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12.4 – Standard Deviation

12.4 – Standard Deviation. Measures of Variation. The range of a set of data is the difference between the greatest and least values. The interquartile range is the difference between the third and first quartiles. median Find the median.

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12.4 – Standard Deviation

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  1. 12.4 – Standard Deviation

  2. Measures of Variation • The range of a set of data is the difference between the greatest and least values. • The interquartile range is the difference between the third and first quartiles

  3. medianFind the median. 16 16 17 192022 24 25 26 (16 + 17) 2 (24 + 25) 2 Q1 = = 16.5Q3 = = 24.5Find Q1 and Q3. Example There are 9 members of the Community Youth Leadership Board. Find the range and interquartile range of their ages: 22, 16, 24, 17, 16, 25, 20, 19, 26. greatest value – least value = 26 – 16 Find the range. = 10 Q3 – Q1 = 24.5 – 16.5 Find the interquartile range. = 8 The range is 10 years. The interquartile range is 8 years.

  4. More Measures of Variation • Standard deviation is a measure of how each value in a data set varies or deviates from the mean

  5. Steps to Finding Standard Deviation 1. Find the mean of the set of data: 2. Find the difference between each value and the mean: 3. Square the difference: 4. Find the average (mean) of these squares: 5. Take the square root to find the standard deviation

  6. x = = 91   Find the mean. (78.2 + 90.5 + 98.1 +93.7 +94.5) 5 x x x – x (x – x)2 Organize the next steps in a table. 78.2 91 –12.8 163.84 90.5 91 –0.5 .25 98.1 91 7.1 50.41 93.7 91 2.7 7.29 94.5 91 3.5 12.25  = Find the standard deviation. (x – x)2 n 234.04 5 = 6.8 Standard Deviation Find the mean and the standard deviation for the values 78.2, 90.5, 98.1, 93.7, 94.5. The mean is 91, and the standard deviation is about 6.8.

  7. x Find the mean. x x x – x (x – x)2 Organize the next steps in a table.  = Find the standard deviation. (x – x)2 n Let’s Try One – No Calculator! Find the mean and the standard deviation for the values 9, 4, 5, 6

  8. x = = 6   Find the mean. (9+4+5+6) 4 x x x – x (x – x)2 Organize the next steps in a table. 9 6 3 9 4 6 -2 4 5 6 -1 1 6 6 0 0 sum 14  = Find the standard deviation. (x – x)2 n Let’s Try One – No Calculator Find the mean and the standard deviation for the values 9, 4, 5, 6 The mean is 6, and the standard deviation is about 1.87.

  9. More Measures of Variation • Z-Score: The Z-Score is the number of standard deviations that a value is from the mean.

  10. value – mean standard deviation z-score = 24 – 22 3 = Substitute. 2 3 = Simplify. = 0.6 Z-Score A set of values has a mean of 22 and a standard deviation of 3. Find the z-score for a value of 24.

  11. value – mean standard deviation z-score = 26 – 34 4 = Substitute. -8 4 = Simplify. = -2 Z-Score A set of values has a mean of 34 and a standard deviation of 4. Find the z-score for a value of 26.

  12. Step 2: Use the CALC menu of STAT to access the 1-Var Stats option. S M T W Th F S 53 52 47 47 50 39 33 40 41 44 47 49 43 39 47 49 54 53 46 36 33 45 45 42 43 39 33 33 40 40 41 42 The mean is x. The mean is about 36.1 MWh; the standard deviation is about 3.6 MWh. The standard deviation is x. Standard Deviation Use the data to find the mean and standard deviation for daily energy demands on the weekends only. Step 1: Use the STAT feature to enter the data as L1.

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