Measures of Dispersion MATH 102 Contemporary Math S. Rook

Overview Section 15.3 in the textbook: – Range & standard deviation

Range & Standard Deviation

Measures of Dispersion Last lesson we discussed describing data using measures of central tendency Sometimes the amount of spread is also useful to describe data especially when we desire consistency – e.g. See battery example in the textbook on page 740 We will discuss two measures of dispersion: – Range – Standard deviation

Range The range of a list of data is the difference between the maximum and minimum values – Is an effective way to describe data when the data values are close together e.g. What is the range of 1, 3, 4, 6, 7, 9? – Is an ineffective way to describe data when one or more data values are spread out from the rest e.g. What is the range of 1, 3, 4, 6, 7, 9, 100?

Standard Deviation The standard deviation of a list of data is the average difference each data item lies from the mean – Very useful to develop a measure of consistency e.g. If we know that the mean of a data set is 50 and its standard deviation is 4, we know that about 68% (over half) of the data values are between 46 and 54

Calculating Standard Deviation by Hand Feasible to calculate the standard deviation of a small list of data (e.g. 4, 6, 8, 10, 12) by hand: – Calculate the mean of the data – Find the difference between each data value and the mean This is the deviation from the mean – Square the deviations and then sum them all – Divide the sum by n – 1 (one less than the number of data items) for a sample or n for a population This is known as the [sample or population] variance – Take the square root of the quotient This is the [sample or population] standard deviation

Computing Standard Deviation by Calculator Compute the [population/sample] standard deviation using the same steps for the mean & median or five-number summary – Use the value next to: σ (sigma) for the population standard deviation s for the sample standard deviation – Make sure you know which standard deviation to use from the context of the problem! They are different!

Range & Standard Deviation (Example) Ex 1: Find the range, mean, and sample standard deviation of the data by using a calculator: a) 8, 4, 7, 6, 5, 5, 4, 9 b) 22, 18, 15, 21, 21, 15, 19, 13

Range & Standard Deviation (Example) Ex 2: The following table lists the state income tax for a person earning $50,000 per year for several states. Find the range, mean, and sample standard deviation: StateIncome Tax (%) AZ3.36 CO4.63 HI8.25 KS6.45 MA5.3 NY6.85 PA3.07 VA5.75

Range & Standard Deviation (Example) Ex 3: The following table gives the weight (in pounds) of the players on an athletic team. Find the range, mean, and population standard deviation: PlayerWeightPlayerWeight Ajavon160Maiga-Ba160 Byears206Shields155 Dixon148Snow158 Holmes155Thompson178 Johnson152Walker253 Kelly190White135 Lyttle175Williams184 Mabika165

Summary After studying these slides, you should know how to do the following: – Calculate the range and [population or sample] standard deviation of a small data set BY HAND – Calculate the range and [population or sample] standard deviation of a data set with a calculator Additional Practice: – See problems in Section 15.3 Next Lesson: – Normal Distribution (Section 15.4)