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Fundamental Statistical Concepts Explained

Apr 23, 2025

Lecture Notes: Statistical Concepts

Mean, Median, Mode, and Range

  1. Arranging Data

    • Organize data in increasing order: 7, 7, 10, 14, 15, 23, 32.
  2. Mean (Average)

    • Formula: Sum of numbers / Total number of numbers.
    • Example: (7 + 7 + 10 + 14 + 15 + 23 + 32) / 7 = 108 / 7 β‰ˆ 15.43.
  3. Median

    • Median is the middle number: Eliminate numbers from both ends to find the middle.
    • Example: Middle number is 14.
  4. Mode

    • Mode is the most frequent number: 7 appears twice.
  5. Range

    • Range is the difference between the highest and lowest numbers: 32 - 7 = 25.

Second Data Set Example

  1. Data Set: 11, 15, 15, 21, 37, 41, 59
  2. Mean
    • (11 + 15 + 15 + 21 + 37 + 41 + 59) / 8 = 258 / 8 = 32.25.
  3. Median
    • Find two middle numbers, average them: (21 + 37) / 2 = 29.
  4. Mode
    • Bimodal: 15 and 59 both appear twice.
  5. Range
    • 59 - 11 = 48.

Quartiles and Interquartile Range

  1. Quartiles

    • Q1: Median of the lower half.
    • Q2: Median of the entire data set.
    • Q3: Median of the upper half.
  2. Interquartile Range (IQR)

    • IQR = Q3 - Q1
  3. Outliers

    • Formula: If a number is outside Q1 - 1.5IQR or Q3 + 1.5IQR, it’s an outlier.

Example Problem

  1. Data Set: 7, 11, 14, 5, 8, 27, 16, 10, 13, 17, 16.
  2. Quartiles Calculation
    • Arrange data, find Q1, Q2, Q3.
    • IQR = Q3 - Q1 = 16 - 8 = 8.
  3. Outliers
    • Check if 27 is an outlier: It is not, as it falls within calculated range.

Box-and-Whisker Plot

  1. Components
    • Minimum, Q1 (25th percentile), Median (Q2, 50th percentile), Q3 (75th percentile), Maximum.
  2. Identifying Outliers
    • Outliers appear as individual points outside whiskers.

Skewness

  1. Symmetric Distribution
    • Mean = Median.
  2. Right Skew (Positive Skew)
    • Tail extends to the right.
    • Mean is greater than the median.
  3. Left Skew (Negative Skew)
    • Tail extends to the left.
    • Mean is less than the median.

Dot Plot Construction

  1. Example Data: 5, 8, 3, 7, 1, 5, 3, 2, 3, 3.
  2. Mode Identification
    • Mode: 3 (most dots in the plot).

Stem-and-Leaf Plot

  1. Example Data: 4, 9, 13, 17, 21, 36, 56.
  2. Construction
    • Separate tens (stem) and units (leaf).

Frequency Table and Histogram

  1. Frequency Table: Number, Frequency, Sum.
  2. Histogram
    • Bars connected, represents frequency distribution.

Relative Frequency Table

  1. Percentiles and Cumulative Frequency
    • Use data to calculate specific percentile values.

Conclusion

  • This lecture covers fundamental statistical concepts such as measures of central tendency, variability, skewness, and data representation methods.