Data Collection & Analysis in Research

• Business Statistics Exam #1 on Monday, February 24, 2024 covers defining and collecting data, organizing and visualizing variables, and numerical descriptive measures

Defining Variables and Types

• Categorical (qualitative) variables take categories as their values, such as "yes," "no," "blue," "brown," or "green."
• Nominal variables are categories lacking a specific order, like gender or brand names.
• Ordinal variables are categories possessing a meaningful order without a consistent difference, such as customer satisfaction ratings.
• Numerical (quantitative) variables have values representing a counted or measured quantity
• Discrete variables come from a counting process.
• Continuous variables come from a measuring process.

Measurement Scales

• Nominal scales categorize data without any order or ranking, e.g., gender or types of cuisine.
• Ordinal scales categorize data with a meaningful order, where intervals between categories are inconsistent, e.g., customer satisfaction or education levels.
• Interval scales measure data with equal intervals but lack a true zero point, e.g., temperature in Celsius or Fahrenheit and IQ scores.
• Ratio scales measure data with equal intervals and have a true zero point, allowing for ratio calculations, e.g., height, weight, income, and sales revenue.

Data Collection Methods

• Surveys and Questionnaires: Data is gathered using structured questions. They provide customer satisfaction and employee feedback forms.
• Interviews: Detailed data is collected through direct (face-to-face or virtual) conversations. For example, in-depth interviews with stakeholders or focus group discussions
• Observations: Behaviors or events are recorded as they naturally occur, such as observing customer behavior in a store or monitoring employee performance.
• Experiments: Controlled tests are conducted to study cause-and-effect relationships, such as A/B testing or product usability testing.
• Existing Records and Databases: Readily available internal or external data, such as company financial records, industry reports, and government databases, is used.
• Online Analytics Tools: Data is collected from digital platforms and websites, such as Google Analytics.

Populations and Samples

• Population: This includes all items or individuals of interest in a study.
• Sample: Only contains a portion of a population.
• Sampling Frame: A listing of items that make up the population

Sampling Methods

• Simple Random Sampling: Every population member has an equal chance of selection, like drawing names from a hat.
• Systematic Sampling: Every nth member of the population is selected, like choosing every 10th customer.
• Stratified Sampling: The population is divided into subgroups (strata), and samples are randomly taken from each subgroup, like sampling employees from different departments.
• Cluster Sampling: The population is divided into clusters, and entire clusters are randomly selected.
• Convenience Sampling: Samples are selected based on ease of access, like surveying people in a shopping mall.
• Judgmental (Purposive) Sampling: Samples are selected based on the researcher's judgment, such as choosing experts in a field.

Sources of Survey Errors

• Coverage Error: Occurs when certain members are excluded from the sampling frame.
• Nonresponse Error: Arises from failure to follow up on non-responses.
• Sampling Error: Involves random differences between the sample and the population.
• Measurement Error: Results from bad or leading questions.

Organizing and Visualizing Categorical Variables

• Summary Table: Tallies frequencies/percentages of items in a set of categories to see differences between categories.
• Contingency Table: Used to study patterns between two or more categorical variables. Cross-tabulates responses, with tallies for one variable in rows and the other in columns.

Organizing and Visualizing Numerical Variables

• Histograms: Bar charts that represent the frequency distribution of numerical data, such as visualizing sales revenue distributions.
• Box Plots (Box-and-Whisker Plots): Summarize data using quartiles, highlighting the median and identifying outliers.
• Line Graphs: Show trends over time by connecting data points with lines, such as plotting monthly sales revenue over a year.
• Scatter Plots: Display the relationship between two numerical variables, such as analyzing correlation between advertising expenses and sales revenue.
• Bar Charts: Compare different categories using bars, such as comparing quarterly sales figures across different product lines.
• Pie Charts: Show the proportion of different categories within a whole, visualizing market share.

Creating and Reading Tables/Diagrams

• Contingency tables can be created and read using absolute and percentage frequencies
• A multidimensional contingency table tallies responses of three or more categorical variables and can be used to discover possible patterns and relationships in multidimensional data that simpler tables and charts would fail to make apparent.
• As a practical rule, tables should be limited to no more than three or four variables.
• Frequency distributions are summaries that represent how often different values occur within a dataset and can be represented using a table format.
• Absolute Frequency: This is using the count of times a particular value or category appears in a data set.
• Percentage Frequency: Is the absolute value/ total number of points * 100.

Central Tendency Measures

• Mean: The average of all data values (sum of values divided by the number of data points) and is calculated in Excel using =average(data set).
• Median: The middle value in a dataset arranged in ascending or descending order, calculated in Excel using =median(data set).
• Mode: The value that occurs most frequently in a dataset, calculated in Excel using =mode.multi(data set).

Applications of Central Tendency Measures:

• Use the Mean when:
• Data is normally distributed (symmetrical with no extreme outliers).
• You want to consider all data points in the calculation.
• Data is on an interval or ratio scale (e.g., height, weight, temperature).
• Use the Median when:
• Data are skewed or contain outliers.
• You want a measure that represents the middle value of the dataset.
• Data is on an ordinal, interval, or ratio scale (e.g., income, house prices).
• Use the mode when:
  • The data are categorical (e.g., favorite color, most common product).
  • You want to identify the most frequently occurring value.
  • The data are on a nominal, ordinal, interval, or ratio scale.

Measures of Variation

• Variance: Measures the average squared deviation of each data point from the mean and quantifies the data points spread.

• Population variance is calculated as =var.p(data set) in Excel.

• Sample variance is calculated as =var.s(data set) in Excel.

• Standard Deviation: The square root of the variance and measures the average distance of values from the mean.

• Population standard deviation is calculated as =stdev.p(data set) in Excel.

• Sample standard deviation is calculated as =stdev.s(data set) in Excel.

• Range: The difference between the maximum and minimum values in the dataset.

• Excel calculates the range using =max(data set) - min(data set).

Outliers (z-score)

• Data points with z-scores greater than 3 or less than -3 are considered outliers.
• Z-score is calculated as (point – mean) / standard deviation in Excel.

Distribution Shape

• Skewness measures the asymmetry of a distribution.
• Positive Skewness: The right tail is longer, with most data on the left.
• Negative Skewness: The left tail is longer, with most data on the right.
• Zero Skewness: The distribution is symmetrical.
• Skewness is calculated as =skew(data set) in Excel.
• Kurtosis indicates the presence of outliers and the sharpness of the peak.
• Positive Kurtosis: Heavy tails and a sharp peak indicate more data points are in the tails.
• Negative Kurtosis: Light tails and a flat peak indicate fewer data points in the tails.
• Zero Kurtosis: Tails are similar to a normal distribution.
• Kurtosis is calculated as =kurt(data set) in Excel.

Quartiles

• First Quartile (Q1): The median of the lower half of the dataset, separating the lowest 25% of the data.
• Excel calculates Q1 using =quartile.inc(data, 1).
• Second Quartile (Q2): The median, divides the dataset in half.
• Third Quartile (Q3): The median of the upper half of the dataset, separates the highest 25% of the data.
• Excel calculates Q3 using =quartile.inc(data, 3).
• Interquartile Range (IQR): The range between the first and third quartiles. Measures the spread of the middle 50% of the data, calculated as IQR = Q3 – Q1.
• Excel calculates lower bound as Q1 – 1.5 * IQR.
• Excel calculates upper bound as Q3 + 1.5 * IQR.

Summary of a Five-Number summary

• Represents the distribution of a dataset, indicates data spread and central tendency:
• Minimum: The smallest value.
• First Quartile (Q1): The median of the lower half (25th percentile).
• Median (Q2): The middle value (50th percentile).
• Third Quartile (Q3): The median of the upper half (75th percentile).
• Maximum: The largest value.
• Symmetrical Distribution: The median (Q2) will be roughly in the center, with similar distances between Q1 and Q2 and between Q2 and Q3.
• Skewed Distribution:
• Right-Skewed (Positively Skewed): The median will be closer to Q1, with a greater distance between Q2 and Q3. The maximum value will be farther from Q3.
• Left-Skewed (Negatively Skewed): The median will be closer to Q3, with a greater distance between Q1 and Q2. The minimum value will be farther from Q1.

Boxplot Components

• Box: Represents the interquartile range (IQR) that is the range between Q1 and Q3 and contains the middle 50% of the data.
• Median Line: Represents the median (Q2) of the dataset inside the box.
• Whiskers: Indicate lines extending from the box to the minimum and maximum values, excluding outliers.
• Outliers: Data points outside the whiskers.

Boxplot Reading Interpretation

• Symmetrical Distribution: The median is roughly in the center of the box, and whiskers are of similar length.
• Right-Skewed Distribution: The median is closer to Q1, and the right whisker is longer.
• Left-Skewed Distribution: The median is closer to Q3, and the left whisker is longer.

Empirical Rule

• Applies to normal distributions and shows percentages:
• 68% of data within one standard deviation of the mean.
• 95% within two standard deviations.
• 99.7% within three standard deviations.

Chebyshev's Rule

• Applies to all distributions, providing a minimum proportion within a certain number of standard deviations.

Covariance

• Measures the degree to which two variables change together and indicates the direction of a linear relationship.
• Excel calculates covariance using =covariance.s(data x, data y)
• A positive covariance: As one variable increases, the other tends to increase.
• A negative covariance: As one variable increases, the other tends to decrease.

Correlation

• Reveals both the strength and direction of the linear relationship (ranges from -1 to 1).
• Calculated in Excel using =correl(data x, data y).

Understand research methodologies with targeted data collection. Learn about methodologies like appropriate sampling techniques. Explore variable measurement scales and data representaion.