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: • Standard Deviation: • Example 1.16 (p.85): Metabolic Rates
Standard Deviation Metabolic Rates: mean=1600 What does this value, s, mean?
Example 1 Which of the following measures of spread are resistant? • Range • Variance • Standard Deviation Not Resistant Not Resistant Not Resistant
Example 2 Given the following set of data:70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62, 45, 60, 38, 73, 45, 51,56, 51, 46, 39, 56, 32, 44, 60, 51, 44, 63, 50, 46, 69, 53, 70, 33, 54, 55, 52What is the range?What is the variance?What is the standard deviation? 73-28 = 45 117.958 10.861
Q1=23 med Q3=29.5 Q1 Q3 med=79 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’.
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
45 50 55 60 65 70 75 80 85 90 95 100 Quiz Scores Outlier? 5-Number Summary, Boxplots • The 5 Number Summary provides a reasonably complete description of the center and spread of distribution • We can visualize the 5 Number Summary with a boxplot.
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: • Find 5 Number Summary • Determine IQR • Multiply 1.5xIQR • Set up “fences” • Lower Fence: Q1-(1.5∙IQR) • Upper Fence: Q3+(1.5∙IQR) • Observations “outside” the fences are outliers.
fence: 19.06-39.99 = -20.93 outliers fence: 45.72+39.99 = 85.71 0 10 20 30 40 50 60 70 80 90 100 Spending ($) IQR=45.72-19.06 IQR=26.66 { } Outlier Example All data on pg 48, #1.6 1.5IQR=1.5(26.66) 1.5IQR=39.99
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? Construct a boxplot for the data above.
Example 4 - Answer Q1 = 182 Q2 = 221.5 Q3 = 319 Min = 111 Max = 439 Range = 328 IQR = 137 UF = 524.5 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: 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 Q2 = 138 Q3 = 145.5 Min = 121 Max = 213 Range = 92 IQR = 15 UF = 168 LF = 108 Mean = 143.6 StDev = 23.91 Extreme Outliers( > 3 IQR from Q3) * * 100 110 120 130 140 150 160 170 180 190 200 210 220 Weight (lbs) Shape: somewhat symmetric Outliers: 2 extreme outliers Center: Median = 138 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
Example 6 - Answer • Convert Pounds to Kg ( 0.4536 ) and add 2 Q1 = 61.19 Q2 = 64.60 Q3 = 68.00 Min = 56.89 Max = 98.62 Range = 41.73 IQR = 6.81 UF = 78.22 LF = 50.98 Mean = 67.14 (143.6 0.4536 + 2) StDev = 10.84 (23.91 0.4536)
Example 6 – Answer cont Extreme Outliers( > 3 IQR from Q3) 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 * * 45 50 55 60 65 70 75 80 85 90 95 100 105 Weight (in Kg)
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
