Unit 2: Modeling Distributions of Quantitative Data

The distribution of measures for individuals within a sample or population describes variation.

The value of a statistic varies from sample to sample. How can we determine whether differences between measures represent random variation or meaningful distinctions?

Statistical methods based on probabilistic reasoning provide the basis for shared understandings about variation and about the likelihood that variation between and among measures, samples, and populations is random or meaningful.

Statistical tools allow us to represent and describe patterns in data and to classify departures from patterns.

Simulation and probabilistic reasoning allow us to anticipate patterns in data and to determine the likelihood of errors in inference.

Measure location: percentiles, cumulative relative frequency graphs, standardized scores

Transform data

Describe density curves and normal distribtions (the empirical rule)

Find areas in a normal distribution and work backward to find values from areas

Assess normality

Students will demonstrate their mastery by transitioning from raw data points to standardized scores. They will prove they can calculate and interpret z-scores, explaining that a z-score represents the number of standard deviations a specific value falls above or below the mean. They will show they understand that standardizing data allows for a "fair comparison" between individuals from different distributions (e.g., comparing an SAT score to an ACT score).

A primary way students show understanding is through the application of the 68-95-99.7 Rule (The Empirical Rule). They will demonstrate the ability to estimate the percentage of observations that fall within one, two, or three standard deviations of the mean for a Normal distribution. They will also prove they can work in reverse: given a percentile or a proportion, they will use a standard Normal table or technology to find the corresponding value in the original units of the data.

Students will further demonstrate their skills by evaluating whether a distribution is actually Normal. They will create and interpret Normal Probability Plots, showing they understand that a linear pattern in the plot confirms the data is approximately Normal, while a curved pattern indicates the data is skewed. They will also prove they can identify the properties of a Density Curve, specifically that the total area under the curve is exactly one and that the mean is the "balance point" while the median is the "equal-areas point."

Finally, students will demonstrate their ability to solve "non-standard" Normal problems. They will prove they can find the mean or standard deviation of a distribution when given other parameters (like a specific percentile). In their final analysis, they will demonstrate a clear understanding of percentiles, explaining that a value at the 80th percentile is greater than or equal to 80% of the other values in the distribution, regardless of the distribution's shape.