Statistics: Parameters and Statistics, Sampling Distribution,

Definitions

  • Parameter: A measurement or quantity that describes a population.
  • Statistic: A measurement or quantity that describes a sample.

Example of Calculating Population Metrics

  • Data Set: Scores in General Mathematics for Grade 11 H.E Knowledge (Summative Test): 25, 35, 34, 29, 28, 41, 27, 38, 39, 54.
    • Population Mean:
      xˉ=ΣxN\bar{x} = \frac{\Sigma x}{N}
    • Population Variance:
      σ2=Σ(x−xˉ)2N\sigma^2 = \frac{\Sigma (x - \bar{x})^2}{N}
    • Population Standard Deviation:
      σ=Σ(x−xˉ)2N\sigma = \sqrt{\frac{\Sigma (x - \bar{x})^2}{N}}

Example of Calculating Sample Metrics

  • Random Sample of 5 (out of the 10 total): 25, 35, 34, 29, 28.
    • Sample Mean:
      xˉ=Σxn\bar{x} = \frac{\Sigma x}{n}
    • Sample Variance:
      s2=Σ(x−xˉ)2n−1s^2 = \frac{\Sigma (x - \bar{x})^2}{n-1}
    • Sample Standard Deviation:
      s=Σ(x−xˉ)2n−1s = \sqrt{\frac{\Sigma (x - \bar{x})^2}{n-1}}

Sampling Distribution of Sample Means

  • Definition: A frequency distribution of sample means derived from all possible samples of a specific size from a population.
  • Sampling Error: The difference between the sample mean and the population mean.

Steps to Construct Sampling Distribution of Means

  1. Determine the number of possible random samples from the population using:
    NCn=N!n!(N−n)!NCn = \frac{N!}{n!(N-n)!}
    where N = population size, n = sample size.
  2. List all the possible samples and compute the mean for each.
  3. Construct the sampling distribution.
  4. Create a histogram of the sampling distribution of means.

Detailed Example

  • Population: 2, 4, 9, 10, 5.
  • Compute all possible samples of size 3, e.g.:
    • 2, 4, 9
    • 2, 4, 10
    • 2, 4, 5
    • 2, 9, 10
    • 2, 9, 5
    • 2, 10, 5
    • 4, 9, 10
    • 4, 9, 5
    • 4, 10, 5
    • 9, 10, 5
  • Calculate the mean for each sample, then compile the means into a frequency table to analyze the sampling distribution.