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. Teacher Introductory Statistics Lesson 2.4 C Objective: SSBAT interpret standard deviation and use the Empirical Rule. Standards: S2.5B Review Review Variation of Data Describes how spread out or scattered the data is Deviation of an Entry in a population Measures of Variation The Difference between an entry in the data set (x) and the mean of the data set Deviation of x = x – ࣆ 1. Range 2. Variance 3. Standard Deviation Salary (thousands $) x Deviation x – 41.5 41 -0.5 38 -3.5 39 -2.5 45 3.5 47 5.5 41 -0.5 44 2.5 41 -0.5 37 -4.5 42 0.5 Standard Deviation: ߪ = 2.97 Deviation is the distance each entry is away from the Mean. We could not average the deviations together because the sum will always be zero. Therefore, we squared each deviation and then found the mean of these values – This was referred to as the Variance Then, we square rooted the variance to come up with the Standard Deviation – This gave us the typical amount an entry deviates from the mean. 1 . Teacher Standard Deviation or s Bell Shape Curve (Normal Distribution) A measure of the Typical amount an entry deviates from the Mean The distribution is symmetrical (The typical difference an entry is away from the mean) The more the entries are spread out, the greater the standard deviation. When all the data entries are the Same, the standard deviation is 0 http://www.tangopadawan.com/2008/05 Empirical Rule Empirical Rule For data with a bell-shape (symmetrical) distribution, the Standard Deviation has the following characteristics: About 68% of the data lie within ONE standard deviation of the mean (1 on each side of mean) About 95% of the data lie within TWO standard deviations of the mean (2 on each side of mean) About 99.7% of the data lie within THREE standard deviations of the mean (3 on each side of mean) Examples Using Empirical Rule 1. The mean rate for satellite television from a sample of households was $49.00 per month, with a standard deviation of $2.50 per month. Between what 2 values does 99.7% of the data lie? 99.7% of the data lies within 3 standard deviations of the mean Therefore, Add 2.50 three times to the mean and then Subtract it three times from the mean to get the two endpoints http://sites.stat.psu.edu/~ajw13/stat200_notes/01_turning/graphics/emp_rule.gif 1. Continued The mean rate for satellite television from a sample of households was $49.00 per month, with a standard deviation of $2.50 per month. Between what 2 values does 99.7% of the data lie? 49 + 2.5 + 2.5 + 2.5 = 56.5 49 – 2.5 – 2.5 – 2.5 = 41.5 Therefore 99.7% of the data lies between $41.50 and $56.50. 2 . Teacher 2. SAT verbal scores in a particular year were normally distributed (bell shaped) with a Mean of 489 and Standard Deviation of 93. Between what two values does 68% of the data lie? 68% of the data lies within 1 standard deviation of the mean Therefore, Add 93 one time to the mean and then Subtract it one time from the mean to get the two endpoints 2. Continued SAT verbal scores in a particular year were normally distributed (bell shaped) with a Mean of 489 and Standard Deviation of 93. Between what two values does 68% of the data lie? 489 + 93 = 582 489 – 93 = 396 Therefore, 68% of the data lies between 396 and 582. Homework: Worksheet 2.4 C 3

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