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

Data Collection

Section 1.1: Introduction to Data

This section introduces the foundational vocabulary and concepts necessary for understanding how data is collected and classified in statistics.

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

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

• Statistic: A numerical summary of a sample.

• Parameter: A numerical summary of a population.

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

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

• Quantitative Variable: Describes quantities or amounts (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 inherent order (qualitative).

  • Ordinal: Categories with a meaningful order but not evenly spaced (qualitative or quantitative).

  • Interval: Quantitative, differences have meaning, but zero is arbitrary (e.g., temperature in °F).

  • Ratio: Quantitative, differences and ratios have meaning, and zero indicates absence (e.g., test scores).

Example: The number of books in a library (discrete, quantitative, ratio); the color of cars (qualitative, nominal).

Section 1.2: Observational Studies vs Designed Experiments

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

• Observational Study: Researchers observe outcomes without assigning treatments.

• Designed Experiment: Researchers assign treatments to study their effects.

• 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.

• Confounding Variable: Variable related to both explanatory and response variables, potentially distorting results.

• Lurking Variable: Unmeasured variable influencing the relationship between studied variables.

Example: Studying the effect of a new drug (experiment) vs observing health outcomes in a population (observational study).

Section 1.3: Simple Random Sampling

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

• Random Sampling: Selection based on chance.

• Simple Random Sample: Every possible sample of a given size has the same probability of selection.

Example: Drawing names from a hat to select a sample.

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.

Systematic Sampling Steps: (1) Number the population, (2) Choose a random starting point, (3) 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 flawed 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: Surveying only daytime shoppers (sampling bias); poorly worded questions (response bias).

Section 1.6: Experimental Design

Experimental design involves planning how to assign treatments and measure outcomes to ensure valid conclusions.

• Experiment: Study where treatments are assigned to observe effects.

• Factors: Explanatory variables in an 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 researchers (double-blind) do not know treatment assignments.

Example: Testing a new medication with a placebo group and double-blind design.

Organizing and Summarizing Data

Section 2.1: Qualitative Data

Qualitative data is summarized using tables and graphical displays to reveal patterns and distributions.

• Raw Data: Unprocessed data as collected.

• 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 relative frequencies as sectors.

Formula:

• Relative frequency = frequency / total number of observations

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

Section 2.2: Quantitative Data

Quantitative data is organized into classes and displayed using histograms to reveal distribution shapes.

• Class: Range of values grouped together for analysis.

• 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, with most values near the center.

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

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

Section 2.3: Other Displays of Quantitative Data

Additional displays help summarize cumulative information and trends over time.

• Cumulative Frequency Distribution: Table showing the number of observations below each class boundary.

• Cumulative Relative Frequency Distribution: Proportion of observations below each class boundary.

• Ogive: 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), for i > 1; for i = 1, it is just the frequency of class 1.

• 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 (\( \mu \)): Mean of all values in the population.

• Sample Mean (\( \overline{x} \)): Mean of values in a sample.

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

• Mode: Value(s) that occur most frequently.

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

• Bimodal/Multimodal: Two or more modes.

Formulas:

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

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

• Median (odd n): $M = x_{\frac{n+1}{2}}$

• Median (even n): $M = \frac{x_{n/2} + x_{n/2+1}}{2}$

Example: Data set: 2, 4, 4, 5, 7. Mean = 4.4, Median = 4, Mode = 4.

Section 3.2: Measures of Spread (Dispersion)

Measures of spread describe the variability in a data set.

• Range (R): Difference between maximum and minimum values.

• Population Standard Deviation (\( \sigma \)): Average distance of data points from the population mean.

• Sample Standard Deviation (s): Average distance of data points from the sample mean.

• Variance: Square of the standard deviation.

• Degrees of Freedom: Number of values free to vary when estimating a parameter (n-1 for sample standard deviation).

Formulas:

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

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

• Sample standard deviation: $s = \sqrt{\frac{\sum (x_i - \overline{x})^2}{n-1}}$

Example: Data set: 2, 4, 4, 5, 7. Range = 5, Standard deviation can be calculated using the above formulas.

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.

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

• Quartiles (Q1, Q3): 25th and 75th percentiles, respectively.

• Interquartile Range (IQR): Difference between Q3 and Q1.

• Outlier: Value outside the range defined by the lower and upper fences.

• Upper Fence: $Q_3 + 1.5 \times IQR$

• Lower Fence: $Q_1 - 1.5 \times IQR$

Formulas:

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

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

• IQR: $IQR = Q_3 - Q_1$

• Upper fence: $Q_3 + 1.5 \times IQR$

• Lower fence: $Q_1 - 1.5 \times IQR$

Example: Data set: 2, 4, 4, 5, 7. Q1 = 4, Q3 = 5, IQR = 1, Upper fence = 6.5, Lower fence = 2.5.

Section 3.5: 5 Number Summaries and Box Plots

The five-number summary and box plots 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 set: 2, 4, 4, 5, 7. Five-number summary: 2, 4, 4, 5, 7.

Describing the Relation Between Two Variables

Section 4.1: Scatter Diagrams and Correlation

Scatter diagrams and correlation coefficients are used to assess 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 with x.

• Linear Correlation Coefficient (r): Measures strength and direction of linear relationship (range: -1 to 1).

• Critical Value: Threshold for determining statistical significance of r.

Formulas:

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

Example: Plotting students' heights vs weights and calculating r to assess the relationship.

Section 4.2: Least Squares Regression Line

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

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

• Residual: Difference between observed and predicted values (\( y_i - \hat{y}_i \)).

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

• Extrapolation: Predicting for x-values outside the observed range (unreliable).

• Interpretation of Slope (b1): Change in y for a one-unit increase in x.

• Interpretation of Intercept (b0): Predicted value when x = 0 (may not always be meaningful).

Formulas:

• Line through two points: $(y - y_1) = m(x - x_1), \quad m = \frac{y_2 - y_1}{x_2 - x_1}$

• Least-squares regression line: $y = b_1 x + b_0$

• Slope: $b_1 = r \frac{s_y}{s_x}$

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

Example: Given data on study hours (x) and test scores (y), calculate the regression line to predict scores based on hours studied.