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P5 ungrouped data

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Statistics
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• 1. Statistics (P5)
• 2.  Data that has not been organized into groups. Ungrouped data looks like a big list of numbers.
• 3.  These statistics describe the “middle region” of the sample. Mean▪ The arithmetic average of the data set. Median▪ The “middle” of the data set. Mode▪ The value in the data set that occurs most frequently. These are almost never the same, unless you have a perfectly symmetricdistribution. Mode MedianMeanMode = Median = Mean
• 4.  The Median is the middle observation of an ordered (fromlow to high) data set Examples: 1, 2, 4, 5, 5, 6, 8▪ Here, the middle observation is 5, so the median is 5 1, 3, 4, 4, 5, 7, 8, 8▪ Here, there is no “middle” observation so we take theaverage of the two observations at the center5.4254Median
• 5.  The Mode is the value of the data set that occurs most frequently Example: 1, 2, 4, 5, 5, 6, 8▪ Here the Mode is 5, since 5 occurred twice and no other valueoccurred more than once Data sets can have more than one mode, while the mean andmedian have one unique value Data sets can also have NO mode, for example: 1, 3, 5, 6, 7, 8, 9▪ Here, no value occurs more frequently than any other,therefore no mode exists▪ You could also argue that this data set contains 7 modes sinceeach value occurs as frequently as every other
• 6.  The mean, ( is calculated by taking the sumof the observed values (yi) divided by thenumber of observations (n).nyyynynnii211y
• 7.  Determine the mean for the set: {2, 3, 7, 5, 5,13, 1, 7, 4, 8, 3, 4, 3} y1 = 2, y2 = 3, y3 = 7, y4 = 5, …, y13 = 3 n = 13 Mean =
• 8.  The Mean, Median and Mode by themselves are notsufficient descriptors of a data set Example: Data Set 1: 48, 49, 50, 51, 52 Data Set 2: 5, 15, 50, 80, 100 Note that the Mean and Median for both data sets areidentical, but the data sets are glaringly different! The difference is in the dispersion of the data points Dispersion Statistics we will discuss are: Range Variance Standard Deviation
• 9.  The Range is simply the difference between thesmallest and largest observation in a data set Example Data Set 1: 48, 49, 50, 51, 52 Data Set 2: 5, 15, 50, 80, 100 The Range of data set 1 is 52 - 48 = 4 The Range of data set 2 is 100 - 5 = 95 So, while both data sets have the same meanand median, the dispersion of the data, asdepicted by the range, is much smaller in dataset 1
• 10.  It gives the amount of dispersion or scatter in the data fromthe central tendency (mean).Nyi2)(
• 11.  Determine the standard deviation of the set:{5, 6, 8, 4, 10, 3} Mean = y = (y - )2 = (5 - )2 + (6 - )2 + (8 - )2 + (4 - )2 +(10 - )2 + (3 - )2 (y - )2 = 12 + 02 + 22 + (-2) 2 + 42 + (-3) 2 (y - )2 = 1 + 0 + 4 + 4 + 16 + 9 = 34 Standard deviation = σ = √( (y - )2 / n) σ = √ 34 / 6 = √ 5.666 = 2.380
• 12.  TheVariance, σ 2, represents the amount of variability ofthe data relative to their meanNyi22)(
• 13.  Determine the variance of the set of the set:{5, 6, 8, 4, 10, 3} As previously shown: Mean = (y - )2 = 34 n = 6 Variance = σ2 = (y - )2 / n = 34/6 = 5.666