1. Looking at Data - Distributions Displaying Distributions with Graphs © 2009 W.H. Freeman and Company

2. Outline Displaying distributions with graphs  Variables  Types of variables  Graphs for categorical variables  Tables  Bar graphs  Pie charts  Graphs for quantitative variables  Histograms  Interpreting histograms

3. Variables In a study, we collect information—data—from individuals. Individuals can be people, animals, plants, or any object of interest. A variable is any characteristic of an individual. A variable varies among individuals. Example: age, height, blood pressure, ethnicity, leaf length, first language The distribution of a variable tells us what values the variable takes and how often it takes these values.

4. Two types of variables  Variables can be either numerical…  Something that takes numerical values for which arithmetic operations, such as adding and averaging, make sense.  Example: How tall you are, your age, your blood cholesterol level, the number of credit cards you own.  … or categorical.  Something that falls into one of several categories. What can be counted is the count or proportion of individuals in each category.  Example: Your blood type (A, B, AB, O), your hair color, your ethnicity, whether you paid income tax last tax year or not.

5. How do you know if a variable is categorical or numerical? Ask:  What is the average value?  If it doesn’t make sense  categorical  Make sense but might not be an answer (e. g. age)  discrete  Otherwise  numerical (quantitative) Individuals in sample DIAGNOSISAGE AT DEATH Patient AHeart disease56 Patient BStroke70 Patient CStroke75 Patient DLung cancer60 Patient EHeart disease80 Patient FAccident73 Patient GDiabetes69 Numerical Each individual is attributed a numerical value. Categorical Each individual is assigned to one of several categories.

6. Ways to chart categorical data Because the variable is categorical, the data in the graph can be ordered any way we want (alphabetical, by increasing value, by year, by personal preference, etc.)  Bar graphs Each category is represented by a bar.  Pie charts The slices must represent the parts of one whole.  Tables Never married Married Widowed Divorced 42 118 16 18

7. Tables  Widely used to summarize data  table() > res=c("Y", "Y", "N", "Y", "N") > table(res) res N Y 2 3  Another example >library(UsingR) >central.park.cloud >table(central.park.cloud)

8. Example: Top 10 causes of death in the United States 2001 RankCauses of deathCounts % of top 10s % of total deaths 1Heart disease700,14237%29% 2Cancer553,76829%23% 3Cerebrovascular163,5389%7% 4Chronic respiratory123,0136%5% 5Accidents101,5375%4% 6Diabetes mellitus71,3724%3% 7Flu and pneumonia62,0343% 8Alzheimer’s disease53,8523%2% 9Kidney disorders39,4802% 10Septicemia32,2382%1% All other causes629,96726% For each individual who died in the United States in 2001, we record what was the cause of death. The table above is a summary of that information.

9. Top 10 causes of deaths in the United States 2001 Bar graphs Each category is represented by one bar. The bar’s height shows the count (or sometimes the percentage) for that particular category. The number of individuals who died of an accident in 2001 is approximately 100,000.

10. Bar graph sorted by rank  Easy to analyze Top 10 causes of deaths in the United States 2001 Sorted alphabetically  Much less useful

11. Barplots in R >beer=scan() 1: 3 4 1 1 3 4 3 3 1 3 2 1 2 1 2 3 2 3 1 1 1 1 4 3 1 26: >barplot(beer) # this isn’t correct >barplot(table(beer),xlab=“beer”,ylab=“frequency”) >barplot(table(beer)/length(beer),xlab=“beer”,ylab=“proportion”)

12. Misleading barplots >sales=c(45,44,46) > names(sales)=c("John", "Jack", "Suzy") > barplot(sales, main="Sales", ylab="Thousands") > barplot(sales, main="Sales", ylab="Thousands",ylim=c(42,46),xpd=FALSE)

13. > sales John Jack Suzy 45 44 46 >pie(sales) Why are pie charts a poor choice? The help page for pie(), (?pie) Pie charts are a very bad way of displaying information. The eye is good at judging linear measures and bad at judging relative areas. A bar chart or dot chart is a preferable way of displaying this type of data. Pie charts Each slice represents a piece of one whole. The size of a slice depends on what percent of the whole this category represents.

14. Ways to chart quantitative data  Histograms These are summary graphs for a single variable. They are very useful to understand the pattern of variability in the data.

15. Histograms The range of values that a variable can take is divided into equal size intervals. The histogram shows the number of individual data points that fall in each interval. The first column represents all states with a Hispanic percent in their population between 0% and 4.99%. The height of the column shows how many states (27) have a percent in this range. The last column represents all states with a Hispanic percent in their population between 40% and 44.99%. There is only one such state: New Mexico, at 42.1% Hispanics.

16. Interpreting histograms When describing the distribution of a quantitative variable, we look for the overall pattern and for striking deviations from that pattern. We can describe the overall pattern of a histogram by its shape, center, and spread. Histogram with a line connecting each column  too detailed Histogram with a smoothed curve highlighting the overall pattern of the distribution

17. Most common distribution shapes  A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. Symmetric distribution Complex, multimodal distribution  Not all distributions have a simple overall shape, especially when there are few observations. Skewed distribution  A distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.

18. AlaskaFlorida Outliers An important kind of deviation is an outlier. Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them. The overall pattern is fairly symmetrical except for 2 states that clearly do not belong to the main trend. Alaska and Florida have unusual representation of the elderly in their population. A large gap in the distribution is typically a sign of an outlier.

19. How to create a histogram It is an iterative process – try and try again. What bin size should you use?  Not too many bins with either 0 or 1 counts  Not overly summarized that you loose all the information  Not so detailed that it is no longer summary  rule of thumb: start with 5 to 10 bins Look at the distribution and refine your bins (There isn’t a unique or “perfect” solution)

20. Not summarized enough Too summarized Same data set

21. IMPORTANT NOTE: Your data are the way they are. Do not try to force them into a particular shape. It is a common misconception that if you have a large enough data set, the data will eventually turn out nice and symmetrical. Histogram of Drydays in 1995

22. Histogram in R  With hist() function > attach(faithful) > hist(waiting) # use defaults > hist(waiting, breaks=10) # suggest 10 breaks, not necessarily 10 > hist(waiting, breaks=seq(43,108,length=10)) # use these breaks > hist(waiting, breaks ="scott” ) #Use “Scott” algorithm >hist(waiting, prob=T) # Use proportions  Estimating the density > hist(waiting, breaks="scott", prob=T, main="", ylab="") > lines(density(waiting))

23. Looking at Data - Distributions Describing distributions with numbers © 2009 W.H. Freeman and Company

24. Outline Describing distributions with numbers  Measures of center: mean, median  Mean versus median  Measures of spread: quartiles, standard deviation  Five-number summary and boxplot  Choosing among summary statistics

25. 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 Measure of center: the mean

26. Mathematical notation: Learn right away how to get the mean using your calculators.

27. Your numerical summary must be meaningful. 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. Height of 25 women in a class

28. 58 60 62 64 66 68 70 72 74 76 78 80 82 84 A single numerical summary here would not make sense.

29. 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. Sort observations by size. n = number of observations ______________________________ n = 24  n/2 = 12 Median = (3.3+3.4) /2 = 3.35 2.b. If n is even, the median is the mean of the two middle observations.  n = 25 (n+1)/2 = 26/2 = 13 Median = 3.4 2.a. If n is odd, the median is observation (n+1)/2 down the list

30. Mean and median for skewed distributions Mean and median for a symmetric distribution Left skewRight skew Mean Median Mean Median Mean Median 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.

31. The median, on the other hand, is only slightly pulled to the right by the outliers (from 3.4 to 3.6). The mean is pulled to the right a lot by the outliers (from 3.4 to 4.2). Percent of people dying Mean and median of a distribution with outliers Without the outliers With the outliers

32. Disease X: Mean and median are the same. Symmetric distribution… Multiple myeloma: … and a right-skewed distribution The mean is pulled toward the skew. Impact of skewed data

33. M = median = 3.4 Q 1 = first quartile = 2.2 Q 3 = third quartile = 4.35 Measure of spread: the quartiles The first quartile, Q 1, 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). The third quartile, Q 3, 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).

34. M = median = 3.4 Q 3 = third quartile = 4.35 Q 1 = first quartile = 2.2 Largest = max = 6.1 Smallest = min = 0.6 Five-number summary: min Q 1 M Q 3 max Five-number summary and boxplot BOXPLOT

35. Comparing box plots for a normal and a right-skewed distribution Boxplots for skewed data Boxplots remain true to the data and depict clearly symmetry or skew.

36. 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 (Q 1 or Q 3 ). We then compare this distance to the interquartile range (distance between Q 1 and Q 3 ). 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.”

37. Q 3 = 4.35 Q 1 = 2.2 8 Interquartile range Q 3 – Q 1 4.35 − 2.2 = 2.15 Distance to Q 3 7.9 − 4.35 = 3.55 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.

38. 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 s 2. 2. Then take the square root to get the standard deviation s. Measure of spread: the standard deviation Mean ± 1 s.d.

39. Properties of Standard Deviation  s measures spread about the mean and should be used only when the mean is the measure of center.  s = 0 only when all observations have the same value and there is no spread. Otherwise, s > 0.  s is not resistant to outliers.  s has the same units of measurement as the original observations.

40. Choosing among summary statistics  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 five number summary which can be plotted as a boxplot. Boxplot Mean ± SD

41. 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.  In a study of standard of living of typical families in Middletown, a sociologist makes a numerical summary of family income in that city.  Mean: Although income is likely to be right-skewed, the city government wants to know about the total tax base.  Median: The sociologist is interested in a “typical” family and wants to lessen the impact of extreme incomes.

42. Boxplot in R  summary() function > attach(cfb) # it is a data frame > summary(VEHIC) Min. 1st Qu. Median Mean 3rd Qu. Max. 0 3875 11000 15400 21300 188000 > hist(VEHIC, br="Scott", prob=T) > lines(density(VEHIC))  Making boxplots in R > attach(alltime.movies) > boxplot(Gross, ylab="All-time gross sales") > f=fivenum(Gross) > text(rep(1.3,5), f, labels=c("min","Q1", "median","Q3","max"))

43. Getting the outliers > the.names=rownames(alltime.movies) > the.names[Gross > f[4] + 1.5*(f[4]-f[2])] [1] "Titanic " [2] "Star Wars " [3] "E.T. " [4] "Star Wars: The Phantom Menace" [5] "Spider-Man " > detach(alltime.movies) #tidy up