Mean and Standard Deviation
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AP Statistics › Mean and Standard Deviation
A discrete random variable $Q$ represents the number of questions a student answers correctly on a 12-question quiz. The distribution of $Q$ has mean $\mu_Q=8.9$ and standard deviation $\sigma_Q=1.4$. Which interpretation of the standard deviation is correct?
The score must always be between 7.5 and 10.3 because $8.9\pm1.4$.
The standard deviation 1.4 is the difference between the highest and lowest possible quiz scores.
The mean number correct is 1.4.
The student will score exactly 8.9 correct on most quizzes.
The standard deviation 1.4 means the score is usually within about 1.4 questions of 8.9 correct.
Explanation
This question focuses on interpreting standard deviation in a quiz score context. The standard deviation σ = 1.4 measures the typical deviation of quiz scores from the mean μ = 8.9. Choice B correctly states "the standard deviation 1.4 means the score is usually within about 1.4 questions of 8.9 correct." This properly interprets standard deviation as describing typical variability around the mean. Choice E incorrectly treats μ ± σ as absolute bounds (7.5 to 10.3), when scores can fall outside this range. Standard deviation describes typical variation, not the range of all possible scores. For a 12-question quiz, scores could range from 0 to 12, well beyond one standard deviation from the mean.
A school counselor records the number of days absent in a semester for each student. Let $X$ be the number of absences for a randomly selected student; $X$ is discrete. The distribution of $X$ has mean $\mu_X=5$ days and standard deviation $\sigma_X=2$ days. Which interpretation of the standard deviation is correct?
The number of absences is usually between 3 and 7 days because $\sigma_X=2$.
A typical student's number of absences differs from 5 days by about 2 days.
The standard deviation 2 means the range of absences is 2 days.
Exactly half of students have 5 or fewer absences.
Most students are absent exactly 2 days.
Explanation
This question evaluates understanding the standard deviation of a discrete random variable in AP Statistics. The SD σ_X = 2 means a typical student's absences deviate from the mean of 5 by about 2 days, as accurately stated in choice C. This reflects the average distance from the mean in terms of root-mean-square. Distractor B incorrectly suggests SD defines a usual range like mean ± SD/2 or something similar, but it doesn't specify bounds that way. Choice A confuses SD with the mode, implying most have exactly 2 absences, which isn't what SD measures. In a mini-lesson, standard deviation is sqrt(Var(X)), where Var(X) = E[(X - $μ)^2$], providing a scale for how spread out the values are around the mean for discrete distributions.
A library tracks the number of books checked out per student in a week. Let $X$ be the number of books checked out by a randomly selected student; $X$ is discrete. The distribution of $X$ has mean $\mu_X=3.2$ books and standard deviation $\sigma_X=1.5$ books. Which interpretation of the standard deviation is correct?
The number of books checked out ranges from 1.7 to 4.7 because the standard deviation is 1.5.
A typical student's number of books checked out differs from 3.2 by about 1.5 books.
The standard deviation 1.5 means the maximum number of books any student can check out is 1.5.
About 1.5% of students check out 3.2 books.
Exactly 1.5 books are checked out by most students.
Explanation
This question assesses the interpretation of the standard deviation for a discrete random variable in AP Statistics. The standard deviation σ_X = 1.5 indicates that the number of books checked out typically deviates from the mean of 3.2 by about 1.5 books, as correctly captured in choice A. This means, on average, the root-mean-square deviation from the mean is 1.5, providing a sense of typical variability. A distractor like choice C incorrectly assumes the range is strictly mean ± SD, but standard deviation doesn't guarantee all values fall within that interval, especially without knowing the distribution shape. Choice B is wrong because it treats SD as a mode or exact value rather than a measure of spread. In a mini-lesson, standard deviation quantifies variability: for discrete X, it's the square root of the variance, where variance is the expected value of (X - $μ)^2$, helping understand how much values fluctuate around the mean.
A delivery app records the number of orders a driver completes in a 4-hour shift. Let $X$ be the number of orders for a randomly selected shift; $X$ is discrete. The distribution of $X$ has mean $\mu_X=14$ orders and standard deviation $\sigma_X=4$ orders. Which interpretation of the mean is correct?
Half of shifts have fewer than 14 orders and half have more than 14 orders.
Most shifts result in exactly 14 orders.
In the long run, drivers complete about 14 orders per 4-hour shift on average.
The probability a driver completes 14 orders is 0.14.
Drivers complete between 10 and 18 orders in every shift because $\sigma_X=4$.
Explanation
This question focuses on interpreting the mean of a discrete random variable in AP Statistics. The mean μ_X = 14 signifies the long-run average number of orders completed per 4-hour shift, correctly interpreted in choice B. Over many shifts, the average would stabilize around 14 orders. Choice E is a distractor that mistakes the mean for the median, which isn't necessarily true unless the distribution is symmetric. Choice C wrongly uses the standard deviation to imply a fixed range of 10 to 18, but SD measures average deviation, not absolute limits. In a mini-lesson, the mean E(X) for a discrete random variable is the sum of x * P(X=x) over all possible x, representing the center of the probability distribution and the expected outcome in repeated trials.
A discrete random variable $T$ is the number of tardy students in a school’s first-period class on a randomly chosen day. The distribution of $T$ has mean $\mu_T=2.0$ and standard deviation $\sigma_T=0.6$. Which interpretation of the mean is correct?
The number tardy is always between 1.4 and 2.6 because $\sigma_T=0.6$.
Exactly 2 students are tardy on a typical day.
Over many days, the average number of tardy students is about 2.0.
Half the days have 2.0 or fewer tardy students.
The number tardy cannot be more than 2.0.
Explanation
This question tests the interpretation of mean for a discrete random variable. The mean μ = 2.0 represents the expected or average number of tardy students over many days. Choice A correctly states "over many days, the average number of tardy students is about 2.0." This is the fundamental interpretation of expected value - it's the long-run average. Choice B incorrectly suggests exactly 2 students are tardy on a typical day, confusing the mean with the mode. The mean tells us about the center of the distribution over many observations, not necessarily what happens most frequently. For discrete distributions, the mean often isn't even a possible value of the variable.
A discrete random variable $S$ represents the number of songs skipped during a 1-hour music streaming session. The distribution of $S$ has mean $\mu_S=7.5$ and standard deviation $\sigma_S=3.0$. Which interpretation of the standard deviation is correct?
The standard deviation 3.0 is the maximum possible number of skipped songs.
The number of skipped songs is always between 4.5 and 10.5.
Most sessions have exactly 7.5 skipped songs.
A typical session’s number of skipped songs is about 3.0 away from the mean of 7.5.
In the long run, the average number of skipped songs is 3.0.
Explanation
This question assesses understanding of standard deviation interpretation. The standard deviation σ = 3.0 measures how much the number of skipped songs typically varies from the mean μ = 7.5. Choice B correctly states "a typical session's number of skipped songs is about 3.0 away from the mean of 7.5." This properly captures that standard deviation measures typical distance from the mean. Choice C incorrectly treats μ ± σ as absolute bounds (4.5 to 10.5), when values can certainly fall outside this range. Standard deviation describes typical variability, not the range of all possible values. Understanding this distinction is crucial for proper statistical interpretation.
A discrete random variable $W$ is the number of games a basketball player makes at least one 3-point shot in during a 10-game stretch. The distribution of $W$ has mean $\mu_W=6.7$ and standard deviation $\sigma_W=1.5$. Which interpretation of the mean is correct?
The standard deviation is 6.7 because that is the typical number of such games.
Over many 10-game stretches, the player averages about 6.7 games with at least one 3-point shot.
Half of all 10-game stretches have 6.7 or fewer such games.
The player’s number of such games is always within 1.5 of 6.7.
In exactly 6.7 of the 10 games, the player will make at least one 3-point shot.
Explanation
This question tests understanding of the mean in a binomial-like context. The mean μ = 6.7 represents the expected number of games with at least one 3-point shot over many 10-game stretches. Choice A correctly states "over many 10-game stretches, the player averages about 6.7 games with at least one 3-point shot." This is the proper interpretation of expected value - it's the long-run average. Choice B incorrectly suggests exactly 6.7 games, which is impossible since the number of games must be a whole number. The mean represents what happens on average across many repetitions of the 10-game experiment, not what happens in any single stretch.
Let $Y$ be a discrete random variable representing the number of text messages a student receives during a 30-minute class period. The distribution of $Y$ has mean $\mu_Y=3.4$ and standard deviation $\sigma_Y=2.1$. Which interpretation of the standard deviation is correct?
In the long run, the average number of texts per class period is 2.1.
The standard deviation 2.1 is the difference between the maximum and minimum possible number of texts.
Most class periods have exactly 3.4 texts.
A typical class period’s number of texts differs from 3.4 by about 2.1 texts.
The number of texts is always between $3.4-2.1$ and $3.4+2.1$.
Explanation
This question focuses on interpreting the standard deviation of a discrete random variable. The standard deviation σ = 2.1 measures the typical deviation from the mean μ = 3.4. Choice C correctly states that "a typical class period's number of texts differs from 3.4 by about 2.1 texts." This captures the essence of standard deviation - it quantifies how spread out the data is from the mean. Choice B incorrectly claims all values fall within one standard deviation of the mean, which is false. Standard deviation describes typical variability, not absolute bounds. Understanding that standard deviation measures typical distance from the mean is crucial for statistical interpretation.
A discrete random variable $X$ represents the number of customer complaints received by a small company in a day. Over many days, the distribution of $X$ has mean $\mu_X=1.8$ complaints and standard deviation $\sigma_X=1.2$ complaints. Which interpretation of the mean is correct?
Half of all days have 1.8 complaints or fewer.
The number of complaints per day is typically between 0.6 and 3.0 because the standard deviation is 1.2.
The company will receive exactly 1.8 complaints on most days.
The maximum possible number of complaints in a day is 1.8.
In the long run, the average number of complaints per day is about 1.8.
Explanation
This question tests understanding of the mean of a discrete random variable. The mean μ = 1.8 represents the long-run average or expected value of the number of complaints per day. Choice C correctly states that "in the long run, the average number of complaints per day is about 1.8." This is the fundamental interpretation of expected value - it's what we expect on average over many observations. Choice A incorrectly suggests the company receives exactly 1.8 complaints most days, which is impossible since complaints must be whole numbers. The mean tells us about the center of the distribution over many days, not what happens on any single day.
A discrete random variable $N$ represents the number of times a website user refreshes a page during a 5-minute visit. The distribution of $N$ has mean $\mu_N=2.6$ and standard deviation $\sigma_N=2.0$. Which interpretation of the mean is correct?
The maximum possible number of refreshes in 5 minutes is 2.6.
Over many 5-minute visits, the average number of refreshes is about 2.6.
The user refreshes the page exactly 2.6 times in a typical 5-minute visit.
The number of refreshes is usually between 0.6 and 4.6 because the standard deviation is 2.0.
Half of all visits have 2.6 or fewer refreshes.
Explanation
This question tests the interpretation of mean for a discrete random variable. The mean μ = 2.6 represents the expected or average number of page refreshes over many 5-minute visits. Choice B correctly states "over many 5-minute visits, the average number of refreshes is about 2.6." This is the fundamental meaning of expected value - it's what we expect on average in the long run. Choice A incorrectly suggests a typical visit has exactly 2.6 refreshes, which is impossible since refreshes must be whole numbers. The mean is a theoretical average that emerges over many observations, not a value that occurs frequently in practice.