Study Notes on Measures of Spread

Course Information

  • Chapter: 3.2 (part B)

  • Topic: Measures of Spread

  • Instructor: Melony Parkhurst

  • Textbook: Essential Statistics, Navidi & Monk

  • Course Code: STAT 1401

Bell-Shaped Histogram

  • Description: To describe a histogram that has a single peak near the center of the data.

    • Shape: Bell-shaped

    • Histogram: This type of histogram is characterized by its symmetry and a peak at the center.

The Empirical Rule

  • Definition: The Empirical Rule is used for data that has a bell-shaped distribution. It provides an approximate description of the data regarding the spread and distribution of data points relative to the mean.

  • Characteristics:

    • The data follows a bell-shaped histogram.

    • Standard Deviations: When population data has a bell-shaped histogram:

    • Approximately 68% of the data falls within 1 standard deviation of the mean.

    • Approximately 95% of the data falls within 2 standard deviations of the mean.

    • Approximately 99.7% of the data falls within 3 standard deviations of the mean.

Symbols in Distribution

  • X-axis Representation: The symbols on the x-axis of the histogram represent the mathematical concepts and data characteristics rather than specific values. The symbol 'μ' is commonly used to denote the mean.

Data Distribution Percentages

  • Distribution: The percentages of data are distributed among the standard deviations in relation to the mean, as described in the context of a bell-shaped histogram. Refer to specific diagrams for highlighted areas and percentage values.

Chebyshev’s Inequality

  • Definition: Chebyshev’s Inequality offers a method to approximate the proportion of data within a certain number (K) of standard deviations from the mean.

  • Application: This inequality is more general as it can be applied to any shape of data distribution.

Comparison of Empirical Rule and Chebyshev’s Inequality

  • Empirical Rule:

    • Applicability: Only applicable to data that follows a bell-shaped distribution.

  • Chebyshev’s Inequality:

    • Applicability: Can apply to any shape of data distribution, hence is more versatile.

    • Disadvantages: Often provides a rough approximation, and the actual proportions of data within K standard deviations are typically larger than the estimates given by Chebyshev’s Inequality.

Exercises

  • Exercise 1:

    • Context: A data set has a population mean of 20 and a standard deviation of 3.

    • Question 1: Is it appropriate to use the Empirical Rule to approximate the proportion of data between 14 and 26? Why or why not?

  • Exercise 2:

    • Context: A data set has a mean of 50 and a standard deviation of 8.

    • Question 2: Is it appropriate to use the Empirical Rule to approximate the proportion of data between 42 and 58? Why or why not?

  • Exercise 3:

    • Context: Assume a sample comes from population data with a bell-shaped histogram; sample mean is 120 and sample standard deviation is 8.

    • Questions:

    1. Label the distribution according to the Empirical Rule.

    2. What are the cut-off values for the middle 95%?

    3. What percent of the values are between 112 and 128?