AP Stats Chapter 2 Review

Percentiles and Graphs

• A percentile indicates the value below which a percentage of observations fall.
• Cumulative relative frequency graphs represent the accumulation of relative frequency for a distribution.
• An ogive is another term for a cumulative relative frequency graph.

Density Curves

• The mean of a density curve represents the balance point of the curve.
• The median is the point that divides the area under the curve into two equal halves.
• Normal distribution is defined by a mean (μ) and a standard deviation (σ), abbreviated as N(μ, σ).

Z-Scores

• A z-score standardizes an observation by calculating how many standard deviations it is from the mean: z = (x - mean) / standard deviation.
• A standard normal distribution specifically has a mean of 0 and a standard deviation of 1.
• Positive z-scores indicate values above average, while negative z-scores indicate values below average.

Data Assessment Techniques

• Initial assessment of data involves checking for skewness or normal distribution.
• Adding or subtracting a constant shifts the center without changing the shape or spread of the data.
• Measures such as the number of data points, standard deviation, IQR, range, and distribution shape remain unchanged when constants are added or subtracted.

Effects of Linear Transformations

• A linear transformation adjusts a variable via an equation: x-new = a + bx, where 'a' is a constant added and 'b' is a multiplier.
• Dividing or multiplying data by a constant retains the shape but alters measures of center and spread proportionally.

Describing Distributions

• A density curve features an area of 1 and is always non-negative.
• Median is the equal-areas point, and the mean is the balance point of the density curve.
• On skewed curves, the mean is pulled toward the long tail, affecting its relationship with the median.

Location and Proportions

• Individual placement within a distribution can be described using percentiles and z-scores.
• Cumulative relative frequency graphs effectively illustrate an individual's location in a distribution.

Statistical Notations

• Symbols include θ for population proportion, p-hat for sample proportion, μ for population mean, x-bar for sample mean, σ for population standard deviation, and s for sample standard deviation.
• Parameters pertain to entire populations, while statistics pertain to samples.

Normal Distribution Characteristics

• Normal curves are symmetric, single-peaked, and bell-shaped.
• All normal curves function as density curves but asserting a dataset as normally distributed should be stated as "approximately" normal.
• A steep normal distribution peak indicates a smaller standard deviation, while a wider peak indicates a larger standard deviation.

Empirical Rule

• The empirical rule outlines probability distributions in normal data: approximately 68% within ±1 standard deviation, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations from the mean.