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Chapter 2.75 Methods Statistics

Chapter 2: Research Methods: Statistics

Measures of Variability

  • Measures of variability describe the difference and spread within a dataset.

  • They are part of descriptive statistics and include:

    • Range

    • Variance

    • Standard Deviation

Range

  • The range is calculated as the difference between the highest and lowest scores in a distribution.

    • Example 1: If the highest score is 75 and the lowest score is 25, the range is 75 - 25 = 50.

    • Example 2: For scores 10 and 1, the range is 10 - 1 = 9.

Variance

  • Variance measures how much scores differ from the mean (average).

  • Understand variance as the dispersion of data points from the mean due to various chance factors.

Calculating Variance

  • Formula for sample variance:[ s^2 = \frac{\Sigma (x - \bar{x})^2}{n - 1} ]

    • Where:

      • ( s^2 ) = variance

      • ( \Sigma ) = sum of squared deviations

      • ( x ) = individual score

      • ( \bar{x} ) = mean

      • ( n ) = number of scores

Example of Variance Calculation

  1. Scores: {2, 3, 4, 5, 6}

  2. Mean: (2 + 3 + 4 + 5 + 6) / 5 = 4

  3. Calculating Differences from the Mean:

    • (2 - 4) = -2 → (-2)² = 4

    • (3 - 4) = -1 → (-1)² = 1

    • (4 - 4) = 0 → (0)² = 0

    • (5 - 4) = 1 → (1)² = 1

    • (6 - 4) = 2 → (2)² = 4

  4. Sum of Squared Deviations: 4 + 1 + 0 + 1 + 4 = 10

  5. Variance Calculation:( s^2 = \frac{10}{4} = 2.5 )

Standard Deviation

  • Reflects how tightly data values cluster around the mean.

  • A small standard deviation indicates that values are close to the mean, while a large standard deviation indicates they are spread out.

Calculating Standard Deviation

  • Formula:[ S = \sqrt{s^2} = \sqrt{\frac{\Sigma(x - \bar{x})^2}{n - 1}} ]

    • Standard deviation is the square root of the variance.

Standard Deviation Example

  • ABC School: Mean GPA = 70%, SD = 4.

  • DEF School: Mean GPA = 70%, SD = 28.

  • Decision on funding: ABC School should get more funding due to lower variability (SD of 4).

Properties of Normal Distribution

  • The normal distribution is bell-shaped with specific characteristics:

    • About 68% of scores lie within ±1 SD.

    • Approximately 95% lie within ±2 SD.

    • Roughly 99% lie within ±3 SD.

Z Scores

  • Z-scores quantify how many standard deviations a score is from the mean.

  • Percentile Ranking: Indicates score distribution relative to others (e.g., someone at the 90th percentile performed better than 90% of test takers).

Statistical Significance

  • Pertains to hypothesis testing; used to validate hypotheses against observed data.

  • Denoted by ( \alpha ) (alpha), usually set at 0.05 (5%).

  • Determine support for hypotheses using the p-value:

    • If p-value ≤ α, hypothesis is supported.

    • If p-value > α, null hypothesis is supported.

Inferential Statistics

  • These methods compare sample data to population data, allowing findings from samples to be generalized.

    • Example: A study on clothing's impact on popularity uses inferential statistics for broader conclusions.

Visual Representation of Normal Distribution

  • Demonstrates standard deviations and cumulative percentages across normal distribution range.

  • Shows data distribution where the mean divides data into symmetrical halves.