Average of the First Column

Oct 13, 2025 by talynt.org 18 views

Introduction

In the field of mathematics, data analysis is a crucial aspect of understanding and interpreting various phenomena. By examining data, we can identify patterns, trends, and correlations that can help us make informed decisions. In this article, we will analyze a given table and determine which of the following statements is true.

The Table

(centimeters)	(centimeters)
105.1	104.8
101.3	87
86.7	95
93.8	92.1
92.4	100

Statement Analysis

Based on the table, we need to determine which of the following statements is true:

The average of the first column is greater than the average of the second column.
The median of the first column is greater than the median of the second column.
The standard deviation of the first column is greater than the standard deviation of the second column.
The correlation coefficient between the two columns is positive.

Calculating Averages

To calculate the average of each column, we need to add up all the values and divide by the number of values.

Average of the First Column

The sum of the values in the first column is:

105.1 + 101.3 + 86.7 + 93.8 + 92.4 = 480.3

There are 5 values in the first column, so the average is:

480.3 / 5 = 96.06

Average of the Second Column

The sum of the values in the second column is:

104.8 + 87 + 95 + 92.1 + 100 = 478.9

There are 5 values in the second column, so the average is:

478.9 / 5 = 95.78

Comparing Averages

Since the average of the first column (96.06) is greater than the average of the second column (95.78), we can conclude that:

The average of the first column is greater than the average of the second column.

Calculating Medians

To calculate the median of each column, we need to arrange the values in order and find the middle value.

Median of the First Column

The values in the first column in order are:

86.7, 93.8, 101.3, 105.1, 92.4

The middle value is the third value, which is 101.3.

Median of the Second Column

The values in the second column in order are:

87, 95, 92.1, 100, 104.8

The middle value is the third value, which is 92.1.

Comparing Medians

Since the median of the first column (101.3) is greater than the median of the second column (92.1), we can conclude that:

The median of the first column is greater than the median of the second column.

Calculating Standard Deviations

To calculate the standard deviation of each column, we need to use the formula:

σ = √((Σ(xi - μ)^2) / (n - 1))

where xi is each value, μ is the mean, and n is the number of values.

Standard Deviation of the First Column

The sum of the squared differences from the mean is:

(105.1 - 96.06)^2 + (101.3 - 96.06)^2 + (86.7 - 96.06)^2 + (93.8 - 96.06)^2 + (92.4 - 96.06)^2 = 9.44 + 25.44 + 121.44 + 8.44 + 16.44 = 181.2

There are 5 values in the first column, so the standard deviation is:

σ = √(181.2 / (5 - 1)) = √(45.3) = 6.74

Standard Deviation of the Second Column

The sum of the squared differences from the mean is:

(104.8 - 95.78)^2 + (87 - 95.78)^2 + (95 - 95.78)^2 + (92.1 - 95.78)^2 + (100 - 95.78)^2 = 9.44 + 81.44 + 0.04 + 6.44 + 16.44 = 113.8

There are 5 values in the second column, so the standard deviation is:

σ = √(113.8 / (5 - 1)) = √(28.45) = 5.33

Comparing Standard Deviations

Since the standard deviation of the first column (6.74) is greater than the standard deviation of the second column (5.33), we can conclude that:

The standard deviation of the first column is greater than the standard deviation of the second column.

Calculating Correlation Coefficients

To calculate the correlation coefficient between the two columns, we need to use the formula:

ρ = Σ((xi - μx)(yi - μy)) / (√(Σ(xi - μx)^2) * √(Σ(yi - μy)^2))

where xi and yi are each value, μx and μy are the means, and Σ denotes the sum.

Correlation Coefficient

The sum of the products of the differences from the mean is:

(105.1 - 96.06)(104.8 - 95.78) + (101.3 - 96.06)(87 - 95.78) + (86.7 - 96.06)(95 - 95.78) + (93.8 - 96.06)(92.1 - 95.78) + (92.4 - 96.06)(100 - 95.78) = 9.44 * 9.02 + 5.24 * -8.78 + -9.36 * -0.78 + 2.74 * -3.68 + -3.66 * 4.22 = 85.31 - 45.81 - 7.28 - 10.08 - 15.45 = 6.69

The sum of the squared differences from the mean for the first column is:

181.2

The sum of the squared differences from the mean for the second column is:

113.8

The correlation coefficient is:

ρ = 6.69 / (√181.2 * √113.8) = 6.69 / (13.43 * 10.64) = 0.49

Interpretation

Since the correlation coefficient (0.49) is positive, we can conclude that:

The correlation coefficient between the two columns is positive.

Conclusion

Based on the analysis, we can conclude that:

The average of the first column is greater than the average of the second column.
The median of the first column is greater than the median of the second column.
The standard deviation of the first column is greater than the standard deviation of the second column.
The correlation coefficient between the two columns is positive.

Therefore, the correct answer is:

The average of the first column is greater than the average of the second column.
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