Statistics Overview: Descriptive & Inferential

Two Kinds of Statistics

• Descriptive Statistics are used to summarize and simplify sample characteristics.
• Inferential Statistics are used to make generalizations about the population based on the sample's characteristics.

Symbols

• Symbols are used to represent population and sample statistics.
• Population Mean (μ) is the average of all values in the population.
• Sample Mean (X) is the average of all values in a sample.
• Population Proportion (π) is the proportion of a specific characteristic in the population.
• Sample Proportion (p) is the proportion of a specific characteristic in a sample.
• Population Variance (σ²) is a measure of the spread of data around the population mean.
• Sample Variance (s²) is a measure of the spread of data around the sample mean.
• Population Standard Deviation (σ) is the square root of the population variance and measures the average distance of each data point from the mean.
• Sample Standard Deviation (s) is the square root of the sample variance and measures the average distance of each data point from the mean.

Types of Data

• Categorical Data is data that can be divided into groups.
  • Qualitative Variables are categorical variables.
  • Examples include: grape variety, social class, gender
• Numerical Data is data that can be measured and ordered and the distance between each value is meaningful.
  • Quantitative Variables are numerical variables.
  • Examples include: Robert Parker ratings, scales, prices, time, height, weight, amount.

Analyzing Categorical Data

• Frequency is the number of times a category appears in a data set.
• Relative Frequency is the proportion of times a category appears in a data set.

Analyzing Numerical Data

• Sample Mean (X) is the average of all observations in a sample.
• Population Mean (μ) is the average of all observations in the population.
• Median is the middle value that divides the data into two equal parts.
• Mode is the most frequent value in a data set.

Distributions of Variables

• Distributions can be characterized by the number of peaks or modes.
  • Unimodal: Single peak
  • Bimodal: Two peaks
  • Multimodal: More than two peaks.
• When a distribution is symmetric, the mean and median are equal.
• When a distribution has a longer lower or upper tail, the mean is generally below or above the median.

Outliers

• Outliers or extreme values are observations that are substantially different from other observations.
• Problematic outliers are not representative of the population and can distort statistical tests.

The Notion of Variance

• Variance measures how far a set of numbers is spread out.
  • Small Variance: Data is close to the mean.
  • Large Variance: Data is spread out around the mean.
• The Standard Deviation (σ) is a measure of variability that describes how much the data spreads out around the mean.
  • Small Standard Deviation: Values are close to the mean.
  • Large Standard Deviation: Values are spread further from the mean.
• The z-score is used to calculate probabilities and tells us how many standard deviations a value is away from the mean.

Z Scores

• Z-score is a measure of how many standard deviations a data point is away from the mean.
• A z-score of 1.4 means that a data point is 1.4 standard deviations above the mean.
• A z-score of -2.1 means that a data point is 2.1 standard deviations below the mean.
• Z-scores can be used with a Standard Normal Distribution Table to calculate probabilities of observing data points within a specific range.

This quiz covers essential concepts of statistics, including descriptive and inferential statistics. It highlights crucial symbols such as the population and sample means, variances, and standard deviations. Test your knowledge on how statistics are used to summarize data and make population generalizations.