Mathematical Average, Average of Position, and Measures of Partition Values

 

Introduction

Statistics is a vital branch of mathematics that helps us understand and interpret data. Among its fundamental concepts are averages and partition values, which summarize data and help in analyzing trends and distributions. This article explores the Mathematical Average, Average of Position, and Measures of Partition Values in detail.

Measure of partition value

1. Mathematical Average

Definition:

A mathematical average is a single value that represents the central tendency of a dataset. It helps to summarize a large set of data using a single representative value.

Types of Mathematical Averages:

a. Arithmetic Mean:

It is the most common type of average, calculated by dividing the sum of all values by the number of values.

$$\text{Arithmetic Mean (AM)} = \frac{\sum x}{n}$$

Example:

Data: 5, 7, 9, 11
AM = (5 + 7 + 9 + 11) / 4 = 32 / 4 = 8

b. Geometric Mean (GM):

It is the nth root of the product of n values.

$$\text{GM} = \sqrt[n]{x_1 \cdot x_2 \cdot x_3 \cdots x_n}$$

• $\sqrt[3]{2 \times 4 \times 8} = \sqrt[3]{64} = 4$

1. Example:
   Data: 2, 4, 8
   GM = $\sqrt[3]{2\times 4\times 8}=\sqrt[3]{64}=4$

c. Harmonic Mean (HM):

It is the reciprocal of the arithmetic mean of the reciprocals of the values.

$$\text{HM} = \frac{n}{\sum \frac{1}{x_i}}$$

1. Example:
   Data: 2, 4, 8
   HM = 3 / (1/2 + 1/4 + 1/8) = 3 / (0.875) ≈ 3.43

2. Average of Position (Positional Averages)

Definition:

Positional averages are values that represent the position of data within a sorted dataset, rather than the numerical computation of all data values.

Types of Positional Averages:

a. Median:

The middle value when data is arranged in ascending or descending order.

If n is odd: Median = middle value

If n is even: Median = average of two middle values

Example:

Data: 3, 5, 7, 9, 11 → Median = 7

Data: 3, 5, 7, 9 → Median = (5+7)/2 = 6

b. Mode:

The most frequently occurring value in a dataset.

Example:

• Data: 2, 4, 4, 5, 6 → Mode = 4

3. Measures of Partition Values

Definition:

Partition values divide the dataset into equal parts or groups. These values are used in descriptive statistics to understand data distribution.

Types of Partition Values:

a. Quartiles (Q):

Divide the data into four equal parts.

1. Q1 (Lower Quartile): 25% of data below it

2. Q2 (Median): 50% of data below it

3. Q3 (Upper Quartile): 75% of data below it

b. Deciles (D):

Divide the data into ten equal parts.

D1 to D9 denote the 10th to 90th percentiles respectively.

c. Percentiles (P):

Divide the data into hundred equal parts.

P1 to P99 represent the 1st to 99th percentiles.

Example (Quartile Calculation):

Data: 5, 10, 15, 20, 25, 30, 35, 40 

Q1 = 12.5, Q2 (Median) = 22.5, Q3 = 32.5

Conclusion

Understanding the various types of averages and partition values is crucial in statistical analysis. Mathematical averages like mean, median, and mode offer quick insights into the central tendency, while positional and partition values like quartiles, deciles, and percentiles provide a deeper understanding of data distribution and variability. These concepts are foundational tools for researchers, economists, biologists, and data analysts alike.

References:

1. Gupta, S. C., & Kapoor, V. K. (2020). Fundamentals of Mathematical Statistics. Sultan Chand & Sons.

2. Spiegel, M. R., Schiller, J., & Srinivasan, R. A. (2017). Schaum's Outline of Statistics. McGraw-Hill Education.

3. Freund, J. E., & Perles, B. M. (2007). Modern Elementary Statistics. Pearson Education.