Statistics: Levels of Measurement and Data Types

Levels of Measurement

• Nominal: Categories without order (e.g., favorite colors, types of pets)
• Ordinal: Ordered categories, differences between them are not meaningful (e.g., movie rankings)
• Interval: Ordered data with meaningful differences, no true zero (e.g., temperature in Celsius)
• Ratio: Ordered data with meaningful differences and a true zero (e.g., age, weight, income)

Data Types

• Quantitative Data: Numerical values
  • Discrete: Countable values (e.g., number of students)
  • Continuous: Measured values (e.g., height, weight)
• Qualitative Data: Categorical data (e.g., types of cars, favorite foods)

Descriptive Statistics

Central Tendency

• Mean: Average value (sum of values / number of values)
• Median: Middle value when data is ordered
• Mode: Most frequent value

Measures of Spread

• Range: Difference between maximum and minimum values
• Standard Deviation: Measures how spread out data is around the mean; smaller standard deviation indicates data more clustered around the mean.

Graphical Representations

• Histograms: Show frequencies of data ranges
• Boxplots: Display 5-number summary (minimum, Q1, median, Q3, maximum)
• Stem-and-Leaf Plots: Organize data to show its shape and distribution

Sampling Methods

• Random Sampling: Every individual has an equal chance of selection
• Systematic Sampling: Selecting every nth individual
• Stratified Sampling: Dividing population into groups and sampling from each
• Cluster Sampling: Dividing population into groups and sampling entire groups

Probability Basics

Complement Rule

• If P(A) is the probability of event A, then P(A') (complement) is the probability that A does not happen: P(A') = 1 - P(A)

Binomial Probability

• Calculates the likelihood of getting a certain number of successes in a fixed number of trials:
  • P(X=k) = ( n k ) * p^k * (1-p)^n-k
  • n: Number of trials
  • k: Number of successes
  • p: Probability of success
  • 1-p: Probability of failure

Confidence Intervals

• Estimates a range for the true population mean:
  • CI = x̄ ± z * σ /√n
  • x̄: Sample mean
  • z: Z-score (based on confidence level)
  • σ: Standard deviation
  • n: Sample size

Linear Regression & Correlation

• Regression: Predicts relationship between two variables using y = mx + b (where 'm' is the slope and 'b' is the intercept)
• Correlation Coefficient (r): Measures strength of relationship between two variables; ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation).

Test your understanding of different levels of measurement in statistics, including nominal, ordinal, interval, and ratio. Explore the distinctions between quantitative and qualitative data, as well as central tendency and measures of spread like mean and standard deviation.