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Unformatted text preview: Numerical descriptors BPS chapter 2 2006 W.H. Freeman and Company Objectives (BPS chapter 2)
Describing distributions with numbers Measure of center: mean and median Measure of spread: quartiles and standard deviation The fivenumber summary and boxplots IQR and outliers Choosing among summary statistics Using technology Organizing a statistical problem 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."
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 Sum of heights is 1598.3 Divided by 25 women = 63.9 inches woman (i) i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i = 10 i = 11 i = 12 i = 13 height (x) x1= 58.2 x2= 59.5 x3= 60.7 x4= 60.9 x5= 61.9 x6= 61.9 x7= 62.2 x8= 62.2 x9= 62.4 x10= 62.9 x11= 63.9 x12= 63.1 x13= 63.9 woman (i) i = 14 i = 15 i = 16 i = 17 i = 18 i = 19 i = 20 i = 21 i = 22 i = 23 i = 24 i = 25 height (x) x14= 64.0 x15= 64.5 x16= 64.1 x17= 64.8 x18= 65.2 x19= 65.7 x20= 66.2 x21= 66.7 x22= 67.1 x23= 67.8 x24= 68.9 x25= 69.6 Mathematical notation: x 1 + x 2 + .... + xn x= n
1 x= xi n i=1
18 5. 93 x = =3 69 . 2 5
n n=25 =1598.3 Learn right away how to get the mean using your calculators. Measure of spread: quartiles
The first quartile, Q1, is the value in the sample that has 25% of the data at or below it.
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 M = median = 3.4 The third quartile, Q3, is the value in the sample that has 75% of the data at or below it. Q3= third quartile = 4.35 Center and spread in boxplots
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 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
7 Q3= third quartile = 4.35
Years until death 6 5 4 3 2 1 M = median = 3.4 Q1= first quartile = 2.2 Smallest = min = 0.6 0
Disease X "Fivenumber summary" Measure of spread: standard deviation
The standard deviation is used to describe the variation around the mean.
2 1) First calculate the variance s . 1 n s2 = n 1 ( xi  x ) 2 1 x 2) Then take the square root to get the standard deviation s. Mean 1 s.d. 1 n s= ( xi  x ) 2 n 1 1 Calculations ... Women's height (inches) 1 s= n 1 n 1 ( xi  x ) 2 Mean = 63.4 Sum of squared deviations from mean = 85.2 Degrees freedom (df) = (n  1) = 13 s2 = variance = 85.2/13 = 6.55 inches squared s = standard deviation = 6.55 = 2.56 inches We'll never calculate these by hand, so make sure you know how to get the standard deviation using your calculator. Your numerical summary must be meaningful
Height of 25 women in a class x = 69.3 The distribution of women's height appears coherent and symmetrical. The mean is a good numerical summary. Here the shape of the distribution is wildly irregular. Why? Could we have more than one plant species or phenotype? x = 69.6 Height of plants by color
5 x = 63.9 x = 70.5 x = 78.3
red pink blue Number of plants 4 3 2 1 0 58 60 62 64 66 68 70 72 74 76 78 80 82 84 Height in centimeters A single numerical summary here would not make sense. Measure of center: the median
The median is the midpoint of a distributionthe 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 1. Sort observations from smallest to largest. n = number of observations ______________________________ 2. 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 3. If n is even, the median is the mean of the two center observations n = 24 n/2 = 12 Median = (3.3+3.4) /2 = 3.35 25 12 6.1 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
x = 3.4
Percent of people dying x = 4.2
Without the outliers With the outliers The mean is pulled to the right a lot by the outliers (from 3.4 to 4.2). The median, on the other hand, is only slightly pulled to the right by the outliers (from 3.4 to 3.6). Impact of skewed data
Mean and median of a symmetric distribution Disease X: x = 3.4 M = 3.4 Mean and median are the same. and a rightskewed distribution Multiple myeloma: x = 3.4 M = 2.5 The mean is pulled toward the skew. Boxplots for skewed data Comparing box plots for a normal and a rightskewed distribution
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Disease X Multiple myeloma Years until death Boxplots remain true to the data and clearly
depict symmetry or skewness. IQR and outliers
The interquartile range (IQR) is the distance between the first and third quartiles (the length of the box in the boxplot) Software output for summary statistics: ExcelFrom Menu:
Tools/Data Analysis/ Descriptive Statistics Give common statistics of your sample data. Minitab Choosing among summary statistics Height in inches Because the mean is not resistant to outliers or skew, use it to describe distributions that are fairly symmetrical and don't have outliers. Plot the mean and use the standard deviation for error bars. Otherwise, use the median in the fivenumber summary, which can be plotted as a boxplot. Height of 30 women 69 68 67 66 65 64 63 62 61 60 59 58 Box plot Mean s.d. Box plot Mean +/ sd 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' incomes to estimate the total tax base. Mean: Although income is likely to be rightskewed, 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. Organizing a statistical problem State: What is the practical question, in the context of a realworld setting? Formulate: What specific statistical operations does this problem call for? Solve: Make the graphs and carry out the calculations needed for this problem. Conclude: Give your practical conclusion in the setting of the realworld setting. ...
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 Summer '08
 AKBAS
 Statistics, Standard Deviation

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