Math 105 - Midterm 1 Study Guide

Sampling Methods and Study Design

  • Stratified Sampling:

    • Definition: A sampling method where the population is divided into separate, non-overlapping sub-groups (strata) based on shared traits or characteristics, and a random sample is selected from each stratum.

    • Example Application: A market researcher selects 500 drivers under 30 years of age and 500 drivers over 30 years of age. The two age categories represent distinct strata.

  • Observational Studies versus Experiments:

    • Experiment:

    • Definition: A study in which researchers actively impose a treatment or intervention on subjects (experimental units) to observe and measure the resulting responses.

    • Example Application: A clinical trial where 10 patients are administered a drug and 10 patients are administered a placebo to assess the drug's effectiveness against an illness.

    • Observational Study:

    • Definition: A study in which researchers observe and collect data on subjects without manipulating variables or imposing treatments.

    • Example Application: A sample of fish is taken from a lake to measure environmental pollution effects caused by a nearby factory.

Parameters versus Statistics

  • Parameter:

    • Definition: A numerical measurement describing a characteristic of an entire population.

    • Example Application: After inspecting all 55,000 kg of meat stored at the Wurst Sausage Company, exactly 45,000 kg was found to be spoiled. Because the entire population of meat at the company was inspected, this calculated quantity is a parameter.

  • Statistic:

    • Definition: A numerical measurement describing a characteristic of a sample drawn from a larger population.

    • Example Application: In a sample of 100,000 randomly selected adults asked whether they drink at least 48 oz of water each day, 45% responded affirmatively. Because this value is calculated from a subset of the population, it is a statistic.

Measures of Central Tendency and Skewness

  • Definition of Central Tendency Measures:

    • Mean: The arithmetic average of a variable, calculated by summing all data values and dividing by the total number of observations NN.

    • Median: The central data value when a data set is ordered sequentially in ascending rank.

  • Relationship Between Mean and Median in Skewed Distributions:

    • Left-Skewed Distributions (Negative Skew):

    • In distributions that are skewed left, the tail extends towards smaller numerical values, pulling the arithmetic mean to the left.

    • Mathematical Relationship: mean<median\text{mean} < \text{median}

    • Symmetric Distributions:

    • Mathematical Relationship: mean=median\text{mean} = \text{median}

    • Right-Skewed Distributions (Positive Skew):

    • In distributions that are skewed right, extreme high values pull the mean to the right.

    • Mathematical Relationship: mean>median\text{mean} > \text{median}

Measures of Variation and Sensitivity to Outliers

  • Measures of Spread (Variability):

    • Range: The total distance between extreme values, computed as Maximum−Minimum\text{Maximum} - \text{Minimum}.

    • Interquartile Range (IQR): The distance covered by the middle 50% of the distribution, computed as Q3−Q1Q_3 - Q_1.

    • Standard Deviation: The typical distance or deviation of observations from the sample mean.

  • Measures of Central Location (Not Variation):

    • Midrange: Computed as Maximum+Minimum2\frac{\text{Maximum} + \text{Minimum}}{2}. The midrange is a measure of center/location, not a measure of variation or spread.

  • Resistance versus Sensitivity to Outliers:

    • Sensitive Statistics: Statistics whose numerical values are significantly altered by extreme observations or outliers.

    • The mean is SENSITIVE to the effects of outliers.

    • The standard deviation is SENSITIVE to the effects of outliers.

    • Resistant Statistics: Statistics whose numerical values remain relatively stable despite the presence of extreme observations or outliers.

    • The median is RESISTANT to the effects of outliers.

    • The IQR is RESISTANT to the effects of outliers.

Analysis of Footwear Ownership Data

Histogram showing frequency of pairs of footwear owned by 40 students

  • Distribution Metrics:

    • Total sample size: N=40N = 40 statistics students.

    • Sample Mean: Mean=16.425\text{Mean} = 16.425

    • Sample Standard Deviation: Std. Dev.=18.53048\text{Std. Dev.} = 18.53048

  • Frequency Table by Interval:

    • Interval [0.00,10.00)[0.00, 10.00): Frequency = 18

    • Interval [10.00,20.00)[10.00, 20.00): Frequency = 10

    • Interval [20.00,30.00)[20.00, 30.00): Frequency = 4

    • Interval [30.00,40.00)[30.00, 40.00): Frequency = 5

    • Interval [40.00,50.00)[40.00, 50.00): Frequency = 1

    • Interval [50.00,60.00)[50.00, 60.00): Frequency = 1

    • Interval [100.00,110.00)[100.00, 110.00): Frequency = 1

  • Locating the Median Interval:

    • For N=40N = 40 observations, the median is the average of the 20th and 21st ordered data points.

    • Cumulative count analysis:

    • Interval [0,10)[0, 10): Covers ranks 1 to 18.

    • Interval [10,20)[10, 20): Covers ranks 19 to 28.

    • Conclusion: Both the 20th and 21st observations fall within the interval 10–20.

  • Evaluating Appropriate Summary Measures:

    • Correct Choice: Median and Interquartile Range (IQR).

    • Analysis: The distribution is heavily right-skewed with an extreme outlier in the interval [100.00,110.00)[100.00, 110.00). It is false that the mean and standard deviation are resistant to outliers. For heavily skewed data containing outliers, the median and IQR must be reported instead of the mean and standard deviation.

Penguin Body Measurement Analysis

Table template for penguin species bill length mean and standard deviation

  • Dataset Components and Units:

    • Observational Units: Individual penguins in the study sample.

    • Variables Analyzed:

    • body_mass_g: Quantitative variable measuring body mass in grams.

    • bill_length_mm: Quantitative variable measuring bill length in millimeters.

    • species: Categorical variable identifying penguin species (Adelie, Chinstrap, or Gentoo).

  • Outlier Detection Procedure using 1.5IQR Rule (body_mass_g):

    1. Calculate First Quartile (Q1Q_1) and Third Quartile (Q3Q_3).

    2. Compute Interquartile Range: IQR=Q3−Q1\text{IQR} = Q_3 - Q_1

    3. Compute Lower Fence: Lower Fence=Q1−1.5×IQR\text{Lower Fence} = Q_1 - 1.5 \times \text{IQR}

    4. Compute Upper Fence: Upper Fence=Q3+1.5×IQR\text{Upper Fence} = Q_3 + 1.5 \times \text{IQR}

    5. Any observation satisfying \text{body_mass_g} < \text{Lower Fence} or \text{body_mass_g} > \text{Upper Fence} is designated as an outlier.

  • Group Summary Statistics Table (bill_length_mm grouped by species):

    • Species Adelie: Compute mean and standard deviation rounded to one decimal place.

    • Species Chinstrap: Compute mean and standard deviation rounded to one decimal place.

    • Species Gentoo: Compute mean and standard deviation rounded to one decimal place.

  • Boxplot and Histogram Structural Insights (body_mass_g):

    • Body Mass Comparison Across Species: Gentoo penguins tend to have the largest body mass, while Chinstrap and Adelie penguins have fairly similar body mass distributions.

    • Histogram Shape: The overall distribution of body_mass_g across all species exhibits a right-skewed shape.

    • Proportion Calculation: The proportion of penguins with body mass less than 3500\n,g is obtained by taking the count of observations where \text{body_mass_g} < 3500\n,g and dividing by the total sample size NN.


In the context of your notes, the lower fence is the threshold boundary used in the 1.5×IQR1.5 \times \text{IQR} rule to detect unusually small data values (low outliers).

It is calculated using the formula: Lower Fence=Q1−1.5×IQR\text{Lower Fence} = Q_1 - 1.5 \times \text{IQR}

Where:

  • Q1Q_1 is the first quartile (25th percentile).

  • IQR\text{IQR} is the Interquartile Range (Q3−Q1Q_3 - Q_1).

If any observed value in your data set is smaller than the lower fence (Value<Lower Fence\text{Value} < \text{Lower Fence}), it is classified as an outlier.