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.