Question

You can use the Empirical Rule to see why the distribution of survival times could not be mound shaped. a. Find the value of \(x\) that is exactly one standard deviation below the mean. b. If the distribution is in fact mound-shaped. approximately what percentage of the measurements should be less than the value of \(x\) found in part a? c. Since the variable being measured is time, is it possible to find any measurements that are more than one standard deviation below the mean? d. Use your answers to parts \(b\) and \(c\) to explain why the data distribution cannot be mound-shaped.

Short answer

Explain why or why not. Answer: No, the survival time distribution cannot be mound-shaped if the mean is very close to 0. This is because, due to the nature of time as a non-negative variable, some measurements will not be allowed to be more than one standard deviation below the mean, which would result in a negative time value. This violates the condition of the Empirical Rule for a normally distributed mound-shaped data distribution, where approximately 34% of the measurements should be below the value of x, which is one standard deviation below the mean.

Step-by-step solution

Understanding the Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 Rule, is a statistical rule that states that for a normally distributed dataset, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. In this exercise, we will focus on the application of the 68% rule.

Finding the value of \(x\) one standard deviation below the mean

Let \(\mu\) be the mean of the distribution and \(\sigma\) be the standard deviation. To find the value of \(x\) that is exactly one standard deviation below the mean, we need to subtract the standard deviation \(\sigma\) from the mean \(\mu\). \(x = \mu - \sigma\)

Percentage of measurements less than \(x\)

If the distribution is mound-shaped and follows the Empirical Rule, approximately 68% of the data should lie within one standard deviation of the mean. Since we are looking for the value one standard deviation below the mean, this represents half of this range. Thus, approximately 34% of the measurements should be less than the value of \(x\) found in part a.

Measurements more than one standard deviation below the mean

Since the variable being measured is time, it can only have non-negative values. Thus, if the mean survival time is very close to 0, it might not be possible for some values to be more than one standard deviation below the mean because it would result in a negative time which is not possible.

Explain why the data distribution cannot be mound-shaped

Based on the answers to parts b and c, we can conclude that the data distribution cannot be mound-shaped because the percentage of measurements below the value of x found in part a cannot be estimated to be approximately 34% (as per the empirical rule) if some measurements are not allowed to be more than one standard deviation below the mean (due to the nature of time as a non-negative variable). This violates the condition of the Empirical Rule for a normally distributed mound-shaped data distribution.

Key concepts

Standard Deviation

The standard deviation is a fundamental concept in statistics that measures the spread or dispersion of a set of data points in relation to its mean. Think of it as a way to quantify how much variation exists from the average. If you have a small standard deviation, this indicates that the data points tend to be close to the mean. Conversely, a large standard deviation means that the data points are spread out over a wider range. Mathematically, the standard deviation \( \sigma \) is calculated as the square root of the average of the squared differences from the mean. This formula might sound complex, but it can be broken down into simpler steps:

• Find the mean (average) of your data set.
• Subtract the mean from each data point and square the result.
• Calculate the average of these squared differences.
• Finally, take the square root of this average to find the standard deviation.

Standard deviation helps in understanding how much your data points deviate from the mean, aiding in analyzing the distribution of data.

Mound-shaped Distribution

A mound-shaped distribution, often referred to as a normal distribution, has a specific characteristic shape. It resembles a symmetrical bell curve when graphed. This shape is prevalent in natural phenomena and is particularly significant in statistics due to its properties. Key features of a mound-shaped distribution include:

• Most data points cluster around a central mean.
• The frequencies of data points decrease as you move away from the mean on either side.
• Approximately 68% of data points fall within one standard deviation of the mean, a key element of the Empirical Rule.

Understanding whether data is mound-shaped can help in predicting patterns and making inferences about a population based on sample data. However, not all data will form a perfect mound shape, especially if there are restrictions like non-negative variables that can’t go below zero, which could distort the expected distribution as in the exercise.

Mean in Statistics

The mean, or average, is one of the most widely used measures of central tendency in statistics. It is calculated by summing all the values in a data set and then dividing by the number of values. The mean gives us a single number representing the center of data, offering a quick snapshot of the entire set. To calculate the mean, you can follow these steps:

• Add up all the data points in the set.
• Divide this total by the number of data points.

Despite being easy to compute, the mean is sensitive to extreme values, called outliers, which can skew the results. In a mound-shaped distribution, the mean typically aligns with the peak of the mound, helping to signify data symmetry. In contexts where data includes only positive values—like the time measurement focused on in the exercise—understanding the mean’s placement relative to zero becomes crucial, particularly when assessing deviation patterns.

Non-negative Variables

Non-negative variables are data values that cannot be less than zero. This constraint is essential for certain types of data, such as measurements of time, length, or countable objects, which never occur in negative quantities. When dealing with non-negative variables, it is crucial to consider the impact on statistical measures and distributions. For example:

• The mean will never be zero or negative, as it is an average of all non-negative values.
• It forces the left tail of a distribution to stop at zero, often creating a skew rather than a symmetric, mound-shaped distribution.
• The standard deviation may also be limited if values don't extend both above and below the mean.

In the exercise scenario, time measured in a non-negative manner further validates why results can't be more than one standard deviation below the mean. Empirical analysis with non-negative variables often requires careful interpretation to avoid incorrect assumptions, as negative deviations from a mean imply impossible results.