INTRO TO STATISTICS (QL)(SSS)
INTRO TO STATISTICS (QL)(SSS) STAT 1040
Utah State University
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This 14 page Class Notes was uploaded by Anita Hettinger on Wednesday October 28, 2015. The Class Notes belongs to STAT 1040 at Utah State University taught by Kady Schneiter in Fall. Since its upload, it has received 19 views. For similar materials see /class/230492/stat-1040-utah-state-university in Statistics at Utah State University.
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Date Created: 10/28/15
Ch 4 The Average and the Standard Deviation Histograms give a good summary of the data but frequently having to use a picture can be incon venient Sometimes we can summarize a data set even more strongly just by reporting numbers to indi cate the and of the data This is especially true if the histogram has only one peak and is at least roughly symmetric The Average The most commonly used measure of the center of a set of numbers is the also called The average of a list of numbers equals their sum divided by how many numbers there are in the list Q What is the average of the following list of numbers 3 6 9 7 Q The Health and Nutrition Examination Sur vey HANES surveyed a cross section of 20322 Americans and found that the men had an av erage height of 539 9quot and an average weight of 171 lbs The women had an average height of 539 35 and an average weight of 146 lbs By reducing data sets to their averages we can compare many groups simultaneously The 2 plots see Figure 3 page 59 from your textbook on additional handout show the height and weight data for both genders for six different age groups 31 Does the height plot mean people are shrinking Intuitively what percentage of people do you think has a weight below average And what percentage do you think has a weight above average The histogram see Figure 4 page 62 from your textbook on additional handout shows a histogram for the weights of the women in the HAN ES sam ple In the histogram of women39s weights only about of the area is to the right of the average rather than the we might expect To see why consider the averages of the follow ing sets of numbers 0 1 2 2 3 average values below average 7 values equal to average 7 values above average 7 o 1 2 2 5 average values below average 7 values equal to average 7 values above average 7 o 1 2 2 7 average values below average 7 values equal to average 7 values above average 7 A histogram balances when supported at the av erage 34 The Median The is the value with 50 of the area above and 50 below How to find the median of a data set 1 Sort the values from smallest to largest 2 If you have an odd number of values the me dian is the center one If you have an even number of values the me dian is the average of the two center values List 1 35180 Sorted list Median List 2 2 1 5 1 Sorted list Center values Median If the histogram of your data is symmetric the median and the average will be close If the histogram has a long right tail the average will be greater than the median If the histogram has a long left tail the average will be less than the median Q What do you think is larger the average or the median body weight of the women in the HANES sample Look at Figure 4 page 62 from your textbook on additional handout If your histogram has an extremely long tail the mean will be strongly influenced by the few cases in the tail and the median will better indicate the center of your data Q Why might we prefer the median to the aver age as a measure of the center of a list of incomes of employees of Microsoft The Standard Deviation The SD is a mea sure of how spread out data values are around the average Most numbers from a data set will be within of their average Few will be more than away from their av erage Almost none will be more than away from their average More precisely 0 about of values will be within 1 SD of the average 0 about of values will be within 2 SD39s of the average 0 about of values will be within 3 SD39s of the average How to calculate the SD 1 Find the average M Find the deviations from the average entry average 0 Square the deviations from the average 4s Average the squared deviations from the av erage U1 Take the square root SD iaverage of deviations from avg2 Q Find the SD of the list 5 12 15 20 In practice we usually use computers or calcula tors with statistical functions to calculate the SD But don39t if you39re asked to show your workquot Be aware there39s another number slightly larger than the SD also sometimes referred to as the standard deviationquot we39ll call it SD Most calculators use the symbol a for the SD and s for the SD Read Section 47 in the textbook to learn how to find out whether your calculator calculates SD or SD Note in calculating the SD we are using the root mean square operation rmslist yaverage of entries2
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