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Frequency Distribution, Chebyshev's Theorem, and Statistics

1. Given the following frequency distribution, find the mean, variance, and standard deviation.

2. Please show all of your work when answering the following:

Use Chebyshev's theorem to find what percent of the values will fall between 222 and 286 for a data set with a mean of 254 and standard deviation of 16.

Use the Empirical Rule to find what two values 95% of the data will fall between for a data set with a mean of 216 and standard deviation of 12.


Solution Preview

1. To determine the mean from the frequency table, you must first find the mid-point of each category. For example, the mid-point of the first category (61-63) is 62. You then multiply each mid-point by the frequency for that category. Sticking with the (61-63) category, you would multiply 62 by 19. Add all of the products together and divide by the sum of all frequency values.

Mean = 4, 221/63

= 67

To determine variance, we will use the following formula:

variance = sum of (x - mean)^2*f/n

where x is a mid-point value of a category, where f is the frequency of that category and n is the total frequency.

variance = [(62 - 67)^2*19] + [(65 - 67)^2]*17] + [(68 - 67)^2]*5] + [(71 - 67)^2]*10] + [(74 - 67)^2]*12]/63

= 1,296/63

= 20.57

Standard deviation is simply the square root of the variance, so ...

Solution Summary

Seven complete solutions for statistics problems related to descriptive statistics (mean, median, mode, range, variance, standard deviation, percentile and more). Includes a step by step solution for a problem related to Chebyshev's theorem.