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III.A. Terms and Definitions

  • Overview: Statistics as a body of knowledge to evaluate variability in all systems; variability exists in measurements, signatures, snowflake patterns, etc.
  • Core topics from the Body of Knowledge overview:
    • Terms and Definitions
    • Data Types and Collection Methods
    • Sampling
    • Measurement Systems Analysis
    • Statistical Process Control (SPC)
    • Advanced Statistical Analysis
  • This section introduces the foundational concepts used throughout Data Analysis for CQPA preparation.

III.A.1. Basic Statistics

  • Descriptive statistics explain characteristics of a sample or population, including:

    • Measures of center: mean, median, mode
    • Measures of variability: range, standard deviation, variance
    • Location and frequency, and cumulative distributions

  • Central tendency (location of data):

    • Mean (x¯): arithmetic average; symbolized as x̄; used with many distributions, especially normal data
    • Median: middle value that splits data into two equal halves; useful with skewed data
    • Mode: most frequent value; can be multiple values (multimodal)

  • Mean example: data set 1, 1, 2, 3, 4, 5, 5 → mean = ar{x} = rac{1+1+2+3+4+5+5}{7} = 3.

  • Median example: data sets illustrated in transcript show medians such as 3, 5, 5.5 depending on data arrangement.

  • Mode example: data sets where the mode is 1 and 5; bimodal distributions discussed.

  • Variability and dispersion concepts:

    • Variability (dispersion) describes how data spread around the center
    • Range: difference between max and min; simple dispersion measure; not always informative about spread in the data
    • Standard deviation (s or σ): average distance of data points from the mean; most commonly used for dispersion; linked to normal distributions
    • Variance: square of the standard deviation; for a population, \sigma^2 = rac{1}{N}\sum{i=1}^{N}(xi-ar{x})^2; for a sample, s^2 = rac{1}{n-1}\sum{i=1}^{n}(xi-ar{x})^2.

  • Example: for data set 1, 2, 3, calculate mean = 2; then variance and standard deviation via the standard deviation formula; demonstration in transcript shows a simple calculation leading to a variance of 1 and standard deviation of 1 for the small example.

  • Summary points:

    • Descriptive statistics describe center and spread
    • Central tendency describes location; dispersion describes spread
    • The sample mean is an unbiased estimator for the population mean (via Central Limit Theorem considerations)

  • Quick formulas (summary):

    • Mean (sample): ar{x} = rac{1}{n}
      extstyle\sum{i=1}^{n} xi
    • Population mean: $$ar{\