Basics of Statistics II Quiz: Chapters 1-3

Course Information

• Course title: Basics of Statistics II
• Chapters covered: 1-3

Review of Lecture 1

• Basic notions of Statistics
  • Descriptive and Inferential Statistics
  • Population, Sample, Variable, Data
  • Statistical model, parameter, statistic
• Organizing Data (table, chart, diagram)
  • Frequency table
  • Pie chart
  • Bar chart
  • Histogram
  • Dotplot
  • Stem-and-leaf diagram
• Shapes of distributions:
  • Modality
  • Symmetry
  • Skewness
  • (Approximate) normal distributions
• Descriptive measures of a dataset
  • Measures of center: mean, median, mode

Populations, Samples, and Variables

• Recall: Population, Sample, Variable: Height x
• Population: Students of the university
• Sample: Students of the class
• Variable: Height x (example of a quantitative variable)
• Statistical model x~N (μ, σ²)
• Parameters: μ, σ²
• Statistics: x, s²
• Confidence interval: 98% confident that 167.5 < μ < 190.4
• Hypothesis testing: Reject σ² = 1.01 at 5% significance level

Qualitative and Quantitative Variable/Data

• Recall: Qualitative (categorical) and quantitative variable / data
• Qualitative variable: Takes non-numerical names or labels
  • Examples: Gender, Birth month, Favorite movie, Major, Campus
• Discrete variable: Takes finite or countable different numbers
  • Examples: Age, Household size, Number of siblings, Shoe size
• Continuous variable: Takes a range/interval of numbers
  • Examples: Height, Weight, Distance, Temperature, Foot length

Organizing Data Using Tables, Charts, and Diagrams

• Recall: Use table, chart, diagram to organize data
• Categorical data: Non-numerical names or labels
  • (Relative) frequency table
  • Pie chart
  • Bar chart
• Discrete data: Finite or countable different numbers
  • Dotplot
  • Stem-and-leaf diagram (stemplot)
• Continuous data: A range/interval of numbers
  • (Relative) frequency table
  • Histogram

Distribution Shapes

• Recall: Distribution Shapes
• Shape of a distribution: modality, symmetry, and skewness
• Normal (or approximately normal) distribution:
  • Has (approximately) bell-shaped density curve

Descriptive Measures

• Descriptive measures
  • Measures of center
    • Mean
    • Median
    • Mode
  • Measures of variation
    • Range
    • Standard deviation
    • Variance
  • Three standard deviation rule
  • Five-number summary
    • Quartiles
    • IQR, Upper and lower limits
    • Boxplot (box-and-whisker plot)
  • Descriptive measures of populations
    • Population mean
    • Population standard deviation

Measures of Center (mean, median, mode)

• Mean of a Data Set (Quantitative data): Sum of observations / number of observations
• Median of a Data Set (Quantitative data)
  • Arrange data in increasing order
  • If odd number of observations, median is the middle observation
  • If even number of observations, median is the mean of the two middle observations
• Mode of a Data Set (Quantitative and Qualitative data)
  • Find the frequency of each value in the data set
  • Value with greatest frequency is the mode
• Example data sets and calculations are presented

Sample Mean Statistic

• Sample Mean
• Summation Notation
• Check

Measures of Center (Midrange)

• Midrange = (maximum data value + minimum data value) / 2

Measures of Variation

• Range of a Data Set = Max – Min
• Sample Standard Deviation
• Sample Variance

Measures of Variation (compute sample mean and variance)

• Formula for variance calculation, using Σ notations

Examples: Heights of Basketball Players (Data and calculations)

• Example data table provided for analysis (Team I and Team II)

Three-Standard-Deviations Rule

• Almost all observations are within 3 standard deviations from the mean

Empirical Rule for Bell-Shaped Distributions

• About 68% of values fall within 1 standard deviation of the mean
• About 95% of values fall within 2 standard deviations of the mean
• About 99.7% of values fall within 3 standard deviations of the mean

The Five-Number Summary

• Quartiles
  • First quartile (Q1): Median of the bottom half of the data
  • Second quartile (Q2): Median of the entire data
  • Third quartile (Q3): Median of the top half of the data
• Example data sets (Weekly TV-Viewing Times example data set and calculations are presented in the slides)
• Lower and Upper Limits calculation, and detecting outliers.

Boxplots

• Steps to construct a boxplot
  • Determine quartiles
  • Determine potential outliers and adjacent values
  • Plot quartiles and adjacent values
  • Connect quartiles to make a box
  • Connect box to adjacent values with lines
  • Plot potential outliers with an asterisk

Boxplots (Applications)

• Comparing Datasets (skin fold thickness of different groups)
• Detecting potential outliers

Boxplots (Applications and Types of Distributions)

• Right-skewed, Symmetric, and Left-skewed distributions

Descriptive Measures for Populations

• Population Mean (Mean of a Variable)
• Population Standard Deviation (Standard Deviation of a Variable)

Descriptive Measures for Populations (cont.)

• Parameter and Statistic
  • Parameter: Descriptive measure for a population
  • Statistic: Descriptive measure for a sample
• Standardized Variable
• Z-score (Sample z-score, formula)

Preview: Probability theory for inferential statistics

• Understanding data generating models (e.g., x ~ N(μ, σ²), x ~ Bern(p), x ~ Multi(P1, ..., Pk, k), y \ x ~ N(β₀ + β₁x, σ²))
• Understanding sampling distributions (e.g., x̄ ~ N(μ, σ²/n), (x̄ – μ) / (s/√n) ~ tₙ₋₁)

Preview: Probability theory for inferential statistics (cont.)

• Formulate and understand inferential statements (e.g., Data Set 31: Commute Times. Example of confidence intervals, and hypothesis tests)

Test your knowledge on the fundamental concepts of Statistics presented in Chapters 1-3 of Basics of Statistics II. This quiz covers key topics such as descriptive and inferential statistics, data organization, and population/sample basics. Challenge yourself with questions on measures of center, distributions, and statistical models.