Lecture Notes: Measures of Central Tendency and Dispersion

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Finding Medians and Modes

Median

• Def: Median splits a data set into two equal parts.
• Grouped Data Example: Values with assigned frequencies:
  • Four 1s, three 2s, two 3s, six 4s, eight 5s.
• Position of Median:
  • Calculate the sum of frequencies: 23
  • Formula: (Sum of frequencies + 1) / 2 = 12th position
  • Line up values in ascending order; 12th position falls under value 4, making the median 4.

Mode

• Def: Mode is the value that occurs most often.
• Examples:
  • Small sample (10 students’ number of siblings): Mode is 3.
  • Frequency distribution: 5 has the highest frequency (8), so the mode is 5.

Stem and Leaf Displays

• Advantage: Allows visibility of actual data.
• Example: Stem and Leaf Display for data values
  • Find mode and median from display:
  • Median position: (Sum of frequencies + 1) / 2 = 11th position → Median is 37.
  • Mode: Highest frequency digit within grouped values (e.g., most frequent digits in the 10's places).

Symmetry in Data Sets

• Symmetric Distribution: Pattern of frequencies is similar on the left and right of a central point.
• Non-symmetric Distribution: Different patterns on the left and right.
  • Uniform: Values are uniformly distributed (e.g., same or nearly the same frequency).
  • Bimodal: Two peaks in the distribution, not necessarily equal.
  • Skewed Left: Tail extends left.
  • Skewed Right: Tail extends right.

Measures of Central Tendency: Summary

• Mean: Sum of values divided by the number of values.
• Median: Middle value when values are ordered; for even numbers, it’s the mean of two central values.
• Mode: Value that occurs most frequently; some data sets may be bimodal or multimodal.

Measures of Dispersion

Range

• Def: Difference between the maximum and minimum values in a data set.
• Example: Set A (Range = 12); Set B (Range = 4)

Standard Deviation

• Def: Measures data spread from the mean.
• Calculation Steps: (Sample)
  1. Calculate mean.
  2. Find deviations from mean.
  3. Square each deviation.
  4. Sum squared deviations.
  5. Divide by (n-1) for samples.
  6. Take the square root.

Comparing Standard Deviations

• Used to compare the spread or consistency between different data sets.
• Example: Comparing sugar pack consistency from two companies using their mean and standard deviation values.

Coefficient of Variation

• Def: Standard deviation expressed as a percentage of the mean.
• Usage: Allows meaningful comparisons between different data sets by normalizing the spread.
• Example: Comparing relative dispersions between two samples with different units.

Measures of Position

Z-Score

• Def: Measures how many standard deviations a value is from the mean.
• Calculation: Z = (X - mean) / standard deviation
• Example: Comparing performance of students (Jen and Joy) in different settings using their Z-scores.

Percentiles

• Def: Value below which a specified percent of observations fall.
• Calculation: Percentile rank position = (Percentage * Total number of data) -> rounding rules apply.
• Example: Finding the 40th percentile in a given test score distribution.

Additional Measures

• Deciles, Quartiles: Will be covered in the next lecture.
• Box Plots: Visualization tool for showing measures of position.

Summary

• Measures of Central Tendency: Mean, Median, Mode.
• Measures of Dispersion: Range, Standard Deviation, Coefficient of Variation.
• Measures of Position: Z-scores, Percentiles, Deciles, Quartiles.
• Understanding symmetry and shape of data distributions.