Problem 56 - Applied Statistics and Probability for Engineers

Question

An adult can lose or gain two pounds of water in the course of a day. Assume that the changes in water weight are uniformly distributed between minus two and plus two pounds in a day. What is the standard deviation of a person's weight over a day?

Step-by-Step Solution

Answer

The standard deviation is approximately 1.15 pounds.

Step 1: Understand the Distribution

The problem states that the changes in water weight are uniformly distributed between -2 and +2 pounds. A uniform distribution means every interval of the same length within the range is equally likely. Hence, we have a uniform distribution over the interval [-2, 2].

Step 2: Calculate the Range of the Distribution

The range of a uniform distribution [a, b] is calculated as b-a. For this problem, a = -2 and b = 2 , thus b-a = 2 - (-2) = 4.

Step 3: Determine the Variance

For a uniform distribution, the variance \( \sigma^2 \) is given by the formula\[ \sigma^2 = \frac{(b-a)^2}{12} \]. Substituting \( a = -2 \) and \( b = 2 \) into the formula gives:\[ \sigma^2 = \frac{(2 - (-2))^2}{12} = \frac{4^2}{12} = \frac{16}{12} = \frac{4}{3} \].

Step 4: Calculate the Standard Deviation

The standard deviation \( \sigma \) is the square root of the variance: \[ \sigma = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \]. For ease of interpretation, you can also write it as approximately\( 1.15 \), which is the approximate numerical value.

Key Concepts

Standard Deviation

Standard deviation is a key measure in statistics that helps us understand how spread out numbers in a data set are from the mean or average. In simple terms, it tells us how much variation or dispersion exists from the average value.
For the uniform distribution of changes in water weight, we've calculated the standard deviation to be approximately 1.15 pounds.
This means that, on average, the amount of water weight change will deviate about 1.15 pounds from the mean change over the course of a day.

• This tells us how consistent or variable the weight changes.
• If the standard deviation were smaller, weight change would be more consistent.
• A larger standard deviation would indicate more variability.

This concept is useful for assessing risk, consistency, or reliability in various data analyses.

Variance

Variance is closely tied to standard deviation, and it provides insight into the spread of a data set. Essentially, it is the average of the squared differences from the mean.
In the context of a uniform distribution, the variance helps us quantify how much the values differ from the average over an entire range.
In this exercise, we used the formula for variance in a uniform distribution: \[ \sigma^2 = \frac{(b-a)^2}{12} \] We found the variance to be \( \frac{4}{3} \), indicating the degree of spread within the possible range of weight changes (-2 to +2 pounds).

• Variance helps us understand the data's volatility.
• Higher variance indicates more data points are far from the mean.
• Lower variance shows a tighter clustering around the mean.

Understanding variance is particularly valuable in fields like finance, where it helps gauge investment risk.

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.
In this exercise, we consider a uniform probability distribution, where every outcome in the specified range is equally likely.
This means any weight change between -2 and +2 pounds has the same likelihood of occurring.

• Uniform distributions are simple and straightforward.
• They apply when each outcome is equally probable, like rolling a fair die.
• These distributions are defined by boundaries, such as \([-2, 2]\).

Recognizing the type of probability distribution involved is useful for determining the right statistical approach.

Continuous Random Variable

A continuous random variable is a type of variable that can take on an infinite number of possible values within a given range.
In this context, the change in water weight, ranging continuously from -2 to +2 pounds, is a perfect example.

• Continuous random variables can represent measurements like weight, height, or temperature.
• The probability of it taking on any exact value is technically zero.
• Instead, we consider the probability of the variable falling within a range.

Understanding continuous random variables is crucial for calculations that involve probability distributions, especially in real-world scenarios where exact values can't be pinpointed.