Trimmed Mean & Percentile ungrouped data

Trimmed Mean

The trimmed mean is a statistical measure of central tendency that involves removing a specified percentage of data points from both ends of a sorted dataset before calculating the mean. This technique is useful for reducing the impact of outliers on the mean.

Calculation:

1. Sort the data: Arrange the data points in ascending or descending order.

2. Trim the data: Remove a specified percentage of data points from both ends. For example, a 10% trimmed mean removes the top 10% and bottom 10% of the data.

3. Calculate the mean: Calculate the average of the remaining data points.

Advantages:

• Less sensitive to outliers than the regular mean.

• Can provide a more robust measure of central tendency.

Disadvantages:

• Can lose information by discarding data points.

• The choice of the trimming percentage can affect the result.

Percentiles

Percentiles divide a dataset into 100 equal parts. A percentile represents the value below which a certain percentage of the data falls.

Common Percentiles:

• 25th percentile (Q1): The value below which 25% of the data falls (first quartile).

• 50th percentile (Q2): The median, or the value below which 50% of the data falls.

• 75th percentile (Q3): The value below which 75% of the data falls (third quartile).

Calculation:

1. Sort the data: Arrange the data points in ascending order.

2. Determine the position of the percentile:

   • For the nth percentile, the position is (n/100) * (N + 1), where N is the number of data points.

3. Find the value at the calculated position:

   • If the position is a whole number, the percentile is the value at that position.

   • If the position is not a whole number, interpolate between the two nearest values.

Applications:

• Measuring the spread of data.

• Identifying outliers.

• Comparing different datasets.

Trimmed Mean Example

Problem: Consider the following dataset representing the daily sales of a small business: 10, 15, 20, 25, 30, 35, 40, 45, 50, 100

Calculate the 10% trimmed mean.

Solution:

1. Sort the data: 10, 15, 20, 25, 30, 35, 40, 45, 50, 100

2. Trim 10% from each end: Remove the lowest 10% (1 value) and the highest 10% (1 value). 15, 20, 25, 30, 35, 40, 45

3. Calculate the mean of the remaining values: (15+20+25+30+35+40+45) / 7 = 30

Therefore, the 10% trimmed mean is 30.

Percentiles Example

Problem: Calculate the 25th, 50th, and 75th percentiles for the following dataset: 12, 15, 18, 22, 25, 28, 31, 34, 37, 40

Solution:

1. Sort the data: 12, 15, 18, 22, 25, 28, 31, 34, 37, 40

2. Calculate the positions:

   • 25th percentile: (25/100) * (10 + 1) = 2.75

   • 50th percentile: (50/100) * (10 + 1) = 5.5

   • 75th percentile: (75/100) * (10 + 1) = 8.25

3. Find the values:

   • 25th percentile: The value at the 3rd position is 18.

   • 50th percentile: The average of the 5th and 6th values is (25+28)/2 = 26.5.

   • 75th percentile: The value at the 9th position is 37.

Therefore, the 25th, 50th, and 75th percentiles are 18, 26.5, and 37, respectively.