2.2.10 - Five Number Summary | STAT 200

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2.2.10 - Five Number Summary

A five number summary can be used to communicate some key descriptive statistics. It is comprised of five values, presented in the following order:

  1. Minimum: Smallest value
  2. First quartile (Q1): 25th percentile (value that separates the bottom 25% of the distribution from the top 75%)
  3. Median: Middle value (50th percentile)
  4. Third quartile (Q3): 75th percentile (value that separates the bottom 75% of the distribution from the top 25%)
  5. Maximum: Largest value
Five Number Summary Minimum, Q1, Median, Q3, Maximum

Five number summaries are used to describe some of the key features of a distribution. Using the values in a five number summary we can also compute the range and interquartile range.

Range The difference between the maximum and minimum values. Range \(Range = Maximum - Minimum\) Note: The range is heavily influenced by outliers. For this reason, the interquartile range is often preferred because it is resistant to outliers. Interquartile range (IQR) The difference between the first and third quartiles. Interquartile Range \(IQR = Q_3 - Q_1\)

Example: Hours Spent Studying Section

Studying

A professor asks a sample of students how many hours they spent studying for the final. The five number summary for their responses is (5, 7, 9, 11, 13).

Range

The maximum is 13 and the minimum is 5.

\(Range = 13 - 5 = 8\)

Interquartile Range

The third quartile is 11 and the first quartile is 7.

\(IQR = Q_3 - Q_1 = 11 - 7 = 4\)

Example: Test Scores Section

A teacher wants to examine students’ test scores. The five number summary for their scores is (74, 80, 89, 90, 98).

Range

The highest score is 98. The lowest score is 74.

\(Range = 98 - 74 = 24\)

Interquartile Range

The third quartile is 90 and the first quartile is 80.

\(IQR = Q3 - Q1 = 90 - 80 = 10\)

  • Previous2.2.9 - Percentiles
  • Next2.3 - Lesson 2 Summary

Lessons

  • Welcome to STAT 200!
  • 0: Prerequisite Skills
    • 0.1 - Review of Algebra
      • 0.1.1 - Order of Operations
      • 0.1.2 - Summations
      • 0.1.3 - Basic Linear Equations
    • 0.2 - Introduction to Minitab
    • 0.3 - Word's Equation Editor
    • 0.4 - Canvas' Equation Editor
  • 1: Collecting Data
    • 1.1 - Cases & Variables
      • 1.1.1 - Categorical & Quantitative Variables
      • 1.1.2 - Explanatory & Response Variables
    • 1.2 - Samples & Populations
      • 1.2.1 - Sampling Bias
      • 1.2.2 - Sampling Methods
        • 1.2.2.1 - Minitab: Simple Random Sampling
    • 1.3 - Other Sources of Bias
    • 1.4 - Research Study Design
      • 1.4.1 - Confounding Variables
      • 1.4.2 - Causal Conclusions
      • 1.4.3 - Independent and Paired Samples
      • 1.4.4 - Control and Placebo Groups
      • 1.4.5 - Blinding
    • 1.5 - Lesson 1 Summary
  • 2: Describing Data, Part 1
    • 2.1 - Categorical Variables
      • 2.1.1 - One Categorical Variable
        • 2.1.1.1 - Risk and Odds
        • 2.1.1.2 - Visual Representations
          • 2.1.1.2.1 - Minitab: Frequency Tables
          • 2.1.1.2.2 - Minitab: Pie Charts
          • 2.1.1.2.3 - Minitab: Bar Charts
      • 2.1.2 - Two Categorical Variables
        • 2.1.2.1 - Minitab: Two-Way Contingency Table
        • 2.1.2.2 - Minitab: Clustered Bar Chart
        • 2.1.2.3 - Minitab: Stacked Bar Chart
      • 2.1.3 - Probability Rules
        • 2.1.3.1 - Range of Probabilities
        • 2.1.3.2 - Combinations of Events
          • 2.1.3.2.1 - Disjoint & Independent Events
          • 2.1.3.2.2 - Intersections
          • 2.1.3.2.3 - Unions
          • 2.1.3.2.4 - Complements
          • 2.1.3.2.5 - Conditional Probability
            • 2.1.3.2.5.1 - Advanced Conditional Probability Applications
    • 2.2 - One Quantitative Variable
      • 2.2.1 - Graphs: Dotplots and Histograms
      • 2.2.2 - Outliers
      • 2.2.3 - Shape
      • 2.2.4 - Measures of Central Tendency
        • 2.2.4.1 - Skewness & Central Tendency
      • 2.2.5 - Measures of Spread
      • 2.2.6 - Minitab: Central Tendency & Variability
      • 2.2.7 - The Empirical Rule
      • 2.2.8 - z-scores
      • 2.2.9 - Percentiles
      • 2.2.10 - Five Number Summary
    • 2.3 - Lesson 2 Summary
  • 3: Describing Data, Part 2
    • 3.1 - Single Boxplot
    • 3.2 - Identifying Outliers: IQR Method
    • 3.3 - One Quantitative and One Categorical Variable
    • 3.4 - Two Quantitative Variables
      • 3.4.1 - Scatterplots
        • 3.4.1.1 - Minitab: Simple Scatterplot
      • 3.4.2 - Correlation
        • 3.4.2.1 - Formulas for Computing Pearson's r
        • 3.4.2.2 - Example of Computing r by Hand (Optional)
        • 3.4.2.3 - Minitab: Compute Pearson's r
      • 3.4.3 - Simple Linear Regression
        • 3.4.3.1 - Minitab: SLR
        • 3.4.3.2 - Example: Interpreting Output
    • 3.5 - Relations between Multiple Variables
      • 3.5.1 - Scatterplot with Groups
      • 3.5.2 - Bubble Plots
      • 3.5.3 - Time Series Plot
    • 3.6 - Lesson 3 Summary
  • 4: Confidence Intervals
    • 4.1 - Sampling Distributions
      • 4.1.1 - StatKey Examples
        • 4.1.1.1 - NFL Salaries (One Mean)
        • 4.1.1.2 - Coin Flipping (One Proportion)
      • 4.1.2 - Copying Data into StatKey
      • 4.1.3 - Impact of Sample Size
    • 4.2 - Introduction to Confidence Intervals
      • 4.2.1 - Interpreting Confidence Intervals
      • 4.2.2 - Applying Confidence Intervals
    • 4.3 - Introduction to Bootstrapping
      • 4.3.1 - Example: Bootstrap Distribution for Proportion of Peanuts
      • 4.3.2 - Example: Bootstrap Distribution for Difference in Mean Exercise
    • 4.4 - Bootstrap Confidence Interval
      • 4.4.1 - StatKey: Standard Error Method
        • 4.4.1.1 - Example: Proportion of Lactose Intolerant German Adults
        • 4.4.1.2 - Example: Difference in Mean Commute Times
      • 4.4.2 - StatKey: Percentile Method
        • 4.4.2.1 - Example: Correlation Between Quiz & Exam Scores
        • 4.4.2.2 - Example: Difference in Dieting by Biological Sex
        • 4.4.2.3 - Example: One sample mean sodium content
    • 4.5 - Paired Samples
    • 4.6 - Impact of Sample Size on Confidence Intervals
    • 4.7 - Lesson 4 Summary
  • 5: Hypothesis Testing, Part 1
    • 5.1 - Introduction to Hypothesis Testing
    • 5.2 - Writing Hypotheses
      • 5.2.1 - Examples
    • 5.3 - Randomization Procedures
      • 5.3.1 - StatKey Randomization Methods (Optional)
    • 5.4 - p-values
    • 5.5 - Randomization Test Examples in StatKey
      • 5.5.1 - Single Proportion Example: PA Residency
      • 5.5.2 - Paired Means Example: Age
      • 5.5.3 - Difference in Means Example: Exercise by Biological Sex
      • 5.5.4 - Correlation Example: Quiz & Exam Scores
    • 5.6 - Lesson 5 Summary
  • 6: Hypothesis Testing, Part 2
    • 6.1 - Type I and Type II Errors
    • 6.2 - Significance Levels
    • 6.3 - Issues with Multiple Testing
    • 6.4 - Practical Significance
    • 6.5 - Power
    • 6.6 - Confidence Intervals & Hypothesis Testing
    • 6.7 - Lesson 6 Summary
  • 7: Normal Distributions
    • 7.1 - Standard Normal Distribution
    • 7.2 - Minitab: Finding Proportions Under a Normal Distribution
      • 7.2.1 - Proportion 'Less Than'
        • 7.2.1.1 - Example: P(Z<-1)
        • 7.2.1.2 - Example: P(SATM<540)
      • 7.2.2 - Proportion 'Greater Than'
        • 7.2.2.1 - Example: P(Z>0.5)
      • 7.2.3 - Proportion 'In between'
        • 7.2.3.1 - Example: Proportion Between z -2 and +2
      • 7.2.4 - Proportion 'More Extreme Than'
    • 7.3 - Minitab: Finding Values Given Proportions
      • 7.3.1 - Top X%
      • 7.3.2 - Bottom X%
      • 7.3.3 - Middle X%
    • 7.4 - Central Limit Theorem
      • 7.4.1 - Hypothesis Testing
        • 7.4.1.1 - Video Example: Mean Body Temperature
        • 7.4.1.2 - Video Example: Correlation Between Printer Price and PPM
        • 7.4.1.3 - Example: Proportion NFL Coin Toss Wins
        • 7.4.1.4 - Example: Proportion of Women Students
        • 7.4.1.5 - Example: Mean Quiz Score
        • 7.4.1.6 - Example: Difference in Mean Commute Times
      • 7.4.2 - Confidence Intervals
        • 7.4.2.1 - Video Example: 98% CI for Mean Atlanta Commute Time
        • 7.4.2.2 - Video Example: 90% CI for the Correlation between Height and Weight
        • 7.4.2.3 - Example: 99% CI for Proportion of Women Students
    • 7.5 - Lesson 7 Summary
  • 8: Inference for One Sample
    • 8.1 - One Sample Proportion
      • 8.1.1 - Confidence Intervals
        • 8.1.1.1 - Normal Approximation Formulas
          • 8.1.1.1.1 - Video Example: PA Residency
          • 8.1.1.1.2 - Video Example: Dog Ownership
          • 8.1.1.1.3 - Video Example: Books
          • 8.1.1.1.4 - Example: Retirement
        • 8.1.1.2 - Minitab: Confidence Interval for a Proportion
          • 8.1.1.2.1 - Example with Summarized Data
          • 8.1.1.2.2 - Example with Summarized Data
        • 8.1.1.3 - Computing Necessary Sample Size
      • 8.1.2 - Hypothesis Testing
        • 8.1.2.1 - Normal Approximation Method Formulas
          • 8.1.2.1.1 - Video Example: Male Babies
          • 8.1.2.1.2 - Example: Handedness
          • 8.1.2.1.3 - Example: Ice Cream
          • 8.1.2.1.4 - Example: Overweight Citizens
        • 8.1.2.2 - Minitab: Hypothesis Tests for One Proportion
          • 8.1.2.2.1 - Minitab: 1 Proportion z Test, Raw Data
          • 8.1.2.2.2 - Minitab: 1 Sample Proportion z test, Summary Data
            • 8.1.2.2.2.1 - Minitab Example: Normal Approx. Method
    • 8.2 - One Sample Mean
      • 8.2.1 - t Distribution
      • 8.2.2 - Confidence Intervals
        • 8.2.2.1 - Formulas
          • 8.2.2.1.1 - Example: MLB Age
          • 8.2.2.1.2- Example: Sleep Deprivation
          • 8.2.2.1.3 - Example: Milk
        • 8.2.2.2 - Minitab: Confidence Interval of a Mean
          • 8.2.2.2.1 - Example: Age of Pitchers (Summarized Data)
          • 8.2.2.2.2 - Example: Coffee Sales (Data in Column)
        • 8.2.2.3 - Computing Necessary Sample Size
          • 8.2.2.3.1 - Example: Estimating IQ
          • 8.2.2.3.2 - Video Example: Age
          • 8.2.2.3.3 - Video Example: Cookie Weights
      • 8.2.3 - Hypothesis Testing
        • 8.2.3.1 - One Sample Mean t Test, Formulas
          • 8.2.3.1.1 - Video Example: Book Costs
          • 8.2.3.1.2 : Example: Pulse Rate
          • 8.2.3.1.3 - Example: Coffee
          • 8.2.3.1.4 - Example: Transportation Costs
        • 8.2.3.2 - Minitab: One Sample Mean t Tests
          • 8.2.3.2.1 - Minitab: 1 Sample Mean t Test, Raw Data
          • 8.2.3.2.2 - Minitab: 1 Sample Mean t Test, Summarized Data
        • 8.2.3.3 - One Sample Mean z Test (Optional)
    • 8.3 - Paired Means
      • 8.3.1 - Confidence Intervals
        • 8.3.1.1. - Example: Change in Knowledge
        • 8.3.1.2 - Video Example: Difference in Exam Scores
      • 8.3.2 - Hypothesis Testing
        • 8.3.2.1 - Example: Quiz Scores
      • 8.3.3 - Minitab: Paired Means Test
        • 8.3.3.1 - Example: SAT Scores
        • 8.3.3.2 - Example: Marriage Age (Summarized Data)
    • 8.4 - Lesson 8 Summary
  • 9: Inference for Two Samples
    • 9.1 - Two Independent Proportions
      • 9.1.1 - Confidence Intervals
        • 9.1.1.1 - Minitab: Confidence Interval for 2 Proportions
      • 9.1.2 - Hypothesis Testing
        • 9.1.2.1 - Normal Approximation Method Formulas
          • 9.1.2.1.1 – Example: Ice Cream
          • 9.1.2.1.2 – Example: Same Sex Marriage
        • 9.1.2.2 - Minitab: Difference Between 2 Independent Proportions
          • 9.1.2.2.1 - Example: Dating
    • 9.2 - Two Independent Means
      • 9.2.1 - Confidence Intervals
        • 9.2.1.1 - Minitab: Confidence Interval Between 2 Independent Means
          • 9.2.1.1.1 - Video Example: Mean Difference in Exam Scores, Summarized Data
      • 9.2.2 - Hypothesis Testing
        • 9.2.2.1 - Minitab: Independent Means t Test
          • 9.2.2.1.1 - Example: Summarized Data
          • 9.2.2.1.3 - Example: Height by Sex
    • 9.3 - Lesson 9 Summary
  • 10: One-Way ANOVA
    • 10.1 - Introduction to the F Distribution
    • 10.2 - Hypothesis Testing
    • 10.3 - Pairwise Comparisons
    • 10.4 - Minitab: One-Way ANOVA
    • 10.5 - Example: SAT-Math Scores by Award Preference
    • 10.6 - Example: Exam Grade by Professor
    • 10.7 - Lesson 10 Summary
  • 11: Chi-Square Tests
    • 11.1 - Reviews
      • 11.1.1 - Frequency Table
      • 11.1.2 - Two-Way Contingency Table
      • 11.1.3 - Probability Distribution Plots
      • 11.1.4 - Conditional Probabilities and Independence
    • 11.2 - Goodness of Fit Test
      • 11.2.1 - Five Step Hypothesis Testing Procedure
        • 11.2.1.1 - Video: Cupcakes (Equal Proportions)
        • 11.2.1.2- Cards (Equal Proportions)
        • 11.2.1.3 - Roulette Wheel (Different Proportions)
      • 11.2.2 - Minitab: Goodness-of-Fit Test
        • 11.2.2.1 - Example: Summarized Data, Equal Proportions
        • 11.2.2.2 - Example: Summarized Data, Different Proportions
    • 11.3 - Chi-Square Test of Independence
      • 11.3.1 - Example: Gender and Online Learning
      • 11.3.2 - Minitab: Test of Independence
        • 11.3.2.1 - Example: Raw Data
        • 11.3.2.2 - Example: Summarized Data
      • 11.3.3 - Relative Risk
    • 11.4 - Lesson 11 Summary
  • 12: Correlation & Simple Linear Regression
    • 12.1 - Review: Scatterplots
    • 12.2 - Correlation
      • 12.2.1 - Hypothesis Testing
        • 12.2.1.1 - Example: Quiz & Exam Scores
        • 12.2.1.2 - Example: Age & Height
        • 12.2.1.3 - Example: Temperature & Coffee Sales
      • 12.2.2 - Correlation Matrix
        • 12.2.2.1 - Example: Student Survey
        • 12.2.2.2 - Example: Body Correlation Matrix
    • 12.3 - Simple Linear Regression
      • 12.3.1 - Formulas
      • 12.3.2 - Assumptions
      • 12.3.3 - Minitab - Simple Linear Regression
      • 12.3.4 - Hypothesis Testing for Slope
        • 12.3.4.1 - Example: Quiz and exam scores
        • 12.3.4.2 - Example: Business Decisions
      • 12.3.5 - Confidence Interval for Slope
        • 12.3.5.1 - Example: Quiz and exam scores
    • 12.4 - Coefficient of Determination
    • 12.5 - Cautions
    • 12.6 - Correlation & Regression Example
    • 12.7 - Lesson 12 Summary

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