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 ≈ 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.