MTH 132 Exam 1 Review: Data Collection and Summarizing Data

Data Collection

Types of Data

Understanding the nature of data is fundamental in statistics. Data can be classified as either quantitative (numerical) or qualitative (categorical).

• Quantitative Data: Represents counts or measurements (e.g., number of cars, height, weight).

• Qualitative Data: Represents categories or labels (e.g., color, type, brand).

Examples:

• The number of cars a person owns – Quantitative

• Whether a person takes a shared transit service – Qualitative

• The height of runners in a competition – Quantitative

Observational Studies vs. Experiments

Statistical studies can be classified based on how data is collected:

• Observational Study: Observes individuals and measures variables without influencing responses.

• Experiment: Imposes a treatment to observe its effect on the response variable.

Example: If researchers assign participants to eat blueberries and measure cholesterol, it is an experiment. If they simply observe existing eating habits, it is an observational study.

Explanatory vs. Response Variables

In studies, variables are classified as:

• Explanatory Variable: The variable that explains or influences changes in another variable.

• Response Variable: The outcome or variable being measured.

Example: In a study on blueberries and cholesterol, eating blueberries is the explanatory variable, and cholesterol level is the response variable.

Sampling Methods

Simple Random Sample

A simple random sample is a subset of a population selected so that every possible sample has an equal chance of being chosen.

• Example: Using a random number generator to select states from a list.

Sampling Bias

Bias occurs when a sample does not accurately represent the population. Common types include:

• Sampling Bias: Certain members of the population are less likely to be included.

• Response Bias: Survey questions lead to inaccurate responses.

Example: Asking, "Don’t you agree that metal music is superior to heavy metal?" introduces response bias.

Summarizing Data in Tables and Graphs

Frequency and Relative Frequency Distributions

Data can be summarized using frequency tables:

Category	Frequency
Always	322
Most of the time	319
Sometimes	183
Rarely	78
Never	98

The relative frequency is calculated as:

• Relative Frequency = (Frequency of category) / (Total number of responses)

Relative frequency tables help compare proportions across categories.

Histograms

A histogram is a graphical representation of the distribution of numerical data. It shows the frequency or relative frequency of data within equal intervals (bins).

• Symmetric Distribution: Both sides are mirror images.

• Skewed Left: Tail is longer on the left side.

• Skewed Right: Tail is longer on the right side.

Example: A histogram with a higher bar on the right is skewed left.

Constructing Frequency Tables and Histograms

Steps:

1. Count the number of occurrences for each category or interval.

2. Calculate relative frequencies.

3. Draw the histogram using frequencies or relative frequencies.

Practice Table: Relative Frequency Distribution

Category	Frequency	Relative Frequency
Always	322	0.322
Most of the time	319	0.319
Sometimes	183	0.183
Rarely	78	0.078
Never	98	0.098

Additional info: Relative frequencies are calculated by dividing each frequency by the total (1000 in this example).

Key Formulas

• Relative Frequency: $\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total}}$

• Percentage: $\text{Percentage} = \text{Relative Frequency} \times 100\%$

Summary

• Identify data types and variables in studies.

• Distinguish between observational studies and experiments.

• Understand sampling methods and potential biases.

• Summarize data using frequency tables, relative frequencies, and histograms.

• Interpret the shape of distributions (symmetric, skewed left, skewed right).