Probability - SAT Math

The mean is 6. Sum all values (30) and divide by count (5).

Mean absolute deviation = 1. Average distance from mean is 1.

z = 2. $(10-8) ÷ 1 = 2$ standard deviations.

Negatively skewed: tail extends to the left. Most data on right, few low values.

Mean. Outliers pull the mean toward extreme values more than median or mode.

Normal distribution. Bell-shaped curve with mean, median, and mode all equal.

Variance = 0.25. Mean is 2.5, deviations squared and averaged.

Range = 17. Difference between maximum and minimum values: $20 - 3 = 17$.

It shows cumulative totals of data points up to each class. Running total increases at each point.

It displays quantitative data to show distribution details. Separates tens and ones digits clearly.

Mean = 5.75. $(4+8+6+5) ÷ 4 = 5.75$

A dot plot uses dots to represent frequency of data values. Each dot equals one data point.

The median is the middle value when data is ordered. Sort values first, then find center.

A scatter plot shows the relationship between two variables. Points reveal correlation patterns.

The mode is the most frequently occurring value. Count frequency of each value.

Range = 2. $10 - 8 = 2$

Range = 14. $19 - 5 = 14$

A box plot displays the distribution, median, and quartiles of a data set. Five-number summary in visual format.

A distribution with two distinct peaks or modes. Two separate frequency peaks exist.

The shape is right-skewed. Long tail extends toward higher values on the right.

z = $\frac{x - \text{mean}}{\text{standard deviation}}$. Standardizes values by measuring distance from mean in standard deviations.

Quartiles divide data into four equal parts. 25%, 50%, 75% split points.

Variance = $\frac{\text{Sum of squared deviations from mean}}{\text{Number of data points}}$. Measures spread by averaging squared distances from mean.

Standard Deviation = 4. Standard deviation is the square root of variance: $\sqrt{16} = 4$.

Median = 7. Middle value when data is ordered from least to greatest.

A symmetric distribution is mirrored around its center. Both sides look identical when folded.

$Variance = \frac{\sum (x_i - \text{mean})^2}{n}$. Square deviations from mean, then average.

Positively skewed: tail extends to the right. Most data on left, few high values.

$z = \frac{x - \text{mean}}{\text{standard deviation}}$. Standardizes distance from mean in units.

$Geometric mean = \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}$. Multiply all values, take nth root.

The median is 4. Middle value when ordered: 2, 3, 4, 7, 9.

Standard deviation = 0.5. Mean is 4.5, deviations squared and averaged.

In skewed data, mean is pulled toward the tail. Median stays centered, mean shifts.

Median = 5. Ordered: 3,5,9; middle value is 5.

A histogram displays the frequency distribution of data. Bars show count in each interval.

The median is the best measure for skewed data. Outliers don't affect median much.

Mode = 6. Value that appears most frequently in the data set.

The median is least affected by outliers. Extreme values don't shift median much.

It represents the frequencies of data points in a line graph. Connected dots show frequency distribution.

An outlier is a data point significantly different from others. Falls far from typical data pattern.

IQR = 10. $Q^3 - Q^1 = 15 - 5 = 10$

Median = 3. Ordered: 1,2,3,4,5; middle is 3.

The mean is most affected. Outliers pull the mean toward extreme values.

It refers to the variability or dispersion of data points. How scattered data points are.

$Mean = \frac{\sum x_i}{n}$. Sum all values, divide by count.

IQR = Q3 - Q1. Third quartile minus first quartile.

A uniform distribution has equal frequency across values. All intervals have same probability.

Range = Maximum value - Minimum value. Subtract smallest from largest value.

The strength and direction of a linear relationship. Values closer to -1 or 1 indicate stronger linear relationships.

The critical value is 1.96. Marks the boundary for 95% of the standard normal distribution.

Estimate range for population parameter. Provides likely bounds where true population value falls with given confidence.

The mode is 8. The value 8 appears most frequently (3 times).

A numerical characteristic of a population. Fixed value describing an entire population (usually unknown).

A subset of the population used for analysis. Representative portion selected from the larger population for study.

IQR = Q3 - Q1. Measures the spread of the middle 50% of data values.

A graph showing the relationship between two variables. Each point represents one observation with two variable values.

The range is 12. $15 - 3 = 12$; difference between largest and smallest values.

The confidence interval is (46.08, 53.92). $50 \pm 1.96 \times \frac{10}{\sqrt{25}} = 50 \pm 3.92$.

Histogram. Shows distribution shape and frequency of values across intervals or bins.

The median is 5. Sort values first: 2, 4, 5, 7, 9; middle value is 5.

The frequency distribution of a data set. Shows how often different values or ranges occur in data.

95% of samples contain the true population parameter. Indicates the reliability of the interval estimation method.

$s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$. Measures spread by calculating square root of variance with $n-1$ denominator.

P-value. Measures probability that observed results occurred by random chance alone.

The mean is 8.6. Sum all values and divide by count: $(3+7+8+10+15)/5 = 43/5 = 8.6$.

From -1 to 1. Perfect positive correlation is 1, perfect negative is -1.

The entire set of individuals or items of interest. All possible subjects or objects being studied in research.

Mean = 6.5. Sum all values and divide by count: $(4+5+7+10) ÷ 4 = 6.5$.

A statement that there is no effect or difference. The baseline assumption that researchers try to reject or fail to reject.

Drawing conclusions about a population based on a sample. Uses sample data to make educated guesses about the entire population.

The outlier is 100. Value that falls far outside the typical range of the data.

Range = Maximum - Minimum. Range measures the spread by finding the difference between extremes.

$H_a: \text{mean} \neq 50$. Alternative hypothesis claims the parameter is not equal to the null value.

Mean. Extreme values pull the mean away from the center more than median or mode.

Mode = 3. The value that appears most frequently in the data set.

Probability of observed result under null hypothesis. Lower p-values suggest stronger evidence against the null hypothesis.

Null hypothesis: no effect or difference. Default assumption that treatment has no impact; what we test against.

A conclusion about a population based on a sample. Uses sample data to make generalizations about the entire population.

A distribution that is not symmetric. Data is pulled more toward one tail than the other.

Bias due to non-random sample selection. Systematic error that makes samples unrepresentative of the population.

Correct: 'The sample result supports the hypothesis.'. 'Supports' is proper; samples provide evidence but never absolute proof.

Variance = $\frac{\text{sum of squared deviations}}{n-1}$. Measures how spread out data points are from the mean.

The range of error in sample estimate of population. Quantifies the uncertainty in survey or poll results.

Causal claim. Asserts one variable directly influences or causes changes in another.

The distribution of sample means approximates normality. For large samples, sample means follow a normal distribution pattern.

A bell-shaped distribution symmetric about the mean. Most data falls within a predictable range around the center.

The claim is that the mean height > 6 feet. States that the population mean exceeds 6 feet.

A graphical summary of data distribution using quartiles. Shows median, quartiles, and potential outliers in data distribution.

A numerical characteristic of a sample. Calculated value describing a sample (used to estimate parameters).

The z-score is 1. $(85-80)/5 = 1$; standardizes the value relative to the distribution.

Range = 20. Difference between maximum and minimum: $22 - 2 = 20$.

Failure to reject a false null hypothesis. Accepting a null hypothesis when the alternative is actually true.

Results unlikely due to chance, given a threshold. Indicates results are unlikely to occur by random chance alone.

Both are equally strong. Both have the same absolute value, so equal strength.

Arrange data, find middle value(s). For odd $n$, middle value; for even $n$, average of two middle values.

Probability = $\frac{13}{52}$ or $\frac{1}{4}$. Thirteen hearts in a standard deck of 52 cards.

$\frac{1}{221}$. First ace: $\frac{4}{52}$, second ace: $\frac{3}{51}$, multiply together.

$0.06$. Multiply independent probabilities: $0.3 \times 0.2 = 0.06$.

$\frac{5}{6}$. Complement of rolling a 3: $1 - \frac{1}{6} = \frac{5}{6}$.

$\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes.

$P(A') = 1 - P(A)$. Complement probability equals 1 minus the original event's probability.

$0.5$. Apply addition rule: $0.3 + 0.4 - 0.2 = 0.5$.

Probability = (Number of favorable outcomes) / (Total number of possible outcomes). This is the fundamental definition of probability in statistics.

$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply individual probabilities.

P(A or B) = P(A) + P(B). For mutually exclusive events, probabilities add since they cannot occur together.

$\frac{11}{36}$. Use complement: $1 - P(\text{no 6s}) = 1 - \frac{25}{36} = \frac{11}{36}$.

$\frac{3}{8}$. Count outcomes with exactly two tails: TTH, THT, HTT.

Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Basic probability definition: favorable outcomes divided by total outcomes.

Probability = $\frac{3}{6}$ or $\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes.

Two events are complementary if one event is the complement of the other. One event is exactly the opposite of the other.

$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply their individual probabilities.

Probability = $\frac{4}{52}$ or $\frac{1}{13}$. Four aces in a standard deck of 52 cards.

$\frac{1}{13}$. 2 red queens + 2 black kings = 4 favorable cards out of 52.

A set of events that cover all possible outcomes. Events whose probabilities sum to 1 (complete sample space).

Probability = $\frac{1}{6}$. One favorable outcome (rolling 3) out of six possible outcomes.

$\frac{11}{26}$. 13 spades + 12 face cards - 3 overlap = 22 favorable outcomes.

Events where the occurrence of one does not affect the other. One event's outcome doesn't change the other's probability.

$0.58$. Using addition rule: $0.3 + 0.4 - (0.3 \times 0.4) = 0.58$.

$\frac{1}{4}$. 13 hearts out of 52 total cards simplifies to $\frac{1}{4}$.

$\frac{5}{13}$. 13 clubs + 12 face cards - 3 club face cards = 22 favorable.

$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, probability of both occurring.

1. All possible outcomes together form the complete sample space.

Probability = $\frac{26}{52}$ or $\frac{1}{2}$. 26 red cards (hearts and diamonds) out of 52 total cards.

$\frac{1}{6}$. One favorable outcome out of six possible outcomes.

$0.5$. Complement of A: $1 - 0.5 = 0.5$.

$\frac{1}{26}$. 2 red aces (hearts and diamonds) out of 52 cards.

$\frac{1}{4}$. 13 diamonds out of 52 total cards.

$\frac{1}{13}$. 4 queens out of 52 cards simplifies to $\frac{1}{13}$.

Probability = $\frac{1}{2}$. One favorable outcome (heads) out of two possible outcomes.

$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$. Probability of A given B has occurred.

$\frac{2}{13}$. 8 favorable cards (4 aces + 4 kings) out of 52 total.

$P(A') = 1 - P(A)$. The complement probability equals 1 minus the event probability.

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Basic probability definition using favorable over total outcomes.

$P(A \text{ and } B) = 0$. Mutually exclusive events cannot happen together.

$\frac{1}{4}$. Multiply independent probabilities: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

$\frac{10}{13}$. 40 non-face cards out of 52 total cards.

1/4. 13 hearts out of 52 total cards in a standard deck.

$\frac{1}{2}$. Prime numbers on die: 2, 3, 5 (three out of six).

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. General formula accounting for overlap between events.

$\frac{1}{2}$. 26 black cards (spades and clubs) out of 52 total.

$\frac{26}{51}$. After removing one black card, 26 red remain out of 51 total.

$\frac{4}{13}$. 4 kings + 13 hearts - 1 king of hearts = 16 favorable.

$\frac{3}{4}$. Complement of all tails: $1 - \frac{1}{4} = \frac{3}{4}$.

1/6. One specific outcome (rolling 4) out of six equally likely outcomes.

Probability = $\frac{6}{36}$ or $\frac{1}{6}$. Six ways to make 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) out of 36 total.

$\frac{1}{6}$. 6 ways to roll 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

$\frac{1}{12}$. 3 ways to roll 4: (1,3), (2,2), (3,1) out of 36 possibilities.

Events where the occurrence of one affects the probability of the other. One event's outcome changes the probability of the other.

IQR = 50. Difference between third and first quartiles: $75 - 25 = 50$.

Mode = 6. 6 appears most frequently (3 times).

Mean = $\frac{\text{Sum of all data points}}{\text{Number of data points}}$. Sum all values and divide by count.

The standard deviation measures data spread around the mean. Shows how far values deviate from center.

The range is the difference between the highest and lowest values. Span from minimum to maximum value.

Skewness indicates the asymmetry of a distribution. Measures if distribution leans left or right.

Mean = 6.5. Sum all values and divide by count: $(4+5+7+10) ÷ 4 = 6.5$.

Causal claim. Asserts one variable directly influences or causes changes in another.

Histogram. Shows distribution shape and frequency of values across intervals or bins.

Null hypothesis: no effect or difference. Default assumption that treatment has no impact; what we test against.

Standard Deviation = 4. Standard deviation is the square root of variance: $\sqrt{16} = 4$.

Median = 7. Middle value when data is ordered from least to greatest.

Mode = 6. Value that appears most frequently in the data set.

IQR = 50. Difference between third and first quartiles: $75 - 25 = 50$.

Normal distribution. Bell-shaped curve with mean, median, and mode all equal.

Range = 17. Difference between maximum and minimum values: $20 - 3 = 17$.

Estimate range for population parameter. Provides likely bounds where true population value falls with given confidence.

Arrange data, find middle value(s). For odd $n$, middle value; for even $n$, average of two middle values.

Range = 20. Difference between maximum and minimum: $22 - 2 = 20$.

Probability = $\frac{6}{36}$ or $\frac{1}{6}$. Six ways to make 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) out of 36 total.

Probability = $\frac{26}{52}$ or $\frac{1}{2}$. 26 red cards (hearts and diamonds) out of 52 total cards.

Probability = $\frac{1}{6}$. One favorable outcome (rolling 3) out of six possible outcomes.

Probability = $\frac{13}{52}$ or $\frac{1}{4}$. Thirteen hearts in a standard deck of 52 cards.

P(A') = 1 - P(A). Complement probability equals 1 minus the original event's probability.

Probability = $\frac{3}{6}$ or $\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes.

Probability = $\frac{4}{52}$ or $\frac{1}{13}$. Four aces in a standard deck of 52 cards.

Probability = $\frac{1}{2}$. One favorable outcome (heads) out of two possible outcomes.

Mean. Extreme values pull the mean away from the center more than median or mode.

Mode = 3. The value that appears most frequently in the data set.

IQR = Q3 - Q1. Measures the spread of the middle 50% of data values.

$0.5$. Apply addition rule: $0.3 + 0.4 - 0.2 = 0.5$.

$\frac{3}{8}$. Count outcomes with exactly two tails: TTH, THT, HTT.

$\frac{5}{6}$. Complement of rolling a 3: $1 - \frac{1}{6} = \frac{5}{6}$.

$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply their individual probabilities.

$\frac{1}{221}$. First ace: $\frac{4}{52}$, second ace: $\frac{3}{51}$, multiply together.

$\frac{1}{6}$. One favorable outcome out of six possible outcomes.

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Basic probability definition using favorable over total outcomes.

$\frac{1}{4}$. 13 hearts out of 52 total cards simplifies to $\frac{1}{4}$.

Events that cannot occur simultaneously. If one occurs, the other cannot happen at the same time.

$P(A \text{ and } B) = 0$. Mutually exclusive events cannot happen together.

$P(A') = 1 - P(A)$. The complement probability equals 1 minus the event probability.

$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$. Probability of A given B has occurred.

$\frac{2}{13}$. 8 favorable cards (4 aces + 4 kings) out of 52 total.

$\frac{26}{51}$. After removing one black card, 26 red remain out of 51 total.

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. General formula accounting for overlap between events.

$\frac{11}{36}$. Use complement: $1 - P(\text{no 6s}) = 1 - \frac{25}{36} = \frac{11}{36}$.

$\frac{1}{4}$. Multiply independent probabilities: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

$\frac{1}{13}$. 2 red queens + 2 black kings = 4 favorable cards out of 52.

$\frac{1}{2}$. Prime numbers on die: 2, 3, 5 (three out of six).

$0.06$. Multiply independent probabilities: $0.3 \times 0.2 = 0.06$.