Statistics Lecture Notes

Key Concepts

• Mean
• Median
• Mode
• Range
• Quartiles and Interquartile Range (IQR)
• Outliers
• Box and Whisker Plot
• Skewness
• Dot Plot
• Stem-and-Leaf Plot
• Frequency Table
• Histogram
• Percentiles

Data Set Analysis

Example Data Set 1: {7, 7, 10, 14, 15, 23, 32}

• Mean:
  • Sum: 108
  • Mean = 108 / 7 = 15.43
• Median:
  • Middle number: 14
• Mode:
  • Most frequent number: 7
• Range:
  • Difference between highest and lowest number: 32 - 7 = 25

Example Data Set 2: {11, 15, 15, 21, 37, 41, 59, 59}

• Mean:
  • Sum: 258
  • Mean = 258 / 8 = 32.25
• Median:
  • Middle numbers: 21 and 37
  • Median = (21 + 37) / 2 = 29
• Mode:
  • Numbers that appear most frequently: 15 and 59 (Bimodal)
• Range:
  • Difference between highest and lowest number: 59 - 11 = 48

Quartiles and Interquartile Range (IQR)

• Quartiles: Dividing a data set into four equal parts
  • Q1: Median of the lower half
  • Q2: Median of the entire data set
  • Q3: Median of the upper half
• IQR: Difference between Q3 and Q1 (IQR = Q3 - Q1)
• Determining Outliers:
  • Not an outlier if within range:
    • Q1 - 1.5 * IQR to Q3 + 1.5 * IQR

Example Data Set: {7, 11, 14, 5, 8, 27, 16, 10, 13, 17, 16}

• Q1: 8
• Q2: 13
• Q3: 16
• IQR: Q3 - Q1 = 16 - 8 = 8
• Outliers:
  • Range: -4 to 28
  • 27 is not an outlier

Box and Whisker Plot

• Components:
  • Minimum
  • Q1
  • Median (Q2)
  • Q3
  • Maximum
• Outliers: Represented by points outside the main box

Skewness

• Symmetric Distribution: Mean = Median
• Skewed Right (Positive Skew):
  • Mean > Median
  • Box and whisker plot: Right side longer
• Skewed Left (Negative Skew):
  • Mean < Median
  • Box and whisker plot: Left side longer

Dot Plot

• Example: Data set {5, 8, 3, 7, 1, 5, 3, 2, 3, 3, 8, 5}
  • Dot plot shows frequency of each number
  • Mode is the number with the most dots

Stem-and-Leaf Plot

• Construction:
  • Separate each number into a stem and leaf
  • Example: 78 -> Stem: 7, Leaf: 8
• Key: Provides explanation of stem-leaf representation
• Decimal Values Representation: Similar method, but stems can represent integer part and leaves decimal part

Frequency Table

• Components:
  • Value
  • Frequency
  • Relative Frequency
  • Cumulative Relative Frequency
• Example Calculation:
  • Data set: {5, 9, 8, 7, 8, 12, 9, 8, 10, 8, 9, 7}
  • Frequency of each value found
  • Relative Frequency: Frequency / Total number of data points
  • Cumulative Relative Frequency: Running total of the relative frequencies
• Mean Calculation from Frequency Table: Sum of (value * frequency) / Total frequency

Histogram

• Construction:
  • Bar graph representation of data
  • Bars are connected
  • Example: Test scores categorized by grade intervals

Percentiles

• Calculation: Using cumulative relative frequency to determine percentile rank
• Example:
  • Data set with frequency and cumulative relative frequency columns
  • 60th percentile: Average of values around 0.60 cumulative relative frequency
  • 80th percentile: Average of values around 0.80 cumulative relative frequency

Summary

• Statistics Basics: Mean, median, mode, range, quartiles, IQR, outliers, skewness, dot plots, stem-and-leaf plots, frequency tables, histograms, percentiles
• Importance: Understanding these concepts is crucial for analyzing and interpreting data effectively