COMPUTATIONAL FORMULAS AND IQR’S Boys Girls 72 61 74 64 70 65 68 66 69 68 70 67 68 65 69 63 67 62 70 65 71 66 67 64 66 Compare the following heights in inches: Objective •Calculate standard deviation, variance and IQR. Relevance •To be able to analyze a set of data and see how far each value is above or below the mean of the set. Computational Formula • This is an easier computational formula that contains no rounding. x 2 s x 2 2 n 1 n Example – Find Var and St Dev 𝒙𝟐 𝒙 8 49 64 22 484 34 1156 42 1764 7 x2 3517 x 113 s2 2 x x2 n 1 n 2 113 3517 5 4 240.8 s 240.8 15.5 You Try One! 𝒙 𝒙𝟐 4 16 64 144 324 484 1156 1444 8 12 18 22 34 38 x 136 s2 2 x x2 n 1 n x 3632 2 2 136 3632 7 6 164.95 s 164.95 12.8 Is the standard deviation affected by outliers? 5 5 7 7 5 5 8 8 9 29 The standard deviation IS sensitive to extreme values. 67 65 68 70 74 60 61 64 67 68 The following data represents the heights of some students in my class. a. Find the mean & standard deviation. x 66.4 s 4.14 b. What % are within 1 standard deviation of the mean? () x 1s : 66.4 4.14 62.26 66.4 4.14 70.54 7 70% 10 c. What % are within 2 standard deviations f the mean? () 10 x 2s : 58.12 74.68 10 100% IQR – Interquartile Range • This is the measure of variability that is not affected by outliers. • It is based on quartiles. 25% 25% 25% Q1 Q1 Q2 Q3 Q2 25% Q3 - This is the median of the lower ½ of the sample. - This is the median. - This is the median of the upper ½ of the sample. IQR Q3 Q1 (18 24 25 27 ) 27 ( 29 30 33 34 ) Q1 = 24.5 Q3 = 31.5 IQR = 31.5 – 24.5 = 7 Find IQR 12 13 16 18 22 24 27 40 Q1 = 14.5 Q3 = 25.5 IQR = 25.5 – 14.5 = 11 Find IQR 8 10 11 14 16 20 22 26 28 32 IQR = 15 Find IQR 4 5 6 IQR = 15 6 7 10 22 If normal distribution – then you can estimate the standard deviation with the IQR. Standard Deviation = 𝑰𝑸𝑹 𝟏.𝟑𝟓 *Can also be use to check to see if a distribution is normal – if much larger then it’s skewed. Which Measure Should You Report? Shape Central Tendency Dispersion (Spread) Symmetric Mean Skewed Left or Right Median Standard Deviation IQR NOTE: When told to “describe the distribution” you will give the shape, the center and the spread of the distribution using the above categories as a guideline. Homework • Worksheet • Use the new formula to compute the standard deviation. • Now – time for the lab!