Statistics Lecture Notes

Key Topics

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

Mean, Median, Mode, and Range

1. Mean (Average)

   • Arrange data in increasing order.
   • Calculate sum of numbers.
   • Divide sum by number of data points.
   • Example: For data set [7, 7, 10, 14, 15, 23, 32], Mean = 108 / 7 ≈ 15.43.

2. Median

   • Middle number when data is ordered.
   • If even data points, median = average of two middle numbers.
   • Example: Median of [7, 10, 14, 15, 23, 32] is 14.

3. Mode

   • Number that occurs most frequently.
   • Example: Mode of [7, 7, 10, 14, 15] is 7.
   • Bimodal if two numbers occur most frequently.

4. Range

   • Difference between highest and lowest numbers.
   • Example: Range = 32 - 7 = 25.

Quartiles and Interquartile Range (IQR)

• Q1: 1st Quartile (25th percentile)

• Q2: 2nd Quartile (Median or 50th percentile)

• Q3: 3rd Quartile (75th percentile)

• Interquartile Range (IQR): Q3 - Q1

• Identifying Outliers:

  • Outliers are numbers outside [Q1 - 1.5(IQR), Q3 + 1.5(IQR)].

Box and Whisker Plot

• Visual representation of data.
• Includes Minimum, Q1, Median (Q2), Q3, Maximum.
• Outliers are plotted as individual points.

Skewness

• Symmetric data: Mean = Median.
• Right Skew (Positive): Mean > Median.
• Left Skew (Negative): Mean < Median.

Dot Plot

• Simple plot using dots to represent frequency above number line.

Stem-and-Leaf Plot

• Divide numbers into "stem" and "leaf".
• Example: 56 is 5 (stem) | 6 (leaf).
• Useful for visualizing distribution.

Frequency Table

• Shows frequency of each number or category.
• Sample Mean Calculation: Use frequency table to multiply frequency by values, sum them and divide by total number of values.

Histogram

• Similar to bar graph with contiguous bars.
• Represents frequency distribution of numerical data.
• Categories (classes) like grades can be used.

Cumulative Relative Frequency

• Running total of relative frequencies.
• Helps find percentiles easily.
• Example: 60th percentile may fall between two numbers in cumulative frequency.

Practice Examples

• Calculate mean, median, mode, range, IQR, identify outliers, and construct box plots.
• Analyze skewness of data using mean and median.
• Construct stem-and-leaf plots and frequency tables.
• Draw histograms and dot plots.

Tips

• Always arrange data in order before calculating statistics.
• Double-check calculations for percentiles and outliers.
• Visual tools like plots and graphs enhance understanding of distribution.