Statistics Unit 1: Descriptive Statistics, Sampling Methods, and Data Analysis Study Guide

Fundamental Concepts of Statistics

• Population: The complete, large group of individuals, objects, or items under study. Its size is represented by $N$.
• Sample: A smaller subset of the population selected for analysis. Its size is represented by $n$.
• Parameter: A numerical summary measure calculated from or describing an entire population.
• Statistic: A numerical summary measure calculated from or describing a sample.
• Variable: A characteristic or attribute that can take on different values for different individuals or items in a population or sample.
• Data: The specific values or observations recorded for a variable.
• Cardiologist Recovery Example:
  • A cardiologist wants to know the mean recovery period of her patients who have had heart attacks.
  • Population: All patients who have had heart attacks.
  • Sample: A small selected group of heart attack patients.
  • Parameter: The average recovery period of the entire population of heart attack patients.
  • Statistic: The average recovery period of the sample of patients.
  • Variable: The possible average recovery period or individual recovery period ($x$).
  • Data: The specific recovery period recorded for an individual patient (e.g., the recovery period of $1$ patient).

Variables and Data Types

• Qualitative Variables: Variables that represent categorical labels or attributes rather than numerical measurements.
• Quantitative Variables: Variables that represent numerical measurements or counts.
  • Discrete Quantitative Data: Numerical data that is countable, consisting of distinct, separate numerical values.
  • Continuous Quantitative Data: Numerical data that is not countable, taking on any continuous value within a given spectrum or range.
• Explanatory Variable: A variable that is manipulated or categorized to observe its effect on a response variable (it changes the response).
• Treatment: The specific manipulation or experimental condition applied to an explanatory variable.

Sampling Methods

• Simple Random Sample: A sampling method where every member of the population has an equal chance of selection, typically chosen using a random number generator.
• Stratified Sample: A sampling method where the population is divided into distinct subgroups or strata, which may be uneven in size, and a proportionate random sample is selected from each group.
• Systematic Sample: A sampling method where every $k\text{th}$ element or individual from an ordered population list is selected.
• Cluster Sample: A sampling method where the population is split into separate groups or clusters, and one or more entire clusters are selected in full.
• Convenience Sample: A non-probability sampling method where the easiest or most accessible sample members are chosen.

Sources of Bias and Misleading Data

• Biased Sample: A sample that is not representative of the population (such as a convenience sample).
• Undue Influence: Bias occurring when the wording of a question influences respondents' answers in a specific direction (frequently seen in political surveys).
• Self-Interest: Bias that occurs when a study is conducted or funded by individuals or organizations with a direct stake or conflict of interest to support a specific claim.
• Self-Selected Sample: Bias resulting from responses provided only by individuals who actively choose to participate or respond (such as call-in polls).
• Non-Response Bias: Bias introduced when selected participants are unable or unwilling to respond to surveys.
• Misleading Data: Data presented with a lack of context or improperly displayed graphically to distort interpretation.
• Small Sampling: Inadequate sample size that fails to provide sufficient data to draw reliable conclusions.
• Causality, Causation, and Correlation: Correlation between variables does not imply causation.
• Confounding: Occurs when an unobserved or related variable distorts the data and obscures the true relationship between the variables under study.

Frequency Distributions and Tables

• Relative Frequency: The proportion or fraction of total observations belonging to a specific category or class.   $\text{Relative Frequency} = \frac{\text{Frequency}}{\sum \text{Frequencies}}$
• Cumulative Frequency: The running total sum of frequencies up to and including the current class interval.
• Cumulative Relative Frequency: The running total sum of relative frequencies up to and including the current class interval.
• Frequency Table Dataset Example:
  • Class Row $1$: Frequency = $27$, Relative Frequency = $0.45$, Cumulative Frequency = $27$, Cumulative Relative Frequency = $0.45$
  • Class Row $2$: Frequency = $18$, Relative Frequency = $0.30$, Cumulative Frequency = $45$, Cumulative Relative Frequency = $0.75$
  • Class Row $3$: Frequency = $3$, Cumulative Frequency = $56$, Cumulative Relative Frequency calculations accumulating to $0.65$
  • Class Row $4$: Frequency = $3$, Cumulative Frequency = $59$
  • Class Row $5$: Frequency = $7$, accumulating to total frequencies such as $983$ or full completion of sample size $N$

Graphical Representations of Data

• Stem-and-Leaf Plot: A display that splits each numerical data value into a stem (leading digit or digits) and a leaf (trailing digit).
  • Example format:
  • Stem $1$ | Leaf $2$, $7$ (representing data values $12$, $17$)
  • Stem $2$ | Leaf $4$, $7$, $1$, $2$, $7$ (representing data values $24$, $27$, $21$, $22$, $27$)
• Histogram: A graphical display of quantitative data using vertical bars where the bars are touching (unlike a bar graph, where bars do not touch).
• Reconstructing Raw Data from a Histogram:
  • To retrieve raw data from a histogram, use the midpoint of each bar paired with its corresponding frequency.
  • Numerical Example:
  • Bar with Midpoint $10$ and Frequency $2$: raw data values $15$, $15$
  • Bar with Midpoint $20$ and Frequency $4$: raw data values $25$, $25$, $25$, $25$
  • Bar with Midpoint $30$ and Frequency $6$: raw data values $35$, $35$, $35$, $35$, $35$, $35$
  • Bar with Midpoint $40$ and Frequency $0$
  • Reconstructed Raw Dataset: $15$, $15$, $25$, $25$, $25$, $25$, $35$, $35$, $35$, $35$, $35$, $35$

Shapes of Distributions

• Symmetric Distribution:
  • Data is evenly distributed on both sides of the central point.
  • Relationship between measures of center:     $\text{Mean} = \text{Median} = \text{Mode}$
• Skewed Right (Positive Skew):
  • Most data points are concentrated on the left side and fall below the mean, with the tail extending to the right.
  • Relationship between measures of center:     $\text{Mode} < \text{Median} < \text{Mean}$
• Skewed Left (Negative Skew):
  • Most data points are concentrated on the right side and fall above the mean, with the tail extending to the left.
  • Relationship between measures of center:     $\text{Mean} < \text{Median} < \text{Mode}$
• Uniform Distribution:
  • Data is equally distributed among the entire range of values.
• Bimodal Distribution:
  • A distribution containing two distinct modes (peaks), indicating that the dataset needs to be split into smaller sub-groups.

Percentiles and the Five-Number Summary

• Percentile ($P$):
  • Represents the percentage of data points falling below a specific value of interest.
  • Percentile Rank Formula:     $P = \frac{x}{N} \times 100$     where $x$ is the number of data points below the point of interest, and $N$ is the total number of data points.
  • Percentile Location Formula:     $i = \frac{p}{100} \times n$
• Five-Number Summary:
  • Consists of five boundary statistics: Minimum ($\text{min}$), First Quartile ($Q_1$), Median ($MD$), Third Quartile ($Q_3$), and Maximum ($\text{max}$).
• Box Plot:
  • A graphical display of the five-number summary displaying a central box from $Q_1$ to $Q_3$, a vertical line at the median ($MD$), and whiskers extending to the minimum ($\text{min}$) and maximum ($\text{max}$).
• Interquartile Range ($IQR$):
  • Measures the spread of the middle $50\%$ of the data.     $IQR = Q_3 - Q_1$
• Range:
  • Measures the total distance between the largest and smallest values.     $\text{Range} = \text{max} - \text{min}$

Measures of Center and Spread

• Measures of Center:
  • Sample Mean ($\bar{x}$): The arithmetic average of a sample data set.
  • Population Mean ($\mu$): The arithmetic average of an entire population.
  • Median ($MD$): The middle value of an ordered data set.
• Measures of Spread:
  • Sample Standard Deviation ($s$): The average distance of a data value from the sample mean.     $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$
  • Population Standard Deviation ($\sigma$): Standard deviation calculated for an entire population.
  • Variance: The squared value of standard deviation ($s^2$ or $\sigma^2$).
• Selecting Appropriate Summary Statistics:
  • If data is symmetric, use the mean ($\bar{x}$) and standard deviation ($s$).
  • If data is skewed, use the median ($MD$) and interquartile range ($IQR$).

Outlier Detection Rules and Sensitivity

• $1.5 \times IQR$ Rule for Outliers:
  • Upper and lower boundary equations:     $\text{Lower Bound} = Q_1 - 1.5 \times IQR$$\text{Upper Bound} = Q_3 + 1.5 \times IQR$
  • Any value within this interval is NOT an outlier; any value falling outside is an outlier.
  • Numerical Example:
  • Given $IQR = 12$, compute $1.5 \times IQR$:       $1.5 \times 12 = 18$
  • Boundary calculations:       $Q_1 - 18 = -2$$Q_3 + 18 = 46.7$
  • Values falling within the range from $-2$ to $46.7$ are NOT outliers.
• Two Standard Deviation Rule for Outliers:
  • Upper and lower threshold equations:     $\text{Upper Threshold} = \bar{x} + 2 \times SD$$\text{Lower Threshold} = \bar{x} - 2 \times SD$
  • Numerical Example:
  • Given sample mean $\bar{x} = 22.8$ and standard deviation $SD = 9.064$:
  • Upper threshold calculation:       $\bar{x} + 2 \times SD = 22.8 + 2 \times 9.064 = 40.93$
  • Lower threshold calculation:       $\bar{x} - 2 \times SD = 22.8 - 2 \times 9.064 = 4.672$
  • Any value falling within the range from $4.672$ to $40.93$ is NOT an outlier.
• Sensitivity of Statistics to Outliers:
  • Range: Very heavily affected by extreme values.
  • Mean ($\bar{x}$): Heavily affected by extreme values (more affected than the median).
  • Standard Deviation ($s$): Somewhat to heavily affected by extreme values.
  • Median ($MD$): Resistant / not heavily affected by extreme values.
  • Interquartile Range ($IQR$): Resistant / not heavily affected by extreme values.