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Lesson 1 - 2 Describing Distributions with Numbers adapted from Mr. Molesky’s Statmonkey website Measures of Spread Variability is the key to Statistics. Without variability, there would be no need for the subject. When describing data, never rely on center alone. Measures of Spread: Range - {rarely used ... why?} Quartiles - InterQuartile Range {IQR=Q3-Q1} Variance and Standard Deviation {var and sx} Like Measures of Center, you must choose the most appropriate measure of spread. Standard Deviation Another common measure of spread is the Standard Deviation: a measure of the “average” deviation of all observations from the mean. To calculate Standard Deviation: Calculate the mean. Determine each observation’s deviation (x - xbar). “Average” the squared-deviations by dividing the total squared deviation by (n-1). This quantity is the Variance. Square root the result to determine the Standard Deviation. Standard Deviation Properties s measures spread about the mean and should be used only when the mean is used as the measure of center s = 0 only when there is no spread/variability. This happens only when all observations have the same value. Otherwise, s > 0. As the observations become more spread out about their mean, s gets larger s, like the mean x-bar, is not resistant. A few outliers can make s very large Standard Deviation Variance: (x1 x ) 2 (x2 x ) 2 ... (xn x ) 2 var n 1 Standard Deviation: sx 2 (x x ) i n 1 Example 1.16 (p.85): Metabolic Rates 1792 1666 1362 1614 1460 1867 1439 Standard Deviation 1792 1666 1362 1614 1460 1867 1439 Metabolic Rates: mean=1600 x (x - x) (x - x)2 1792 192 36864 1666 66 4356 1362 -238 56644 1614 14 196 1460 -140 19600 1867 267 71289 1439 -161 25921 Totals: 0 214870 Total Squared Deviation 214870 Variance var=214870/6 var=35811.66 Standard Deviation s=√35811.66 s=189.24 cal What does this value, s, mean? Example 1 Which of the following measures of spread are resistant? 1. Range Not Resistant 2. Variance Not Resistant 3. Standard Deviation Not Resistant Example 2 Given the following set of data: 70, 28, 56, 63, 56, 35, 51, 50, 48, 58, 46, 46, 48, 62, 39, 69, 53, 45, 56, 53, 52, 60, 32, 70, 66, 38, 44, 33, 48, 73, 60, 54, 36, 45, 51, 55, 49, 51, 44, 52 What is the range? 73-28 = 45 What is the variance? 117.958 What is the standard deviation? 10.861 Quartiles Quartiles Q1 and Q3 represent the 25th and 75th percentiles. To find them, order data from min to max. Determine the median - average if necessary. The first quartile is the middle of the ‘bottom half’. The third quartile is the middle of the ‘top half’. 19 22 23 23 23 68 74 Q1 26 27 28 med Q1=23 45 26 75 76 29 30 31 32 Q3=29.5 82 med=79 82 91 Q3 93 98 Using the TI-83 • Enter the test data into List, L1 – STAT, EDIT enter data into L1 • Calculate 5 Number Summary – Hit STAT go over to CALC and select 1-Var Stats and hitt 2nd 1 (L1) • Use 2nd Y= (STAT PLOT) to graph the box plot – – – – – Turn plot1 ON Select BOX PLOT (4th option, first in second row) Xlist: L1 Freq: 1 Hit ZOOM 9:ZoomStat to graph the box plot • Copy graph with appropriate labels and titles 5-Number Summary, Boxplots The 5 Number Summary provides a reasonably complete description of the center and spread of distribution MIN Q1 MED Q3 MAX We can visualize the 5 Number Summary with a boxplot. min=45 45 50 Q1=74 55 Outlier? 60 med=79 65 70 75 Q3=91 80 Quiz Scores 85 max=98 90 95 100 Determining Outliers “1.5 • IQR Rule” InterQuartile Range “IQR”: Distance between Q1 and Q3. Resistant measure of spread...only measures middle 50% of data. IQR = Q3 - Q1 {width of the “box” in a boxplot} 1.5 IQR Rule: If an observation falls more than 1.5 IQRs above Q3 or below Q1, it is an outlier. Why 1.5? According to John Tukey, 1 IQR seemed like too little and 2 IQRs seemed like too much... Outliers: 1.5 • IQR Rule To determine outliers: 1. Find 5 Number Summary 2. Determine IQR 3. Multiply 1.5xIQR 4. Set up “fences” A. B. 5. Lower Fence: Q1-(1.5∙IQR) Upper Fence: Q3+(1.5∙IQR) Observations “outside” the fences are outliers. Outlier Example All data on pg 48, #1.6 IQR=45.72-19.06 IQR=26.66 fence: 19.06-39.99 = -20.93 1.5IQR=1.5(26.66) 1.5IQR=39.99 fence: 45.72+39.99 = 85.71 { } 0 10 20 30 40 50 60 70 Spending ($) 80 outliers 90 100 Example 4 Consumer Reports did a study of ice cream bars (sigh, only vanilla flavored) in their August 1989 issue. Twenty-seven bars having a taste-test rating of at least “fair” were listed, and calories per bar was included. Calories vary quite a bit partly because bars are not of uniform size. Just how many calories should an ice cream bar contain? 342 377 319 353 295 234 294 286 377 182 310 439 111 201 182 197 209 147 190 151 131 151 Construct a boxplot for the data above. Example 4 - Answer Q1 = 182 Min = 111 IQR = 137 Q2 = 221.5 Max = 439 UF = 524.5 Q3 = 319 Range = 328 LF = -23.5 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 Calories Example 5 The weights of 20 randomly selected juniors at MSHS are recorded below: 121 126 130 132 143 137 141 144 148 205 125 128 131 133 135 139 141 147 153 213 a) Construct a boxplot of the data b) Determine if there are any mild or extreme outliers c) Comment on the distribution Example 5 - Answer Q1 = 130.5 Min = 121 IQR = 15 Mean = 143.6 StDev = 23.91 Q2 = 138 Max = 213 UF = 168 Q3 = 145.5 Range = 92 LF = 108 Extreme Outliers ( > 3 IQR from Q3) * 100 110 120 130 140 150 160 170 180 190 200 * 210 220 Weight (lbs) Shape: somewhat symmetric Center: Median = 138 Outliers: 2 extreme outliers Spread: IQR = 15 Linear Transformations Variables can be measured in different units (feet vs meters, pounds vs kilograms, etc) When converting units, the measures of center and spread will change Linear Transformations (xnew = a+bx) do not change the overall shape of a distribution Multiplying each observation by b multiplies both the measure of center and spread by b Adding a to each observation adds a to the measure of center, but does not affect spread If the distribution was symmetric, its transformation is symmetric. If the distribution was skewed, its transformation maintains the same skewness Transformation Example 6 • Using the data from example #5 – a) Change the weight from pounds to kilograms and add 2 kg (for a special band uniform) – b) Get summary statistics and compare with example 5 – c) Draw a box plot 121 126 130 132 143 137 141 144 148 205 125 128 131 133 135 139 141 147 153 213 Example 6 - Answer • Convert Pounds to Kg ( 0.4536 ) and add 2 121 126 130 132 143 137 141 144 148 205 125 128 131 133 135 139 141 147 153 213 56.88 59.15 60.97 61.88 66.87 64.14 65.96 67.32 69.13 94.99 58.7 60.06 61.42 62.33 63.24 65.05 65.96 68.68 71.40 98.62 Q1 = 61.19 Min = 56.89 IQR = 6.81 Mean = 67.14 StDev = 10.84 Q2 = 64.60 Q3 = 68.00 Max = 98.62 Range = 41.73 UF = 78.22 LF = 50.98 (143.6 0.4536 + 2) (23.91 0.4536) Example 6 – Answer cont Extreme Outliers ( > 3 IQR from Q3) * 45 50 55 60 65 70 75 80 85 90 95 * 100 105 Weight (in Kg) Transformation follows what we expect: Multiplying each observation by b multiplies both the measure of center and spread by b Adding a to each observation adds a to the measure of center, but does not affect spread If the distribution was symmetric, its transformation is symmetric. If the distribution was skewed, its transformation maintains the same skewness Day 2 Summary and Homework • Summary – Sample variance is found by dividing by (n – 1) to keep it an unbiased (since we estimate the population mean, μ, by using the sample mean, x‾) estimator of population variance – The larger the standard deviation, the more dispersion the distribution has – Boxplots can be used to check outliers and distributions – Use comparative boxplots for two datasets – Identifying a distribution from boxplots or histograms is subjective! • Homework – pg 82: prob 33; pg 89 probs 40, 41; pg 97 probs 45, 46