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Descriptive Statistics

  • Understand key measures of central tendency (mean, median, mode).
  • Learn about measures of dispersion (range, variance, standard deviation).
  • Explore data visualization techniques.

Measures of Central Tendency

Central tendency measures describe the center of a dataset. The most common measures are:

Measure Formula Definition
Mean (Arithmetic Average) \( \bar{x} = \frac{\sum x_i}{n} \) Sum of all values divided by the number of values.
Median Middle value when data is ordered If \( n \) is odd, the middle value; if even, the average of two middle values.
Mode Most frequently occurring value A dataset can have no mode, one mode, or multiple modes.

Measures of Dispersion

Dispersion measures how spread out the data points are. Key measures include:

Measure Formula Definition
Range \( \text{Range} = \max(x) - \min(x) \) Difference between the largest and smallest value.
Variance \( \sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 \) Measures the squared deviation of each value from the mean.
Standard Deviation \( \sigma = \sqrt{\sigma^2} \) Square root of the variance, indicating the typical deviation from the mean.

Exercises

  • Question 1: Find the mean, median, and mode of {5, 8, 9, 7, 6, 5, 5, 10}.
  • Question 2: Compute the range, variance, and standard deviation for {4, 5, 6, 7, 8, 9, 10}.
  • Question 3: Draw a histogram for the dataset {2, 3, 3, 4, 4, 4, 5, 5, 6}.
  • Answer 1: Mean = 6.875, Median = 6.5, Mode = 5.
  • Answer 2: Range = 6, Variance = 4.67, Standard Deviation ≈ 2.16.
  • Answer 3: The histogram below represents the dataset from Question 3:

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