Stat301 L2

Descriptive Statistics and Data Distribution

Distribution Definitions

  • Distribution of data can be summarized using quartiles (Q1, Q3).

    • Q1: 25th percentile, minimum data point to Q1 contains approximately 25% of the data.

    • Median (Q2): Second quartile that divides the dataset into two equal halves.

    • Q3: 75th percentile, the upper range where the last 25% of the data lies.

Skewness

  • Skewness: Measures asymmetry in the distribution of data.

    • If skewness is approximately 0, data is symmetric.

    • Right (Positive) Skew: Long tail on the right.

    • Left (Negative) Skew: Long tail on the left.

Sample vs. Population

  • Population Mean (μ): Average of a complete set of data.

  • Sample Mean (x̄): Average of a subset from the population.

  • Symbols differ when calculating skewness:

    • Population: uses μ;

    • Sample: uses x̄.

Histogram Construction

  • Histogram: Graphical representation of frequency distribution.

    • Frequencies calculated for each class.

  • Analyze symmetry through the shape of the histogram.

Outliers and Influential Points

Definitions

  • Outliers: Values significantly different from the rest of the data.

    • Influence statistical results if included or excluded.

  • Influential Points: Observations that can overly affect the result of statistical analysis.

Identifying Outliers

  • Use IQR (Interquartile Range) for detection:

    • Outlier Criterion: < Q1 - 1.5IQR or > Q3 + 1.5IQR.

    • Example Calculation:

      • Data: Q1 = 22, Q3 = 38, IQR = 38 - 22 = 16.

      • Lower Limit = Q1 - 1.5*IQR = 22 - 24 = -2.

      • Upper Limit = Q3 + 1.5*IQR = 38 + 24 = 62.

      • Outliers identified are values beyond these limits.

Graphical and Terminal Measures

Relative Frequency

  • Total relative frequency of a dataset equals 1.

    • Example: If relative frequency is defined for four intervals, their sum must equal 1.

    • Class Density: May require calculating areas under rectangles in histograms to determine frequency.

Handling Continuous Data

  • Class Boundaries: Determine equal class sizes for continuous data (e.g., -1 to 0, 0 to 1).

  • Ensure that each interval represents relevant class data accurately in histograms.