The algebraic sum of the deviations of 20 observations, measured from 30, is 4. Find the arithmetic mean of these observations.

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Points to Remember:

  • The algebraic sum of deviations from the mean is always zero.
  • The arithmetic mean is the sum of observations divided by the number of observations.
  • We can use the property of deviations to solve this problem.

Introduction:

This question involves a fundamental concept in statistics: the arithmetic mean and the properties of deviations from the mean. The arithmetic mean (or average) is a measure of central tendency, representing the central value of a dataset. A key property is that the sum of the deviations of individual data points from their mean is always zero. This question tests our understanding of this property to find the arithmetic mean of a dataset given the sum of deviations from a specific value.

Body:

Understanding Deviations:

Let’s denote the 20 observations as x₁, x₂, …, x₂₀. The question states that the algebraic sum of deviations of these observations from 30 is 4. Mathematically, this can be expressed as:

∑ᵢ₌₁²⁰ (xáµ¢ – 30) = 4

Finding the Arithmetic Mean:

We can rewrite the above equation as:

∑ᵢ₌₁²⁰ xáµ¢ – ∑ᵢ₌₁²⁰ 30 = 4

Since ∑ᵢ₌₁²⁰ 30 = 20 * 30 = 600, we have:

∑ᵢ₌₁²⁰ xáµ¢ – 600 = 4

Therefore, the sum of the observations is:

∑ᵢ₌₁²⁰ xᵢ = 604

The arithmetic mean (denoted by x̄) is calculated by dividing the sum of observations by the number of observations:

x̄ = (∑ᵢ₌₁²⁰ xᵢ) / 20 = 604 / 20 = 30.2

Verification:

The sum of deviations from the mean (30.2) should be zero. Let’s verify:

∑ᵢ₌₁²⁰ (xáµ¢ – 30.2) = ∑ᵢ₌₁²⁰ xáµ¢ – ∑ᵢ₌₁²⁰ 30.2 = 604 – (20 * 30.2) = 604 – 604 = 0

This confirms our calculation of the arithmetic mean.

Conclusion:

The arithmetic mean of the 20 observations is 30.2. We arrived at this conclusion by utilizing the fundamental property that the sum of deviations from the mean is always zero. By manipulating the given information about the sum of deviations from 30, we were able to determine the sum of the observations and subsequently calculate the arithmetic mean. This problem highlights the importance of understanding the relationship between the mean and deviations in statistical analysis. No further policy recommendations or best practices are needed as this is a purely mathematical problem. The solution demonstrates a clear and concise application of statistical principles.

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