# 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.

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#### 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.