Statistical Measures Summary

Chapter 1 – Statistical Measures

1.1 What are Statistics?

  • Statistics encompass the study of:

    • Collection of data

    • Processing of data

    • Summarization of data

    • Analysis of data

    • Interpretation of data

What is Data?

  • Quantitative Data: Numbers referring to a characteristic (e.g., ages).

    • Example: Ages of 500 individuals: 16, 45, 62, 81, …, 54.

  • Qualitative Data: Symbols/letters referring to a characteristic (e.g., gender).

    • Example: Genders of 500 individuals: M, F, F, M, …, M, F.

  • Focus on quantitative data throughout the course.

1.2 Populations and Samples

  • Population: Entire set of measurements under consideration.

  • Sample: Subset of the population.

    • A random sample ensures each population entry has an equal chance of selection.

1.3 Statistical Measures

  • Measures are computed numbers that summarize data characteristics.

  • Types of Measures:

    • Measures of Central Tendency (1.4)

    • Measures of Variation (1.5)

1.4 Measures of Central Tendency

  • Describes the center of data; key types:

The Mean

  • Most common measure of central tendency.

  • Calculation:

    • Each data entry is represented by x, with n as the number of entries.

    • Mean = (Sum of data entries) / n

  • Example Calculation:

    • Sample Data (scores): 35, 82, 61, 39, 92, 53, 72

    • Sample Mean Calculation:

    • n = 7

    • Mean = (35 + 82 + 61 + 39 + 92 + 53 + 72) / 7 = 62.

The Weighted Mean

  • Accounts for varying importance of data entries.

  • Example: John's Course Score Calculation:

    • Participation: 87 (weight 0.1)

    • Midterm: 72 (weight 0.4)

    • Final: 47 (weight 0.5)

    • Weighted Mean = (87 * 0.1 + 72 * 0.4 + 47 * 0.5) / (0.1 + 0.4 + 0.5) = 61.

The Median

  • Middle value when data is ordered.

  • Calculation Steps:

    1. Order data from smallest to largest.

    2. Identify median.

  • For an even number of entries, average the two middle values.

  • Example:

    • Data: 25, 24, 25, 22, 23, 20, 20, 24, 21, 22, 25, 24 (Total 12 entries);

    • Ordered: 20, 20, 21, 22, 22, 23, 24, 24, 24, 25, 25, 25;

    • Median = (22 + 23) / 2 = 22.5.

1.5 Measures of Variation

  • Measures extent and nature of data spread.

The Range

  • Formula: Range = Largest data entry - Smallest data entry.

  • Example: Range of Example 8 = 25 - 20 = 5.

Variance

  • Represents average squared distance of each data entry from the mean.

  • Formula:

    • Variance (s²) = (Sum of (x - mean)²) / (n - 1).

  • Standard Deviation (s): Square root of variance.

    • Indicates average deviation from the mean.

  • Example Calculation:

    • Given temperatures: 20, 23, 18, 25, 21, 25;

    • Compute mean, followed by variance and standard deviation.

Interpretation of Standard Deviation

  • Low standard deviation indicates data entries are close to the mean.

  • High standard deviation indicates a wider spread of data entries.

  • It's useful for comparing variations across samples only if they share similar means.

  • Example 10 tasks include finding mean, median, variance, and standard deviation of six numbers.