Calculating the Arithmetic Mean

Oct 13, 2025 by miodeya.com 33 views

Introduction

In this article, we will explore the concepts of arithmetic mean and median, and how to calculate them using a given data set. The arithmetic mean is a measure of central tendency that represents the average value of a data set, while the median is the middle value of a data set when it is arranged in order.

The Given Data Set

The given data set is a 2x5 matrix, where each row represents a set of five numbers.

{ \begin{array}{ccccc} 68.2 & 77.4 & 109.2 & 74.3 & 109.2 \\ 88.4 & 83 & 85.2 & 92.7 & 109.2 \end{array} \}

Calculating the Arithmetic Mean

The arithmetic mean is calculated by summing up all the values in the data set and then dividing by the total number of values.

Step 1: Sum up all the values in the data set

To calculate the sum of all the values in the data set, we need to add up all the numbers in both rows.

{ \begin{array}{ccccc} 68.2 & 77.4 & 109.2 & 74.3 & 109.2 \\ 88.4 & 83 & 85.2 & 92.7 & 109.2 \end{array} \}

Sum of first row = 68.2 + 77.4 + 109.2 + 74.3 + 109.2 = 438.3 Sum of second row = 88.4 + 83 + 85.2 + 92.7 + 109.2 = 458.5

Total sum = 438.3 + 458.5 = 896.8

Step 2: Divide the total sum by the total number of values

The total number of values in the data set is 10 (5 values in each row).

Arithmetic mean = Total sum / Total number of values = 896.8 / 10 = 89.68

Calculating the Median

The median is the middle value of a data set when it is arranged in order. Since the data set has an even number of values, the median will be the average of the two middle values.

Step 1: Arrange the data set in order

To calculate the median, we need to arrange the data set in order from smallest to largest.

{ \begin{array}{ccccc} 68.2 & 74.3 & 77.4 & 83 & 85.2 \\ 88.4 & 92.7 & 109.2 & 109.2 & 109.2 \end{array} \}

Step 2: Identify the two middle values

The two middle values are the third and fourth values in the ordered data set.

Third value = 77.4 Fourth value = 83

Step 3: Calculate the median

Median = (Third value + Fourth value) / 2 = (77.4 + 83) / 2 = 160.4 / 2 = 80.2

Conclusion

In this article, we calculated the arithmetic mean and median of a given data set. The arithmetic mean is a measure of central tendency that represents the average value of a data set, while the median is the middle value of a data set when it is arranged in order. We used the given data set to calculate the arithmetic mean and median, and found that the arithmetic mean is 89.68 and the median is 80.2.

References

[1] Wikipedia. (n.d.). Arithmetic mean. Retrieved from https://en.wikipedia.org/wiki/Arithmetic_mean
[2] Wikipedia. (n.d.). Median. Retrieved from https://en.wikipedia.org/wiki/Median

Mathematical Formulas

Arithmetic mean = (Sum of all values) / (Total number of values)
Median = (Middle value 1 + Middle value 2) / 2

Code

import numpy as np

# Given data set
data_set = np.array([[68.2, 77.4, 109.2, 74.3, 109.2],
                      [88.4, 83, 85.2, 92.7, 109.2]])

# Calculate the sum of all values
sum_of_values = np.sum(data_set)

# Calculate the total number of values
total_number_of_values = np.size(data_set)

# Calculate the arithmetic mean
arithmetic_mean = sum_of_values / total_number_of_values

# Calculate the median
ordered_data_set = np.sort(data_set)
middle_index = total_number_of_values // 2
median = (ordered_data_set[middle_index - 1] + ordered_data_set[middle_index]) / 2

print("Arithmetic mean:", arithmetic_mean)
print("Median:", median)

Note: The code is written in Python and uses the NumPy library to perform the calculations.