect.

Flashcard 25: What is the variance of a Poisson random variable with rate $\text{lambda}$?

Answer: $Var(X) = \text{lambda}$. For Poisson distribution, variance equals the rate parameter.

Flashcard 26: Identify the mean of a uniformly distributed random variable $X$ over $a$ to $b$.

Answer: $E(X) = \frac{a+b}{2}$. Uniform mean is the midpoint of the interval.

Flashcard 27: What is the mean of a binomial random variable $X$ with $n$ trials and probability $p$?

Answer: $E(X) = n \times p$. Binomial mean equals number of trials times success probability.

Flashcard 28: State the formula for the mean of a linear transformation $Y = a + bX$.

Answer: $E(Y) = a + bE(X)$. Linear transformation: add constant $a$, multiply by $b$.

Flashcard 29: What is the variance of the difference of two independent random variables $X$ and $Y$?

Answer: $Var(X-Y) = Var(X) + Var(Y)$. Variances still add even when subtracting variables.

Flashcard 30: What is the variance of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$?

Answer: $Var(X) = \text{sigma}^2$. Normal distribution variance is $\sigma^2$ by definition.

Flashcard 31: What is the variance of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?

Answer: $Var(X) = \frac{1}{\text{lambda}^2}$. Exponential variance is reciprocal of rate squared.

Flashcard 32: For a random variable $X$, what happens to the standard deviation if $X$ is multiplied by $b$?

Answer: Standard deviation is multiplied by $|b|$. Multiplying by $b$ scales standard deviation by $|b|$.

Flashcard 33: What is the variance of $X$ if $X$ is a geometric random variable with probability $p$?

Answer: $Var(X) = \frac{1-p}{p^2}$. Geometric variance uses $(1-p)$ in numerator, $p^2$ in denominator.

Flashcard 34: What is the effect on the mean if $X$ is shifted by $c$, i.e., $Y = X + c$?

Answer: Mean increases by $c$, $E(Y) = E(X) + c$. Adding constant shifts mean but doesn't change spread.

Flashcard 35: State the formula for the variance of a discrete random variable $X$.

Answer: $Var(X) = \text{sum of } [(x_i - \bar{x})^2 \times p_i]$. Each squared deviation from mean times its probability, then sum.

Flashcard 36: Calculate the variance of $X$ with $Var(X)=16$ after $Y=-X$.

Answer: $Var(Y) = 16$. Multiplying by $-1$ doesn't change variance: $(-1)^2 = 1$.

Flashcard 37: For a random variable $X$, what happens to the standard deviation if $X$ is multiplied by $b$?

Answer: Standard deviation is multiplied by $|b|$. Multiplying by $b$ scales standard deviation by $|b|$.

Flashcard 38: Calculate the mean of $X$ if $X$ is uniform from 1 to 5.

Answer: $E(X) = 3$. Uniform from 1 to 5: mean = $(1+5)/2 = 3$.

Flashcard 39: Find the mean of a random variable $X$ with $E(X)=4$ after $Y=2X-3$.

Answer: $E(Y) = 5$. Apply transformation: $E(Y) = 2(4) - 3 = 5$.

Flashcard 40: Identify the mean of a uniformly distributed random variable $X$ over $a$ to $b$.

Answer: $E(X) = \frac{a+b}{2}$. Uniform mean is the midpoint of the interval.

Flashcard 41: For $X$ uniform from 2 to 8, calculate the variance.

Answer: Variance is $3$. Uniform variance: $\frac{(8-2)^2}{12} = \frac{36}{12} = 3$.

Flashcard 42: Find the mean of a random variable $X$ given $P(X=1)=0.2$, $P(X=2)=0.5$, $P(X=3)=0.3$.

Answer: $E(X) = 2.1$. Sum each value times its probability: $1(0.2) + 2(0.5) + 3(0.3)$.

Flashcard 43: What is the formula for the expected value of a discrete random variable $X$?

Answer: $E(X) = \text{sum of } (x_i \times p_i)$. Each value times its probability, then sum all products.

Flashcard 44: Find the mean of a random variable $X$ given $P(X=1)=0.2$, $P(X=2)=0.5$, $P(X=3)=0.3$.

Answer: $E(X) = 2.1$. Sum each value times its probability: $1(0.2) + 2(0.5) + 3(0.3)$.

Flashcard 45: State the formula for the mean of a linear transformation $Y = a + bX$.

Answer: $E(Y) = a + bE(X)$. Linear transformation: add constant $a$, multiply by $b$.

Flashcard 46: What is the variance of a uniformly distributed random variable $X$ over $a$ to $b$?

Answer: $Var(X) = \frac{(b-a)^2}{12}$. Uniform variance uses range squared divided by 12.

Flashcard 47: What is the standard deviation of a binomial random variable $X$ with $n$ trials and probability $p$?

Answer: $\text{SD}(X) = \sqrt{n \times p \times (1-p)}$. Binomial standard deviation uses $n$, $p$, and $(1-p)$ under square root.

Flashcard 48: How do you calculate the standard deviation from variance?

Answer: Take the square root of the variance, $\text{SD}(X) = \text{sqrt}(Var(X))$. Standard deviation is always the positive square root of variance.

Flashcard 49: Find the standard deviation for $Var(X)=4$.

Answer: $\text{SD}(X) = 2$. Standard deviation is the square root of variance.

Flashcard 50: What is the mean of $X$ if $X$ is a geometric random variable with probability $p$?

Answer: $E(X) = \frac{1}{p}$. Geometric mean is reciprocal of success probability.

Flashcard 51: Calculate the mean of $X$ if $X$ is uniform from 1 to 5.

Answer: $E(X) = 3$. Uniform from 1 to 5: mean = $(1+5)/2 = 3$.

Flashcard 52: Calculate the mean of $X$ with $E(X)=3$ after $Y=-X$.

Answer: $E(Y) = -3$. Multiplying by $-1$ changes sign: $E(-X) = -E(X)$.

Flashcard 53: What is the mean of the difference of two independent random variables $X$ and $Y$?

Answer: $E(X-Y) = E(X) - E(Y)$. Expected values subtract when finding difference.

Flashcard 54: What is the mean of $X$ if $X$ is a geometric random variable with probability $p$?

Answer: $E(X) = \frac{1}{p}$. Geometric mean is reciprocal of success probability.

Flashcard 55: What is the mean of $X$ if $X$ follows a normal distribution with mean $\text{mu}$?

Answer: $E(X) = \text{mu}$. Normal distribution mean is the location parameter $\mu$.

Flashcard 56: What is the standard deviation of a binomial random variable $X$ with $n$ trials and probability $p$?

Answer: $\text{SD}(X) = \text{sqrt}(n \times p \times (1-p))$. Binomial standard deviation uses $n$, $p$, and $(1-p)$ under square root.

Flashcard 57: What is the variance of $X$ if $X$ is a geometric random variable with probability $p$?

Answer: $Var(X) = \frac{1-p}{p^2}$. Geometric variance uses $(1-p)$ in numerator, $p^2$ in denominator.

Flashcard 58: Calculate the variance for $X$ with $Var(X) = 4$ after transformation $Y = 3X + 2$.

Answer: $Var(Y) = 36$. Apply variance rule: $Var(Y) = 3^2 \times 4 = 36$.

Flashcard 59: Calculate the mean of $X$ with $E(X)=3$ after $Y=-X$.

Answer: $E(Y) = -3$. Multiplying by $-1$ changes sign: $E(-X) = -E(X)$.

Flashcard 60: If $X$ is a random variable with variance 25, what is $Var(2X + 3)$?

Answer: $Var(2X + 3) = 100$. Constant doesn't affect variance: $Var(2X + 3) = 4(25)$.

Flashcard 61: Find the mean of a random variable $X$ with $E(X)=4$ after $Y=2X-3$.

Answer: $E(Y) = 5$. Apply transformation: $E(Y) = 2(4) - 3 = 5$.

Flashcard 62: Find the variance of a random variable $X$ with $Var(X)=9$ after $Y=2X-3$.

Answer: $Var(Y) = 36$. Apply variance rule: $Var(Y) = 2^2 \times 9 = 36$.

Flashcard 63: Identify the standard deviation of a Poisson random variable with rate $\lambda$.

Answer: Standard deviation is $\sqrt{\lambda}$. Poisson standard deviation is square root of rate parameter.

Flashcard 64: What is the mean of a Poisson random variable with rate $\text{lambda}$?

Answer: $E(X) = \text{lambda}$. For Poisson distribution, mean equals the rate parameter.

Flashcard 65: What is the mean of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?

Answer: $E(X) = \frac{1}{\text{lambda}}$. Exponential mean is reciprocal of rate parameter.

Flashcard 66: What is the variance of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?

Answer: $Var(X) = \frac{1}{\text{lambda}^2}$. Exponential variance is reciprocal of rate squared.

Flashcard 67: What is the variance of the sum of two independent random variables $X$ and $Y$?

Answer: $Var(X+Y) = Var(X) + Var(Y)$. Variances add for independent variables (never subtract).

Flashcard 68: If $X$ is a random variable with mean 10, what is $E(2X + 3)$?

Answer: $E(2X + 3) = 23$. Apply linear transformation: $E(2X + 3) = 2(10) + 3$.

Flashcard 69: If $X$ is a random variable with mean 10, what is $E(2X + 3)$?

Answer: $E(2X + 3) = 23$. Apply linear transformation: $E(2X + 3) = 2(10) + 3$.

Flashcard 70: What is the variance of a uniformly distributed random variable $X$ over $a$ to $b$?

Answer: $Var(X) = \frac{(b-a)^2}{12}$. Uniform variance uses range squared divided by 12.

Flashcard 71: What is the variance of the sum of two independent random variables $X$ and $Y$?

Answer: $Var(X+Y) = Var(X) + Var(Y)$. Variances add for independent variables (never subtract).

Flashcard 72: Identify the standard deviation of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$.

Answer: Standard deviation is $\text{sigma}$. Standard deviation is square root of variance: $\sqrt{\sigma^2} = \sigma$.

Flashcard 73: What is the formula for the expected value of a discrete random variable $X$?

Answer: $E(X) = \text{sum of } (x_i \times p_i)$. Each value times its probability, then sum all products.

Flashcard 74: What is the mean of a binomial random variable $X$ with $n$ trials and probability $p$?

Answer: $E(X) = n \times p$. Binomial mean equals number of trials times success probability.

Flashcard 75: What is the mean of the sum of two independent random variables $X$ and $Y$?

Answer: $E(X+Y) = E(X) + E(Y)$. Expected values always add for independent variables.

Flashcard 76: Find the standard deviation for $Var(X)=4$.

Answer: $\text{SD}(X) = 2$. Standard deviation is the square root of variance.

Flashcard 77: What is the mean of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?

Answer: $E(X) = \frac{1}{\text{lambda}}$. Exponential mean is reciprocal of rate parameter.