Introducing Statistics: Learning from Data - AP Statistics

A null hypothesis is a statement that there is no effect or no difference, used as a starting point. Assumes no relationship exists; the default position to test against.

A p-value indicates the probability of observing the test results under the null hypothesis. Measures how likely our results are if the null hypothesis is true.

The significance level is the probability of a Type I error, typically denoted by $\text{alpha}$. Sets the threshold for rejecting the null hypothesis.

A confidence interval is a range of values that is likely to contain a population parameter. Provides a range estimate with a specified level of confidence.

A scatter plot is used to show the relationship between two quantitative variables. Each point represents one observation with two measurements.

The correlation coefficient measures the strength and direction of a linear relationship. Ranges from -1 to +1, indicating weak to strong relationships.

A correlation coefficient of 0 indicates no linear relationship between variables. The variables move independently with no linear pattern.

A residual is the difference between an observed value and the value predicted by a model. Shows how far off the model's prediction was from reality.

A residual plot is used to assess the fit of a regression model. Helps identify patterns in errors and model appropriateness.

Line of best fit: $y = mx + b$, where $m$ is the slope, $b$ is the y-intercept. Standard linear equation form for the regression line.

Extrapolation involves predicting beyond the range of observed data. Making predictions outside the observed data range.

A categorical variable is a variable that represents categories or groups. Takes on distinct labels or names rather than numerical values.

A continuous variable can take an infinite number of values within a given range. Can be measured to any desired precision within its range.

A contingency table displays the frequency distribution of variables. Cross-tabulates two categorical variables to show relationships.

Relative frequency is calculated as the frequency of an event divided by the total number of observations. Converts counts to proportions for easier comparison.

A normal distribution is a bell-shaped distribution that is symmetric about the mean. The classic bell curve with equal tails on both sides.

Skewness describes asymmetry in the distribution of values in a data set. Indicates whether data leans left or right from center.

An outlier is a data point that differs significantly from other observations. An unusual value that stands apart from the typical pattern.

A box plot visually displays the distribution and identifies outliers of a data set. Shows five-number summary and highlights unusual values.

A histogram is a graphical representation of the distribution of numerical data. Shows frequency of data values using bars or bins.

Standard deviation is the square root of the variance. Provides a measure of spread in the same units as the data.

Variance = $\frac{\text{sum of squared deviations from mean}}{n-1}$. Uses $n-1$ for sample variance to correct for bias.

Range measures the difference between the maximum and minimum values. Simple measure of spread calculated as max - min.

The mode is the value that appears most frequently in a data set. Identifies the most common or repeated value in the dataset.

The median is the middle value when data points are arranged in order. Half the values are above and half are below this central value.

To summarize or describe the characteristics of a data set. Focus is on organizing and presenting data, not making predictions.

A sample is a subset of the population from which data is actually collected. A smaller group selected from the population for practical data collection.

A z-score measures how many standard deviations an element is from the mean. Standardizes values for comparison across different datasets.

The empirical rule states that 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations from the mean, respectively. Applies specifically to normal distributions with predictable percentages.