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. 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 – 16Find the range. = 10 Q 3 – Q 1 = 24.5 – 16.5Find the interquartile range. = 8 The range is 10 years. The interquartile range is 8 years. medianFind the median. 16 16 17 19 20 22 24 25 26 Q 1 = = 16.5 Q 3 = = 24.5 Find Q 1 and Q 3. (16 + 17) 2 (24 + 25) 2 Example

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

6. 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. 234.04 5 = 6.8 = = 91 Find the mean. (78.2 + 90.5 + 98.1 +93.7 +94.5) 5 x  = Find the standard deviation.  (x – x) 2 n Organize the next steps in a table. 78.291–12.8163.84 90.591–0.5.25 98.1917.150.41 93.7912.77.29 94.5913.512.25 x x x – x (x – x) 2

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

8. 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. = = 6 Find the mean. (9+4+5+6) 4 x  = Find the standard deviation.  (x – x) 2 n Organize the next steps in a table. 9639 46-24 56-11 6600 sum14 x x x – x (x – x) 2

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

10. A set of values has a mean of 22 and a standard deviation of 3. Find the z-score for a value of 24. Z-Score = 0.6 = Simplify. 2323 =Substitute. 24 – 22 3 z-score = value – mean standard deviation

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

12. 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. SMTWThFS 535247475039 33404144474943 39474954534636 33454542433933 3340404142 Step 2: Use the CALC menu of STAT to access the 1-Var Stats option. The mean is about 36.1 MWh; The mean is x. the standard deviation is about 3.6 MWh. The standard deviation is  x.