Section 2.2 Graphical Displays of Distributions

Graphical Displays Always plot your data first! To see shape of distribution of data set, you need a suitable graph.

Graphical Displays In this section, we’ll discuss: 1) dot plots 2) histograms 3) stem plots 4) bar charts

Case vs Variable Before we can talk about graphs, we must talk about Cases and Variables: Case: a subject or unit you are measuring Examples: people, cities, mammals, etc In a data chart, the cases are the rows

Case vs Variable Variable: a characteristic that differs from case to case Defines what is to be measured or classified Also called an attribute

Case vs Variable Turn to Page 43, Display 2.24  What are the cases?  What are the variables?

Case vs Variable Turn to Page 43, Display 2.24  Each row is a case (type of mammal)  Each column is a variable for each case

Two Types of Variables 1. Quantitative variable  Must be represented by a number (can calculate the mean)  Examples:  x=3,  y=1.6,  offset=-33,  score=55%. 2. Categorical variable:  Must be represented by a choice between words or groups of words  Examples:  X= blonde or brunette,  y={blue, red, green},  reading level = {illiterate, basic literacy, high school literacy, post-graduate literacy}

Two Types of Variables Categorical variable: variable that groups cases into categories Sometimes called qualitative variable Has no numerical value Examples: eye color; gender; religion; political party; sizes such as small, medium, and large

Quantitative or Categorical? What about the grades students are in high school? Quantitative or Categorical?

What about the grades students are in high school? Quantitative if expressed as grade 9, 10, 11, or 12 Categorical if expressed as freshman, sophomore, junior, or senior

Quantitative or Categorical? Turn to Page 43, Display 2.24 again. Identify each variable as quantitative or categorical.

Graphs: Dot Plots Shows shape, center, and spread Mammal Gestation Periods

Graphs: Dot Plots Tend to work best when: 1) have relatively small number of values to plot 2) want to see individual values at least approximately 3) want to see shape of distribution 4) have one group or small number of groups you want to compare

Graphs: Histograms Think of a histogram as a dot plot with bars drawn around the dots and then the dots are erased.

Graphs: Histograms To make histogram, divide number line into consistent intervals called bins. Over each bin construct a bar that has height equal to number of cases in that bin.

Graphs: Histograms Longevity of Mammals

Graphs: Histograms Work best when: 1) have large number of values to plot 2) do not need to see individual values exactly 3) want to see general shape of distribution 4) have only one distribution or small number of distributions to compare 5) can use calculator/computer to make

Graphs: Histograms Borderline values go into the bin to the right 20 would go into bin to right of 20 on scale

Graphs: Histograms How do you determine how wide to make the bars?

Graphs: Histograms How do you determine how wide to make the bars? Answer: Wide enough to show important features of the distribution If bars too wide, you may miss gaps and clusters If bars too narrow, you would have essentially a dot plot

Graphs: Histograms May need to be like Goldilocks and the Three Bears! Try changing intervals until you’re able to see the overall shape of the distribution.

Graphs: Histograms Histogram shows frequencies on vertical scale For large data sets (i.e. tall boxes),we may be better to use relative frequency histogram where height is a percentage of total number of data points (rows).

Graphs: Histograms To change histogram into relative frequency histogram,  divide the frequency (height) of each bar  by the total number of values in data set. Four of 18 mammals have speeds from 30 mph up to 35 mph. Convert the frequency of 4 animals to a relative frequency. Answer: 4/18 or approximately 0.22 or 22%

Graphs: Histograms Does using relative frequencies change the shape of a histogram? What information is lost and gained by using a relative frequency histogram rather than a frequency histogram?

Graphs: Histograms Does using relative frequencies change the shape of a histogram? Answer: The shape remains the same, only the scale on vertical axis changes. What information is lost and gained by using a relative frequency histogram rather than a histogram? Answer: We lose the ability to see the actual cases or total number of cases. Answer: We gain the ability to compare two groups with a large number of different members.

Graphs: Stem plot Stem plot can also be used to plot quantitative data (numbers+units). Also known as a “stem-and-leaf” plot

Graphs: Stemplot Stemplots useful when: Plotting a single quantitative variable Have relatively small number of values Would like to see individual values exactly Want to see shape of distribution clearly You can also have two groups you want to compare (back-to-back stem plot)

Graphs: Stem plot Every number can be represented by a stem and leaf Leaf is digit furthest to right Stem is rest of number after leaf removed

Graphs: Stem plot Identify stem and leaf for each of these numbers: StemLeaf 1) ) ) 909

Graphs: Stem plot Must have a key that “represents” or explains units

Graphs: Stem plot

Stem plot: Statistical Software

Categorical Variables Can use dot plot for categorical variables Another choice is a bar chart or bar graph

Graphs: Bar Chart Bar chart has categories on horizontal axis whereas histogram has values from quantitative variable Bar charts show frequencies of categorical data as heights of bars

Graphs: Bar Chart

Summary Always plot the data to display the distribution of values. This aids in identifying the shape, center, and spread of the distribution.

Summary For quantitative variables, you can use: 1. dot plots 2. histograms 3. stem plots For categorical variables, you can use: 1. bar chart

Summary For these different plots, amount of detail varies. Choose a plot that fits your data and your reason for analyzing it.

Summary Dot plots best used with small number of values and show roughly where values lie on a number line Histograms show only intervals of values, losing actual data values, and are most appropriate for large data sets Stemplots can show actual data values

Questions to Answer Looking at Plot 1) Where did data set come from? 2) What are cases and variables? 3) What are shape, center, and spread? Any unusual characteristics like outliers, clusters, or gaps? 4) What are possible explanations of patterns in distribution?