MA 113: Statistics Midterm 1 Study Guide – Data Collection, Summarizing, and Relationships

Data Collection

Section 1.1: Introduction to Data

This section introduces the foundational concepts of statistics, focusing on types of data, variables, and levels of measurement.

• Population: The entire group of individuals or items of interest in a study.

• Sample: A subset of the population selected for analysis.

• Individual: A single member of the population or sample.

• Parameter: A numerical summary describing a population.

• Statistic: A numerical summary describing a sample.

• Variable: A characteristic or attribute that can assume different values among individuals.

• Qualitative (Categorical) Variable: Describes qualities or categories (e.g., color, type).

• Quantitative Variable: Describes numerical measurements (e.g., height, age).

• Discrete Variable: Quantitative variable with countable values (e.g., number of clocks).

• Continuous Variable: Quantitative variable with infinitely many possible values within a range (e.g., temperature).

• Levels of Measurement:

  • Nominal: Categories with no order (e.g., gender).

  • Ordinal: Categories with a meaningful order but not equal intervals (e.g., rankings).

  • Interval: Ordered, equal intervals, but no true zero (e.g., temperature in °F).

  • Ratio: Like interval, but with a meaningful zero (e.g., weight, height).

Example: The number of books on a shelf (discrete, quantitative, ratio); temperature outside (continuous, quantitative, interval).

Section 1.2: Observational Studies vs Designed Experiments

This section distinguishes between methods of data collection and the implications for causation.

• Observational Study: Observes individuals without influencing them.

• Designed Experiment: Applies treatments to individuals and observes responses.

• Explanatory Variable: Variable manipulated or categorized to observe its effect.

• Response Variable: Outcome measured in the study.

• Causation vs Association: Experiments can establish causation; observational studies can only suggest association (correlation).

• Confounding: When effects of two variables cannot be distinguished.

• Lurking Variable: A variable not included in the study that affects the response variable.

Example: A clinical trial (experiment) vs a survey of eating habits (observational study).

Section 1.3: Simple Random Sampling

Simple random sampling ensures every member of the population has an equal chance of selection.

• Random Sampling: Selection based on chance.

• Simple Random Sample: Every possible sample of a given size has the same chance of being chosen.

Example: Drawing names from a hat or using a random number generator.

Section 1.4: Other Sampling Methods

Alternative sampling methods are used when simple random sampling is impractical.

• Stratified Sample: Population divided into subgroups (strata), and random samples taken from each.

• Systematic Sample: Every k-th individual is selected from a list after a random start.

• Cluster Sample: Population divided into clusters, some clusters are randomly selected, and all individuals in chosen clusters are sampled.

• Convenience Sample: Individuals are chosen based on ease of access.

Example: Surveying every 10th person entering a store (systematic sampling).

Formula: Systematic sampling steps: Number the population, choose a random starting point, then select every k-th member.

Section 1.5: Bias

Bias refers to systematic errors in data collection that can affect the validity of results.

• Sampling Bias: Sample is not representative due to poor methodology.

• Nonresponse Bias: Individuals selected do not respond.

• Response Bias: Responses are inaccurate due to question wording or respondent behavior.

• Undercoverage: Some groups are inadequately represented.

• Sampling Error: Natural variability from using a sample to estimate a population parameter.

• Non-sampling Error: Errors from poor data collection, processing, or measurement.

Example: Only surveying people at a gym about exercise habits (sampling bias).

Section 1.6: Experimental Design

Experimental design involves planning how to conduct an experiment to ensure valid and reliable results.

• Experiment: Study where treatments are applied to subjects.

• Factors: Explanatory variables manipulated in the experiment.

• Treatment: Specific condition applied to subjects.

• Experimental Unit/Subject: The individual receiving the treatment.

• Control Group: Group receiving no treatment or a standard treatment for comparison.

• Placebo: Inactive treatment used to control for psychological effects.

• Blinding: Subjects (single-blind) or both subjects and experimenters (double-blind) do not know treatment assignments.

Example: Testing a new drug with a placebo group and double-blinding.

Organizing and Summarizing Data

Section 2.1: Qualitative Data

Qualitative data is summarized using tables and graphical displays.

• Raw Data: Original, unprocessed data.

• Frequency Distribution: Table showing counts for each category.

• Relative Frequency: Proportion of observations in each category.

• Bar Graph: Visual display of frequencies for categories.

• Pareto Chart: Bar graph with categories ordered by frequency.

• Pie Chart: Circular chart showing proportions.

• Side-by-Side Bar Graphs: Compare frequencies across groups.

Formula: Relative frequency = frequency / total number of observations.

Example: Survey results on favorite ice cream flavors displayed in a bar graph.

Section 2.2: Quantitative Data

Quantitative data is organized into classes and displayed using histograms and frequency tables.

• Class: Interval grouping of data values.

• Histogram: Bar graph for quantitative data; bars touch to indicate continuous intervals.

• Lower/Upper Class Limit: Smallest/largest value in a class.

• Class Width: Difference between lower limits of consecutive classes.

• Uniform Distribution: All values occur with similar frequency.

• Bell-Shaped Distribution: Symmetrical, most data near the center.

• Left/Right Skew: Tail extends to the left (negative) or right (positive).

Example: Heights of students grouped into intervals and displayed in a histogram.

Note: All classes in a histogram should have equal width.

Section 2.3: Other Displays of Quantitative Data

Additional methods for summarizing quantitative data include cumulative tables and time-series graphs.

• Cumulative Frequency Distribution: Shows the total number of observations up to each class.

• Cumulative Relative Frequency Distribution: Shows the proportion of observations up to each class.

• Ogive: Line graph of cumulative frequencies.

• Time-Series Data: Data collected over time.

• Time-Series Graph: Line graph showing trends over time.

Formulas:

• Cumulative frequency of class i = cumulative frequency of class (i-1) + frequency of class i

• Cumulative relative frequency = cumulative frequency / total number of observations

Example: Monthly sales data plotted as a time-series graph.

Numerically Summarizing Data

Section 3.1: Measures of Central Tendency

Measures of central tendency describe the center of a data set.

• Arithmetic Mean (Average): Sum of all values divided by the number of values.

  • Population mean: $\mu = \frac{\sum x_i}{N}$

  • Sample mean: $\bar{x} = \frac{\sum x_i}{n}$

• Median (M): Middle value when data is ordered.

  • If n is odd: $M = x_{(n+1)/2}$

  • If n is even: $M = \frac{x_{n/2} + x_{n/2+1}}{2}$

• Mode: Value(s) that occur most frequently. Data can be bimodal, multimodal, or have no mode.

• Resistant Statistic: Not affected by extreme values (e.g., median).

Example: Test scores: 70, 80, 80, 90. Mean = 80, Median = 80, Mode = 80.

Note: Mean is sensitive to outliers; median is preferred for skewed data.

Section 3.2: Measures of Spread (Dispersion)

Measures of spread describe the variability in a data set.

• Range (R): $R = \text{max} - \text{min}$

• Population Standard Deviation ($\sigma$): $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$

• Sample Standard Deviation (s): $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

• Variance: Square of the standard deviation.

• Degrees of Freedom: For sample variance, denominator is n-1.

• Bias: Sample variance is an unbiased estimator of population variance when dividing by n-1.

Example: Data: 2, 4, 6. Mean = 4, s = $\sqrt{\frac{(2-4)^2 + (4-4)^2 + (6-4)^2}{2}} = 2$

Section 3.4: Measures of Position and Outliers

Measures of position locate a value within a data set and help identify outliers.

• z-score: Number of standard deviations a value is from the mean.

  • Population: $z = \frac{x - \mu}{\sigma}$

  • Sample: $z = \frac{x - \bar{x}}{s}$

• Percentile (Pk): Value below which k% of data falls.

• Quartiles: Q1 (25th percentile), Q3 (75th percentile), Median (50th percentile).

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• Outliers: Values outside the fences.

  • Upper fence: $Q_3 + 1.5 \times IQR$

  • Lower fence: $Q_1 - 1.5 \times IQR$

Example: Data: 1, 2, 3, 4, 5, 6, 100. 100 is an outlier.

Section 3.5: 5 Number Summaries and Box Plots

The five-number summary and boxplots provide a concise summary and visual representation of data distribution.

• Five-Number Summary: Minimum, Q1, Median, Q3, Maximum.

• Boxplot: Graphical display of the five-number summary, showing spread and skewness.

Example: Data: 2, 4, 6, 8, 10. Five-number summary: 2, 4, 6, 8, 10.

Describing the Relation Between Two Variables

Section 4.1: Scatter Diagrams and Correlation

Scatter diagrams and correlation coefficients are used to analyze the relationship between two quantitative variables.

• Scatter Plot/Diagram: Graph of paired (x, y) data points.

• Linear vs Nonlinear Relations: Linear if points form a straight pattern; nonlinear otherwise.

• Positive/Negative Association: Positive if y increases with x; negative if y decreases as x increases.

• Linear Correlation Coefficient (r): Measures strength and direction of linear relationship.

  • $r = \frac{\sum \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)}{n-1}$

• Critical Value: Used to assess significance of r; if $|r| > c$, the correlation is significant.

• Properties of r: $-1 \leq r \leq 1$; r = 1 or -1 indicates perfect linear relation; r = 0 indicates no linear relation.

• Correlation ≠ Causation: A significant r does not imply causation.

Example: Height and weight plotted on a scatter diagram; r = 0.85 indicates strong positive association.

Section 4.2: Least Squares Regression

The least-squares regression line models the linear relationship between two variables and is used for prediction.

• Least-Squares Regression Line: Line that minimizes the sum of squared residuals.

  • Equation: $y = b_1 x + b_0$

  • Slope: $b_1 = r \frac{s_y}{s_x}$

  • Intercept: $b_0 = \bar{y} - b_1 \bar{x}$

• Predicted Value ($\hat{y}$): Value on the regression line for a given x.

• Residual: Difference between observed and predicted value: $y_i - \hat{y}_i$

• Interpretation of Slope: Change in y for a one-unit increase in x.

• Extrapolation: Predicting for x-values outside the data range is unreliable.

Example: Given data on study hours (x) and test scores (y), the regression line can predict test scores for a given number of study hours.

Appendix: Key Tables

Level of Measurement	Type	Example
Nominal	Qualitative	Gender, Color
Ordinal	Qualitative/Quantitative	Rankings, Letter Grades
Interval	Quantitative	Temperature (°F)
Ratio	Quantitative	Height, Weight

Additional info: This table summarizes the main levels of measurement, their types, and examples for quick reference.