3.2 Measures of Spread In some data sets the observations are close together, while in others they are more spread out. In addition to measures of the center, it's often important to measure the spread of the data. One measurement of variation that is easy to calculate is the range. Range = (Largest value) - (Smallest value) What are some disadvantages about this measurement? We would like to find the spread/variation of all data, not only between the minimum and maximum value. The measurements Variance and Standard Deviation find the deviation of all of the data values from the mean. Ex. Given the following data: 13, 14, 24, 24, 25, 26 Find the mean of the data: Value of x 13 14 24 24 25 26 Total Deviation from mean x-µ The Population Variance is given by The Population Standard Deviation is given by σ 2 = σ = The Sample Variance is given The Sample Standard Deviation is given by s 2 = s = Note: Do NOT use the other formulas for the variance and standard deviation that are also in the book.
- The standard deviation is the measure of variation of all values from the mean. - The variance is the measure of variation equal to the square of the standard deviation. - The value of the standard deviation is usually positive, and in rare cases it could also equal to zero. Describe how the data set would look if the standard deviation is zero: - The unit of the standard deviation is the same unit as the units of the original data. - It is unusual that data fall more than... standard deviations from the mean. Is the standard deviation resistant? Explain. - A general round-off rule for variation: Carry one more decimal place than is present in the original set of data. Round only the final answer, and not values in the middle of a calculation. How to find Mean, Median, Mode, and Standard Deviation using our calculators:
Variance and Standard Deviation for Grouped Data For each class, assume all data values are equal to the class midpoint. The following data give the frequency distribution of the test scores of all the students in a class. Find the standard deviation of these test scores. Unless it is clear that a data set is from a population (it will usually use the word ALL) we will assume it is a sample. Test Scores Frequency 90-100 7 80-89 10 70-79 12 60-69 5 50-59 2
Use of Standard Deviation Empirical Rule For a bell-shaped distribution -about 68% of all values fall within 1 standard deviation of the mean -about 95% of all values fall within 2 standard deviations of the mean - almost all values (about 99.7%) fall within 3 standard deviations of the mean (a) The prices of all college textbooks follow a bell-shaped distribution with a mean of $105 and a standard deviation of $20. Using the empirical rule, find the interval that contains the prices of about 99.7% of college textbooks. (b) Using the empirical rule, find the percentage of all college textbooks with their prices between $85 and $125. (c) Using the empirical rule, find the percentage of all college textbooks with their prices between $65 and $145.
Chebyshev s Theorem 1 At least 1 k 2 of the data values lie within k standard deviations of the mean, for any k>1. Note that the distribution does NOT have to be bell-shaped in order to use this theorem. Use above formula for k=2, and interpret the result. Use above formula for k=3, and interpret the result. Use above formula for k=1.5, and interpret the result. Suppose the average credit card debt for households is $9,500 with a standard deviation of $2,600. (a) Using Chebyshevs theorem, find at least what percentage of current credit card debts for all households are between $3,000 and $16,000. (b) Using Chebyshevs theorem, find the interval that contains credit card debts of at least 89% of all the households. Range Rule of Thumb Most values fall within 2 standard deviations of the mean. Values that fall outside of this interval would be considered unusual. From this we could also say that range SD (unless we have extreme outliers) 4 The Wechsler Adult Intelligence Scale involves an IQ test designed so that the mean score is 100 and the standard deviation is 15. Use the range rule of thumb to find the minimum and maximum "usual" IQ scores. Then determine whether an IQ score of 135 would be considered "unusual."