Question

A set of data has a mean of 75 and a standard deviation of \(5 .\) You know nothing else about the size of the data set or the shape of the data distribution. a. What can you say about the proportion of measurements that fall between 60 and \(90 ?\) b. What can you say about the proportion of measurements that fall between 65 and \(85 ?\) c. What can you say about the proportion of measurements that are less than \(65 ?\)

Short answer

Question: Based on the given data set with a mean of 75 and a standard deviation of 5, determine the minimum proportion of measurements within the following ranges: a) between 60 and 90, b) between 65 and 85, and c) less than 65. Answer: a) At least 89% of the measurements fall between 60 and 90. b) At least 75% of the measurements fall between 65 and 85. c) At most 25% of the measurements are less than 65.

Step-by-step solution

a. Proportion of measurements between 60 and 90

Using Chebyshev's inequality, we can find the minimum proportion of measurements between 60 and 90. First, calculate how many standard deviations each boundary is from the mean: Lower boundary: (60 - 75) / 5 = -3 standard deviations Upper boundary: (90 - 75) / 5 = 3 standard deviations Now, apply Chebyshev's inequality formula: \(1 - 1/k^2\), where k = number of standard deviations: Minimum proportion: \(1 - 1/3^2 = 1 - 1/9 = 8/9 = 0.89 \) (approximately) So, at least 89% of the measurements fall between 60 and 90.

b. Proportion of measurements between 65 and 85

Using the same method as in part a, we'll calculate the number of standard deviations each boundary is from the mean and then apply Chebyshev's inequality: Lower boundary: (65 - 75) / 5 = -2 standard deviations Upper boundary: (85 - 75) / 5 = 2 standard deviations Minimum proportion: \(1 - 1/2^2 = 1 - 1/4 = 3/4 = 0.75\) So, at least 75% of the measurements fall between 65 and 85.

c. Proportion of measurements less than 65

We can find the proportion of measurements less than 65 by calculating the complement of the proportion greater than or equal to 65 (with reference to the mean value of 75). The lower boundary is the same as in part b (65 - 75) / 5 = -2 standard deviations. Apply Chebyshev's inequality formula to find the minimum proportion of measurements greater than or equal to 65: Minimum proportion: \(1 - 1/2^2 = 1 - 1/4 = 3/4 = 0.75\) Since the proportion of measurements greater than or equal to 65 is at least 75%, the proportion of the measurements less than 65 is at most 25% (100% - 75%).

Key concepts

Proportion of Measurements

Chebyshev's Inequality helps us understand the proportion of measurements within a specified range in a dataset, irrespective of the distribution's shape. This can be especially useful when the dataset does not follow a normal distribution. By using Chebyshev's formula: \(1 - 1/k^2\), we can calculate the minimum proportion of data within \(k\) standard deviations from the mean.
For any arbitrary dataset, we utilize this to estimate proportions:

• If measurements fall between 60 and 90, they are within 3 standard deviations from the mean. Thus, the proportion is \(1 - 1/3^2 = 8/9\), meaning at least 89% of the data.
• If measurements are between 65 and 85, within 2 standard deviations, the proportion is \(3/4\), or 75% of the data.
• For values less than 65, being more than 2 standard deviations below the mean, Chebyshev's Inequality indicates that no more than 25% of the data would be less than 65.

This ensures a reliable minimum proportion of data coverage for each specified range.

Standard Deviation

The standard deviation is a critical statistical measure that indicates how spread out the numbers in a data set are. When analyzing data, it provides insight into the variability relative to the mean. In the context of Chebyshev's Inequality, standard deviation tells us how much the individual data points deviate from the mean.
Calculating the number of standard deviations each boundary falls from the mean, we can determine how spread the measurements are. A lower standard deviation implies that data points are closer to the mean, which can affect the results of the inequality by shortening the range of coverage.

• For example, with a mean value of 75 and a standard deviation of 5, if a measurement falls between 65 and 85, it spans 2 standard deviations from the mean.
• It is important to remember that standard deviation doesn't inform us about the exact shape of the distribution but serves as an indicator of dispersion or spread.

Thus, standard deviation underpins the application of inequalities like Chebyshev's to predict proportions within a certain range around the mean.

Mean Value

The mean value is essentially the average of all data points in a dataset. It acts as a central point from which the standard deviations are calculated when applying Chebyshev's Inequality. Understanding the mean is crucial because it centers the data distribution, enabling us to assess proportions in relation to it.
A mean value of 75 means the dataset's central tendency, or balance point, is at this number. By knowing this, we can determine:

• The boundaries of the range (like 60 to 90) and how far they are from the average using standard deviation.
• The proportion of data within these boundaries with the help of Chebyshev's Inequality.

The mean offers a reference point that standard deviation uses to assess how spread out or clustered around the mean data points are. It is vital not just for calculating ranges but also for grasping the general tendency of the dataset.