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Looking at data: distributions - Describing distributions with numbers IPS chapter 1.2 © 2006 W.H. Freeman and Company Objectives (IPS chapter 1.2) Describing distributions with numbers Measure of center: the mean Measure of center: the median Measure of spread: the quartiles Five-number summary and boxplot Measure of spread: the standard deviation Choosing among summary statistics Changing the unit of measurement Measure of center: the mean The mean or arithmetic average To calculate the average, or mean, add all values, then divide by the number of individuals. It is the “center of mass.” Sum of heights is 1598.3 divided by 25 women = 63.9 inches 58.2 59.5 60.7 60.9 61.9 61.9 62.2 62.2 62.4 62.9 63.9 63.1 63.9 64.0 64.5 64.1 64.8 65.2 65.7 66.2 66.7 67.1 67.8 68.9 69.6 woman (i) height (x) woman (i) height (x) i=1 x1= 58.2 i = 14 x14= 64.0 i=2 x2= 59.5 i = 15 x15= 64.5 i=3 x3= 60.7 i = 16 x16= 64.1 i=4 x4= 60.9 i = 17 x17= 64.8 i=5 x5= 61.9 i = 18 x18= 65.2 i=6 x6= 61.9 i = 19 x19= 65.7 i=7 x7= 62.2 i = 20 x20= 66.2 i=8 x8= 62.2 i = 21 x21= 66.7 i=9 x9= 62.4 i = 22 x22= 67.1 i = 10 x10= 62.9 i = 23 x23= 67.8 i = 11 x11= 63.9 i = 24 x24= 68.9 i = 12 x12= 63.1 i = 25 x25= 69.6 i = 13 x13= 63.9 n=25 =1598.3 Mathematical notation: x1 x2 ... xn x n 1 n x xi n i 1 1598.3 x 63.9 25 Learn right away how to get the mean using your calculators. Your numerical summary must be meaningful. Height of 25 women in a class x 69.3 Here the shape of the distribution is wildly irregular. Why? Could we have more than one plant species or phenotype? The distribution of women’s heights appears coherent and symmetrical. The mean is a good numerical summary. x 69.6 Height of Plants by Color x 63.9 5 x 70.5 x 78.3 red Number of Plants 4 pink blue 3 2 1 0 58 60 62 64 66 68 70 72 74 76 78 80 82 Height in centimeters A single numerical summary here would not make sense. 84 Measure of center: the median The median is the midpoint of a distribution—the number such that half of the observations are smaller and half are larger. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 0.6 1.2 1.6 1.9 1.5 2.1 2.3 2.3 2.5 2.8 2.9 3.3 3.4 3.6 3.7 3.8 3.9 4.1 4.2 4.5 4.7 4.9 5.3 5.6 25 12 6.1 1. Sort observations by size. n = number of observations ______________________________ 2.a. If n is odd, the median is observation (n+1)/2 down the list n = 25 (n+1)/2 = 26/2 = 13 Median = 3.4 2.b. If n is even, the median is the mean of the two middle observations. n = 24 n/2 = 12 Median = (3.3+3.4) /2 = 3.35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 0.6 1.2 1.6 1.9 1.5 2.1 2.3 2.3 2.5 2.8 2.9 3.3 3.4 3.6 3.7 3.8 3.9 4.1 4.2 4.5 4.7 4.9 5.3 5.6 Comparing the mean and the median The mean and the median are the same only if the distribution is symmetrical. The median is a measure of center that is resistant to skew and outliers. The mean is not. Mean and median for a symmetric distribution Mean Median Mean and median for skewed distributions Left skew Mean Median Mean Median Right skew Mean and median of a distribution with outliers Percent of people dying x 3.4 x 4.2 Without the outliers With the outliers The mean is pulled to the The median, on the other hand, right a lot by the outliers is only slightly pulled to the right (from 3.4 to 4.2). by the outliers (from 3.4 to 3.6). Impact of skewed data Mean and median of a symmetric Disease X: x 3.4 M 3.4 Mean and median are the same. … and a right-skewed distribution Multiple myeloma: x 3.4 M 2.5 The mean is pulled toward the skew. Measure of spread: the quartiles The first quartile, Q1, is the value in the sample that has 25% of the data at or below it ( it is the median of the lower half of the sorted data, excluding M). M = median = 3.4 The third quartile, Q3, is the value in the sample that has 75% of the data at or below it ( it is the median of the upper half of the sorted data, excluding M). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 1 2 3 4 5 1 2 3 4 5 6 7 1 2 3 4 5 0.6 1.2 1.6 1.9 1.5 2.1 2.3 2.3 2.5 2.8 2.9 3.3 3.4 3.6 3.7 3.8 3.9 4.1 4.2 4.5 4.7 4.9 5.3 5.6 6.1 Q1= first quartile = 2.2 Q3= third quartile = 4.35 Five-number summary and boxplot 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6.1 5.6 5.3 4.9 4.7 4.5 4.2 4.1 3.9 3.8 3.7 3.6 3.4 3.3 2.9 2.8 2.5 2.3 2.3 2.1 1.5 1.9 1.6 1.2 0.6 Largest = max = 6.1 BOXPLOT 7 Q3= third quartile = 4.35 M = median = 3.4 6 Years until death 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 5 4 3 2 1 Q1= first quartile = 2.2 Smallest = min = 0.6 0 Disease X Five-number summary: min Q1 M Q3 max Boxplots for skewed data Years until death Comparing box plots for a normal and a right-skewed distribution 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Boxplots remain true to the data and depict clearly symmetry or skew. Disease X Multiple Myeloma Suspected outliers Outliers are troublesome data points, and it is important to be able to identify them. One way to raise the flag for a suspected outlier is to compare the distance from the suspicious data point to the nearest quartile (Q1 or Q3). We then compare this distance to the interquartile range (distance between Q1 and Q3). We call an observation a suspected outlier if it falls more than 1.5 times the size of the interquartile range (IQR) above the first quartile or below the third quartile. This is called the “1.5 * IQR rule for outliers.” 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 7.9 6.1 5.3 4.9 4.7 4.5 4.2 4.1 3.9 3.8 3.7 3.6 3.4 3.3 2.9 2.8 2.5 2.3 2.3 2.1 1.5 1.9 1.6 1.2 0.6 8 7 Q3 = 4.35 Distance to Q3 7.9 − 4.35 = 3.55 6 Years until death 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 5 Interquartile range Q3 – Q1 4.35 − 2.2 = 2.15 4 3 2 1 Q1 = 2.2 0 Disease X Individual #25 has a value of 7.9 years, which is 3.55 years above the third quartile. This is more than 3.225 years, 1.5 * IQR. Thus, individual #25 is a suspected outlier. Measure of spread: the standard deviation The standard deviation “s” is used to describe the variation around the mean. Like the mean, it is not resistant to skew or outliers. 1. First calculate the variance s2. n 1 2 s2 ( x x ) i n 1 1 2. Then take the square root to get the standard deviation s. x Mean ± 1 s.d. 1 n 2 s ( x x ) i n 1 1 Calculations … Women height (inches) i xi x (xi-x) (xi-x)2 1 59 63.4 -4.4 19.0 2 60 63.4 -3.4 11.3 3 61 63.4 -2.4 5.6 4 62 63.4 -1.4 1.8 5 62 63.4 -1.4 1.8 6 63 63.4 -0.4 0.1 7 63 63.4 -0.4 0.1 8 63 63.4 -0.4 0.1 9 64 63.4 0.6 0.4 10 64 63.4 0.6 0.4 11 65 63.4 1.6 2.7 Degrees freedom (df) = (n − 1) = 13 12 66 63.4 2.6 7.0 s2 = variance = 85.2/13 = 6.55 inches squared 13 67 63.4 3.6 13.3 14 68 63.4 4.6 21.6 Sum 0.0 Sum 85.2 s 1 df n (x i x) 2 1 Mean = 63.4 Sum of squared deviations from mean = 85.2 s = standard deviation = √6.55 = 2.56 inches Mean 63.4 We’ll never calculate these by hand, so make sure to know how to get the standard deviation using your calculator. Software output for summary statistics: Excel - From Menu: Tools/Data Analysis/ Descriptive Statistics Give common statistics of your sample data. Minitab Choosing among summary statistics Because the mean is not Height of 30 Women resistant to outliers or skew, use it 69 to describe distributions that are 68 fairly symmetrical and don’t have Plot the mean and use the standard deviation for error bars. Otherwise use the median in the five number summary which can be plotted as a boxplot. Height in Inches outliers. 67 66 65 64 63 62 61 60 59 58 Box Plot Boxplot Mean ± +/-SD SD Mean What should you use, when, and why? Arithmetic mean or median? Middletown is considering imposing an income tax on citizens. City hall wants a numerical summary of its citizens income to estimate the total tax base. Mean: Although income is likely to be right-skewed, the city government wants to know about the total tax base. In a study of standard of living of typical families in Middletown, a sociologist makes a numerical summary of family income in that city. Median: The sociologist is interested in a “typical” family and wants to lessen the impact of extreme incomes. Changing the unit of measurement Variables can be recorded in different units of measurement. Most often, one measurement unit is a linear transformation of another measurement unit: xnew = a + bx. Temperatures can be expressed in degrees Fahrenheit or degrees Celsius. TemperatureFahrenheit = 32 + (9/5)* TemperatureCelsius a + bx. Linear transformations do not change the basic shape of a distribution (skew, symmetry, multimodal). But they do change the measures of center and spread: Multiplying each observation by a positive number b multiplies both measures of center (mean, median) and spread (IQR, s) by b. Adding the same number a (positive or negative) to each observation adds a to measures of center and to quartiles but it does not change measures of spread (IQR, s).