Chapter 2. Summarizing and Graphing Data

The Big Picture of Statistics Chapter 2 Summarizing and Graphing Data 1 2 Characteristics of Data




Variables Categorical Center: a representative value Variation: a measure of the amount that the data values vary Distribution: the nature or shape of the spread of the data Outliers: sample values that lie very far from the majority of the other values Time: changing characteristics of the data over time Places an individual into one of several categories Examples: Gender, color, favorite movie, type of car, religion, etc. Quantitative Takes numerical values (in a unit of measurement) for which arithmetic operations make sense. Examples: Height, weight, MPG, age, salary, etc. Note: Quantitative variables continuous can take on any numerical value in a range discrete can take on only fixed values 3 4 Graphical Displays Categorical Quantitative

Pie charts
Bar graphs


Dotplot Stemplots Histograms Boxplots Show individual data points. Okay for small data sets. Graphical Displays for Categorical Variables Better for large data sets. (details later) 5 6 1 1. Pie Charts


2. Bar Graphs Pie charts are useful for summarizing a single categorical variable if not too many categories A pie chart must include all the categories that make up a whole. Use a pie chart when you want to emphasize each category’s relation to the whole.

Bar graph uses a horizontal or vertical rectangular bar that levels off at the appropriate level. Bar graphs are useful for summarizing one or two categorical variables and particularly useful for making comparisons when there are two categorical variables. 7 8 1. Dotplots Graphical Displays for Quantitative Variables
Work best when     You have a relatively small data sets Want to see (approximately) individual values Want to see shape Have one group or small number of groups to compare 9 1. Dotplots



1. Dotplots One Axis: only a horizontal axis Scale  10
Tick marks with numerical labels Equally spaced
Simply record a dot for each data point above an appropriate axis. If the data value repeats, the dots are piled up at that location, one dot for each repetition. 11 Example: As part of a study on the effects of calcium on blood pressure, the following 21 blood pressure readings were recorded 107,123,102,110,112,98,136,112,119,109,111,112, 129,117,110,102,130,112,123,114,107 12 2 1. Dotplots 1. Dotplots 13 2. Stemplots 14 Stemplot example are used for relatively small data sets of quantitative variables.
show exact values

Suppose we examine the following data: 55, 65, 66, 69, 71, 73, 79, 81, 83, 84, 84, 85, 86, 88, 89, 90, and 94
The stems for these data are 5, 6, 7, 8, and 9 since the data start in the 50’s and end in the 90’s 16 15 Making the stemplot 5 6 7 8 9 Making the stemplot Now, we record the leaves, the ones digit for each value 17 Data set: 55, 65, 66, 69, 71, 73, 79, 81, 83, 84, 84, 85, 86, 88, 89, 90, and 94 Stemplot: 5 6 7 8 9 5 569 139 13445689 04 18 3 Stemplots with split stem
Making the split stemplot Data set: 55, 65, 66, 69, 71, 73, 79, 81, 83, 84, 84, 85, 86, 88, 89, 90, and 94 Split the stems so that the original stem becomes two stems   One for the digits 0, 1, 2, 3, 4 --placed on first line of the stem One for digits 5, 6, 7, 8, 9 --placed on second line of the stem Stemplot Split Stemplot: 5 5 6 7 8 9 5 569 139 13445689 04 5 6 6 7 7 8 8 9 9 5 569 13 9 1344 5689 04 20 19 Stemplots to compare distributions

Stemplots: Example Back-to to-back stemplots Example: speed of predators and nonpredators Predator 900 2 0 0 1 2 3 4 5 6 7 Nonpredator 12 05 025 00058
The age of Best Actress Oscar winners:  Nonpredator: 11, 12, 20, 25, 30, 32, 35, 40, 40, 40, 45, 48
34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33 To make a stemplot: 1. Separate each observation into a stem and a leaf. 2. Write the stems in a vertical column with the smallest at the top, and draw a vertical line at the right of this column. 3. Go through the data points, and write each leaf in the row to the right of its stem. 4. Rearrange the leaves in an increasing order. Predator: 30, 30, 39, 42, 50, 70 21 Stemplots: Example
22 3. Histograms The age of Best Actress Oscar winners:  34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33



Shows groups of cases as rectangles or bars A dot plot with bars All bars must be same width Bars must be touching Split stemplot 23 24 4 Histogram Histograms Work best  With a large number of values to plot  Do not need to see individual values exactly  Want to see general shape  One distribution or small number of distributions to examine Two Axes

Horizontal Axis : The variable that you are analyzing Vertical Axis: Frequency (or Relative Frequency-percent) 26 25 To make a histogram Note about the bins To make a histogram we choose the classes or bins (usually use 5-15 bins, depending on the number of items, and make bin sizes equal)
tally the data into the classes or bins (put each data point into one and only one bin)
count the number of items in each bin
and then draw each bar with its height proportional to the number of items in its bin frequency histogram How many bins should we use?
No one right choice
 

Too few bins will give a “skycraper” graph Too many bins will give a “pancake” graph Neither choice will give a good picture of the shape of the distribution Use your judgment 27 28 Example Step 1: Choose the bins (classes) The data in Table 1.1 range from 17.0 to 44.2, so we decide to use 6 bins:
15.1-20.0 20.1-25.0 25.1-30.0 30.1-35.0 35.1-40.0 40.1-45.0 29 30 5 Example
Example Step 2: count the individuals in each class
Step 3: draw the histogram  Class Count 15.1-20.0 5 20.1-25.0 21 25.1-30.0 14 30.1-35.0 9 35.1-40.0 1 40.1-45.0 1    Mark the scale for the percent of the state’s adults with college degree (horizontal axis). The scale runs from 1545. The vertical axis contains the scale of counts. The scale runs from 0-21 (21 is the maximum count) Each bar represents a class—the bar height is the class count No space between the bars unless a class is empty 32 31 Summary: Histograms




Shows groups of cases as rectangles or bars A dot plot with bars All bars must be same width Bars must be touching No one right choice for the bins  
Too few bins will give a “skycraper” graph Too many bins will give a “pancake” graph Use your judgment 33 Graphical Displays for Quantitative Variables Interpreting Graphical Displays
5 6 7 8 9 Once the distribution has been displayed graphically, we can describe the overall pattern of the distribution and mention any striking deviations from that pattern. 5 569 139 13445689 04 6 Shape Examples of Symmetric Distributions Look for:

Symmetry/skewness of the distribution Peakedness (modality) - the number of peaks (modes) the distribution has. Examples of Symmetric Distributions

Symmetric, bimodal distribution (two peaks) Example: life expectancy in Europe and Asia Examples of Skewed Distributions
Right –skewed distribution 
Example: salary Left–skewed distribution  Example: age of death from natural causes

Symmetric, unimodal distribution (one peak) Example: test scores Examples of Symmetric Distributions

Symmetric, uniform distribution (no peak) Example: random numbers between 1 and 10 generated by computer Center
The center of the distribution is its midpoint the value that divides the distribution so that approximately half the observations take smaller values, and approximately half the observations take larger values.  Note that from looking at the histogram we can get only a rough estimate for the center of the distribution. 7 Spread

Outliers The spread (also called variability) of the distribution can be described by the approximate range covered by the data. From looking at the histogram we can approximate the smallest observation (min), and the largest observation (Max), and thus approximate the range. Range=Max-Min. 
Outliers are observations that fall outside the overall pattern.
One high outlier (More exact ways of finding measures of spread will be discussed in the next section.) Another example Example





Shape: Roughly symmetric Center is about 70 Spread: range=approximate max-approximate min=95-45=50 Outliers: no outliers Another example
Individuals vs. variables


Center: about $22,000 Spread: range=32,5005,500=$27,000 Note: the center and spread nor very useful here. Airport administrators take a sample of airline baggage and record the number of bags that weigh more than 75 pounds. What is the individual? a) b) c) d)
Shape: rightskewed Center: about 6-7% Spread: the range is 27.52.5=25% Number of bags weighing more than 75 pounds. Average weight of the bags. Each piece of baggage. The airport administrators. Shape: neither symmetric nor skewed. There are three clusters. 8 Individuals vs. variables Individuals vs. variables Airport administrators take a sample of airline baggage and record the number of bags that weigh more than 75 pounds. What is the variable of interest? In a study of commuting patterns of people in a large metropolitan area, respondents were asked to report the time they took to travel to their work on a specific day of the week. What is the individual? a) b) c) d) Number of bags weighing more than 75 pounds. Average weight of the bags. Each piece of baggage. The airport administrators. a) b) c) d) Travel time. A person. Day of the week. City in which they lived. Individuals vs. variables Categorical vs. quantitative variables In a study of commuting patterns of people in a large metropolitan area, respondents were asked to report the time they took to travel to their work on a specific day of the week. What is the variable of interest? Would the variable “monthly rainfall in Michigan” be considered a categorical or quantitative variable? a) b) c) d) a) b) categorical quantitative Travel time. A person. Day of the week. City in which they lived. Categorical vs. quantitative variables Categorical vs. quantitative variables If we asked people to report their “weight,” would that variable be considered a categorical or quantitative variable? We then asked people to classify their weight as underweight, normal, overweight, or obese. Would this variable now be categorical or quantitative variable? a) b) categorical quantitative a) b) categorical quantitative 9 Categorical vs. quantitative variables Graphing What type of data is produced by the answer choices for this question? How many times have you For the Internet access data in the previous question, what is the BEST method of displaying the data? accessed the Internet this week? 1) 2) 3) 4) None Once or twice Three or four times More than four times a) b) c) a) b) categorical quantitative d) bar graph boxplot histogram scatterplot Stemplots Histograms In the dataset represented by the following stemplot, how many times does the number “28” occur? Leaf unit = 1.0. 0 9 Look at the following histogram. How many baseball players report a salary of less than $1,441,000? 1 246999 2 111134567888999 a) b) c) d) 0 1 3 4 3 000112222345666699 5 0014 6 7 c) d) e) Bi-modal Left-skewed Right-skewed Symmetric Uniform c) d) 50 170 220 350 7 3 Look at the following histogram for salaries of baseball players. What shape would you say the data take? b) b) 4 001445 Histograms a) a) Chapter 3 Statistics for Describing, Exploring, and Comparing Data 60 10 Describing Distributions with Numbers


Center and Spread Wide spread The overall pattern of the distribution of a quantitative variable is described by its shape, center, and spread. By inspecting the histogram we can describe the shape of the distribution, but as we saw, we can only get a rough estimate for the center and spread. We need a more precise numerical description of the center and spread of the distribution. Narrow spread About the same center In this chapter you will learn:


how to quantify the center and spread of a distribution with various numerical measures some of the properties of these numerical measures, and how to choose the appropriate numerical measures of center and spread to supplement the histogram. Chapter 3 Section3-2: Measures of Center 64 Notation
Suppose we are making a series of observations, n of them, to be exact. Then we write x1 , x 2 , x 3, K , x n as the values we observe. Thus n is the total number of data points, and x 4 (say) is the value of the fourth data point. Example Suppose we ask five people how many hours of television they watch in a week, and get the following data: Observation 1 2 3 4 5 Data value 5 7 3 38 7 That is, x1 = 5, x 2 = 7, x3, = 3, x 4 = 38, x5 = 7 What is the “center” of these data, and what is the “spread” about that value? 11 Mean (Sample mean) Measures of Center and Spread CENTER SPREAD





Mean Median Mode Midrange


Range Inter-quartile range (IQR) Variance/Standard deviation Mean: x “x-bar” the average of a set of observations (the sum of the observations divided by the number of observations). If the n observations are x1 , x 2 , x 3, K , x n n then x + x 2 + x 3 +K+ x n x= 1 = n Example: Mean ∑x i i =1 n Mean is the balance point of the distribution x1 = 5, x 2 = 7, x 3, = 3, x 4 = 38, x5 = 7 The mean is x + x 2 + x 3 + x 4 + x5 x= 1 = n 5 + 7 + 3 + 38 + 7 x= = 12 5 5 ∑x i i =1 n Median
The median M is the midpoint of the distribution (like the median strip in a road)
It is the number such that half of the observations fall above and half fall below. mean How to find the median? 1.Order the data from smallest to largest. 2. If n is odd, the 2. If n is even, the median M is the median M is the mean center observation in of the two center the ordered list. This observations in the observation is the one ordered list. These two "sitting" in the observations are the (n+1)/2 spot in the ones "sitting" in the ordered list. n/2 and n/2 + 1 spots in the ordered list. 12 Median Example: Median x1 = 5, x 2 = 7, x 3, = 3, x 4 = 38, x5 = 7
Step 1: order the data: 3 5 7 7 38 Median = 7 Example 2: Median Example If the data are Data set A: 64 65 66 68 70 71 73 x1 = 5, x 2 = 7, x 3, = 3, x 4 = 38

Step 1: order the data: 3 5 7 Median = 5+ 7 =6 2 Mean
median is 68 mean is 68.1

median is still 68 mean is 162 38 The mean is very sensitive to outliers, while the median is resistant to outliers. Comparing the mean and the median

outlier Data set B: 64 65 66 68 70 71 730 describes the center as an average value, where the actual values of the data points play an important role Sensitive to outliers Symmetric distributions with no outliers Median

locates the middle value as the center, and the order of the data is the key to finding it. Not sensitive to outliers Mean ≈Median 13 Left-skewed distributions Right-skewed distributions Median=68 Mean < Median Which measure of center to use?
We will therefore use the mean as a measure of center for symmetric distributions with no outliers. Mean=72 Median< Mean Another measure of center: Mode
Mode is the most frequent value in the data set
Data: 2, 4, 4, 4, 5, 5, 6, 7, 8, 10, 15
Data: 2, 4, 4, 4, 5, 5, 5, 6, 7, 8, 10, 15
Data: 2, 4, 5, 6, 7, 8, 10, 15 Mode = 4
Otherwise, the median will be a more appropriate measure of the center of our data. Mode = 4, 5 No mode Summary: Measures of Center




The two main numerical measures for the center of a distribution are the mean x and the median M. The mean is the average value, while the median is the middle value. The mean is very sensitive to outliers, while the median is resistant to outliers. The mean is an appropriate measure of center only for symmetric distributions with no outliers. In all other cases, the median should be used to describe the center of the distribution. The mode is the most frequent value. 14