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Statistical Analysis

Definition

Statistical analysis involves collecting, analyzing, interpreting, presenting, and organizing data. Key concepts include measures of center (mean, median, mode) and measures of spread (range, standard deviation).

Prerequisites

  1. Data Interpretation
  2. Basic Statistics
  3. Ratios and Proportions
  4. Percentages

Learning Objectives

  1. Calculate mean, median, and mode for a data set.
  2. Calculate range and understand basic concepts of spread.
  3. Understand the concept of standard deviation (calculation often not required on SAT, but understanding is).
  4. Interpret box plots and histograms.
  5. Understand concepts of sampling, bias, and drawing conclusions from data.

Examples

Example 1: Measures of Center

Problem: Find the mean, median, and mode of the data set: {2, 3, 5, 5, 7, 8} Solution: * Mean: (2+3+5+5+7+8) / 6 = 30 / 6 = 5 * Median: The middle two numbers are 5 and 5. The average is (5+5)/2 = 5. * Mode: The number 5 appears most often.

Example 2: Range

Problem: Find the range of the data set: {2, 3, 5, 5, 7, 8} Solution: Range = Maximum - Minimum = 8 - 2 = 6.

Common Misconceptions

  1. Confusing mean, median, and mode.
  2. Assuming correlation implies causation.
  3. Misinterpreting the impact of outliers on mean vs. median.
  4. Errors in calculating weighted averages.

Applications in SAT

This is a core component of the Problem Solving and Data Analysis section. Questions test understanding of mean, median, mode, range, standard deviation (conceptually), box plots, histograms, and drawing conclusions from statistical data.

Advanced Connections

  1. Probability
  2. Inferential Statistics (Hypothesis testing, confidence intervals)
  3. Data Science

Practice Problems

  1. Basic: Find the median of {10, 4, 8, 4, 9}.
  2. Intermediate: A student's scores are {85, 90, 75, 90, 80}. If they score 100 on the next test, how much will their mean score increase?
  3. Advanced: (Given a box plot) Identify the median, interquartile range, and potential outliers.
  4. SAT-Level: A survey reported a mean score of 80 with a margin of error of +/- 3 points. What is the interval containing the plausible true mean score?