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Chapter 3: Numerically Summarizing Data

Section 3.1: Measures of Central Tendency

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • parameter
      • statistic
      • population
      • sample
      • quantitative data
    2. Section 2.2:
      • distribution
      • skewness of a distribution
      • shape of a distribution
  2. procedural knowledge
    1. Section 1.3:
      • how to find a SRS
  3. conditional knowledge
    1. Section 1.1:
      • how to distinguish between a population and a sample
    2. Section 2.2:
      • identify and explain why a distribution is symmetric, skewed left, or skewed right

Learning Goals

  1. declarative knowledge (definitions)
    1. arithmetic mean
    2. sample mean, x bar
    3. population mean, μ
    4. median
    5. mode (only appropriate for survey data)
    6. resistant measure of center
  2. procedural knowledge
    1. how to calculate sample and population mean using TI83/84: http://stats.jjw3.com/math1431/ti83sum.htm
    2. how to calculate sample and population median using TI83/84: http://stats.jjw3.com/math1431/ti83sum.htm
  3. conditional knowledge
    1. know the data and numerical summaries
    2. know the importance of using the correct symbol for population mean and sample mean
    3. interpret the mean and median of a distribution, i.e., explain what they describe about a distribution
    4. explain why a measure is a resistant measure
    5. explain the effect of an extreme measure on the mean and median of a distribution
    6. how to identify the approximate locations of the mean and median from symmetric or skewed distributions
    7. how to identify whether the mean or median is more appropriate for data
    8. how to compare two samples using the mean or median
    9. know which measure of center is a resistant measure and explain why
    10. explain what happens to the sample mean as sample size increases
    11. explain when mode is appropriate
    12. identify when there is no mode of a distribution
  4. important notes
    1. you are required to use the correct symbols for population mean, μ, and sample mean, x bar, in all answers to questions

Section 3.2: Measures of Dispersion (a.k.a., Spread)

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • parameter
      • statistic
      • population
      • sample
      • quantitative data
    2. Section 2.2:
      • distribution
    3. Section 3.1:
      • mean
      • resistant measure
  2. procedural knowledge
    1. Section 1.3:
      • how to find a SRS
  3. conditional knowledge
    1. Section 1.1:
      • how to distinguish between a population and a sample

Learning Goals

  1. declarative knowledge (definitions)
    1. dispersion
    2. range, R
    3. deviation about the mean
    4. population standard deviation, σ
    5. sample standard deviation, s
    6. degrees of freedom
    7. population variance, σ2
    8. sample variance, s2
    9. biased statistic
    10. Empirical Rule (a.k.a., 68-95-99.7 Rule)
  2. procedural knowledge
    1. how to calculate range
    2. how to calculate sample and population standard deviation using TI83/84: http://stats.jjw3.com/math1431/ti83sum.htm
    3. how to calculate sample and population variance using TI83/84: http://stats.jjw3.com/math1431/ti83sum.htm
    4. how to calculate sample and population standard deviation by hand (Honors Only)
    5. how to calculate sample and population variance by hand (Honors Only)
    6. how to use the Empirical Rule to estimate percentage of data
  3. conditional knowledge
    1. know the importance of using the correct symbol for population standard deviation and sample standard deviation
    2. know the importance of using the correct symbol for population variance and sample variance
    3. explain what measure of dispersion describes about a distribution
    4. know which measure of spread is a resistant measure and explain why
    5. how to identify which measure of spread is more appropriate for data
    6. how to compare two distributions using measures of spread
    7. interpret the measure spread of a distribution, i.e., explain what the measure describes about a distribution
    8. explain the effect of an extreme measure on the a measure of spread of a distribution
  4. important notes
    1. you are required to use the correct symbols for population standard deviation, σ, and sample standard deviation, s, in all answers to questions
    2. you are required to use the correct symbols for population variance, σ2, and sample variance, s2, in all answers to questions

Section 3.4: Measures of Position and Outliers

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • parameter
      • statistic
      • population
      • sample
      • quantitative data
    2. Section 2.2:
      • distribution
      • outlier
      • skewness of a distribution
    3. Section 3.1:
      • mean
      • median
      • resistant measure
    4. Section 3.2:
      • standard deviation
  2. procedural knowledge
    1. Section 3.1:
      • how to calculate median
  3. conditional knowledge
    1. Section 1.1:
      • how to distinguish between a population and a sample
    2. Section 2.2:
      • identify and explain why a distribution is symmetric, skewed left, or skewed right

Learning Goals

  1. declarative knowledge (definitions)
    1. sample z-score
    2. population z-score
    3. mean and standard deviation of z
    4. percentiles
    5. quartiles: Q1, Q3
    6. interquartile range, IQR
    7. lower fence
    8. upper fence
  2. procedural knowledge
    1. how to calculate z-score
    2. how to calculate Q1 and Q3 using the TI83/84: http://stats.jjw3.com/math1431/ti83sum.htm
    3. how to calculate IQR
    4. how to use the lower fence and the upper fence to determine if an observation is an outlier
  3. conditional knowledge
    1. how to interpret percentiles
    2. how to interpret quartiles
    3. how to interpret z-scores
    4. how to use z-scores to relatively compare observations from two different samples or populations
    5. how to use quartiles to determine the shape of a distribution
  4. important notes
    1. in homework question #9, lower era is better

Section 3.5: The Five-Number Summary and Boxplots

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • quantitative data
    2. Section 2.2:
      • distribution
      • outlier
      • skewness of a distribution
    3. Section 3.1:
      • mean
      • median
      • resistant measure
    4. Section 3.4:
      • quartiles
  2. procedural knowledge
    1. Section 3.1:
      • how to calculate median
    2. Section 3.4:
      • how to use the lower fence and the upper fence to determine if an observation is an outlier
  3. conditional knowledge
    1. Section 2.2:
      • identify and explain why a distribution is symmetric, skewed left, or skewed right
    2. Section 2.3:
      • identify graphical misrepresentations of data

Learning Goals

  1. declarative knowledge (definitions)
    1. exploratory data analysis (EDA)
    2. five-number summary [NOTE: this is a list of five numbers separated by a space only]
    3. boxplot (a.k.a., box-n-whiskers plot)
  2. procedural knowledge
    1. how to construct a boxplot on TI83/84: http://stats.jjw3.com/math1431/ti83box.htm
    2. how to calculate the five-number summary on TI83/84: http://stats.jjw3.com/math1431/ti83sum.htm
    3. how to find the five-number summary using a boxplot on TI83/84: http://stats.jjw3.com/math1431/ti83box.htm
    4. how to describe a distribution (i.e., shape; center; spread; outliers - explicitly identify and list outliers)
  3. conditional knowledge
    1. how to determine the shape of a distribution from a boxplot
    2. how to compare two different distributions using boxplots

Chapter 3: Required Formulas – Need to Know for Tests

  1. Population Mean: Population Mean
  2. Sample Mean: Sample Mean
  3. Population Variance [square the Population Standard Deviation]: σ2
  4. Sample Variance [square the Sample Standard Deviation]: s2
  5. Population z-Score: Population z-Score
  6. Sample z-Score: Sample z-Score
  7. Interquartile Range: IQR = Q3 – Q1
  8. Lower Fence: Lower Fence = Q1 – 1.5(IQR)
  9. Upper Fence: Upper Fence = Q3 + 1.5(IQR)