Section 1: Data (Major Concept Review)

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1 Section 1: Data (Major Concept Review) Individuals = the objects described by a set of data variable = characteristic of an individual weight height age IQ hair color eye color major social security # categorical variable = each individual falls into exactly one of several categories quantitative variable = each individual has an associated numerical value units = the way we measure a quantitative variable (feet vs. inches vs. centimeters) distribution of a variable = what values it takes and how often (pattern of variation) Displaying categorical data: bar graph/chart or pie chart Frequency table for eye color: frequency percent blue green brown other total gaps between bars o The categories are separate entities; there is no nuance or flowing from one to another, if there ever was in the underling concept. no horizontal scale o The width of the bars and of the gaps don t signify anything, but all bars should be the same width, and all gaps should be the same width, for aesthetic purposes. no logical order to categories o It s the same bar graph if we list the categories in some other order.

2 The following all mean essentially the same thing: relative frequency (out of 1) o The relative frequency of people with blue eyes is proportion (out of 1) o The proportion of people with blue eyes is percent (out of 100) o 29.4% of people have blue eyes. probability (out of 1) o If we pick a person a random, there s a probability that they ll have blue eyes. percent chance (out of 100) o If we pick a person a random, there s a 29.4% chance that they ll have blue eyes. A pie chart is another option to display categorical data, but it requires the relative counts: Displaying categorical data: stemplot (stem-and-leaf display) or histogram 69,86,70,65,51,66,65,88,71,72,72,81,84,31,86,67,54,72,82,67,86,67,92,74,66,78,76,68,66,73,76,70,65,40,70,92,78,43,30, 91,65,73,52,69,69,52,88,66,79,41,86,47,80,86,61,52,77,73,74,65,62,80,92,99,58,72,75,84,85,48,93,75,81,82,66,52,71,54, 92,71,98,72,69,36,92,74,89,84,73,81,31,70,72,52,94,80,91,89,54,82,82,84,72,72,49,74,57,74,96,65 Do the stemplot in two stages: 1. Pile up the last digit on each line. 2. Remake the stemplot, ordering each line from low to high

3 The stemplot is an informal display, so we can go increase or decrease as we go down, whichever looks better A split-stem stemplot has a low (0-4) and high (5-9) row for each. We can use this choice to spread out the display if it looks a little too clumped up: Notice that these choices will affect the visual features: a valley appears in the high 50 s/low 60 s. A back-to-back stemplot will display a second set of data going the other way, on the left ( ) (Our book goes for both sides, but this is not typical.) A histogram is created by tallying in a frequency table: Frequency percent [90,100) [80,90) [70,80) [60,70) [50,60) [40,50) [30,40) total

4 Observations: The usual conventional is to include the low end and not the high end. It s important that each piece of data goes in only one place. A 70 for instance would go in [70,80). As before, we can compute some percentages. Notice that we have a little rounding error because (through the luck of the draw) most of these happen to round down. We could also make a relative frequency histogram or a percentage histogram by using the relative amounts. Like any other unit change, it won t change the basic shape. We have more flexibility. A stemplot can only go by 10 s, 5 s, or maybe 2 s. A histogram can have intervals of length 7 if we wish, or anything else. Here s the corresponding histogram: In this histogram, each bar represents the ranges or intervals above. Features of a histogram: no gaps between bars o A 49.9 wouldn t be that different from a 50, even if in a different bar. There is a horizontal scale. o The histogram is a formal construction, so larger numbers should go. o The width of the bar signifies an increase (in this case) of 10 units (gradepoints, feet, gallons, etc.). There is logical order to the intervals: they increase in size to the right. There are many, many choices in construction which can affect the visual pattern we see, ranging from the obvious to the not-so-obvious: If we start at height 0 versus some other height, we can affect the shape we see. You ll see some of these in the practice problems. If our intervals are some other length, the shape may change (as when we split the stems). If we go by 10 s but [23,33), [33, 43), etc., the shape may change. How physically far apart we draw the tick-marks may change our visual impressions, particularly if the shape is somewhat borderline (e.g., Would you call this a peak?, etc.).

5 Here are some other histogram choices for the same set of data: Intervals of length 5, starting from 30: Notice (as with the split stemplot) how the valley appears in the high 50 s/low 60 s. Intervals of length 10, but [23,33), [33, 43), etc.:

6 A distribution of quantitative data (set of numbers) has three major characteristics: shape center spread These are not technical terms, but broad concepts. We will discuss center and spread in section 2. Here are two histograms with the same shape and spread, but different centers: Here are two histograms with the same shape and center, but different spreads:

7 Properties of shape: symmetric o Would the distribution (approximately) land on itself if reflected across its center? right-skewed: long tail going off toward larger numbers o Income, waiting times tend to be right-skewed: probably not big, a few very big left-skewed: long tail going off toward smaller numbers o grades tend to be left-skewed: probably not small, a few very small unimodal o one peak; that is, up, peaks out, down bimodal o two peaks; that is, up, down, up, down uniform : all bars the same height outliers = individuals outside the general pattern Note that right-and left-skewed are defined in terms of the numbers. Note also that a distribution could have more than one of these properties: unimodal right-skewed, symmetric bimodal, etc. Skewed, however, implies asymmetrical, a longer tail in one direction than the other. Example: many, many rolls of a fair die would have a uniform distribution: (Note that it is also an example of a symmetric distribution.)