Displaying Quantitative Data

AP STATS

NHS

Mr. Unruh

Slide 4- 5

Dotplots• A dotplot is a simple display. It just places a dot along an axis for each case in the data.

• The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot.

• You might see a dotplot displayed horizontally or vertically.

Slide 4- 9

Constructing a Stem-and-Leaf Display

• First, cut each data value into leading digits (“stems”) and trailing digits (“leaves”).

• Use the stems to label the bins.

• Use only one digit for each leaf—either round or truncate the data values to one decimal place after the stem.

Slide 4- 10

Stem-and-Leaf Example

• Compare the histogram and stem-and-leaf display for the pulse rates of 24 women at a health clinic. Which graphical display do you prefer?

Dealing With a Lot of Numbers…

• Summarizing the data will help us when we look at large sets of quantitative data.

• Without summaries of the data, it’s hard to grasp what the data tell us.

• The best thing to do is to make a picture…

• Bar charts and Pie Charts are for qualitative data

Slide 4- 13

Histograms: Displaying the Distributionof Price Changes (cont.)

• A histogram plots the bin counts as the heights of bars (like a bar chart).

• Here is a histogram of the monthly price changes in Enron stock:

Slide 4- 14

Slide 4- 15

Histograms: Displaying the Distributionof Price Changes (cont.)

• A relative frequency histogram displays the percentage of cases in each bin instead of the count.

• In this way, relative

frequency histograms are faithful to the area principle.

• Here is a relative frequency histogram of the monthly price changes in Enron stock:

Slide 4- 16

Stem-and-Leaf Displays

• Stem-and-leaf displays show the distribution of a quantitative variable, like histograms do, while preserving the individual values.

• Stem-and-leaf displays contain all the information found in a histogram and, when carefully drawn, satisfy the area principle and show the distribution.

Examining a Distribution

• Look for the following:

• Shape - Does the distribution have one or more peaks (modes) or is it unimodal?

• Outliers – an individual value that falls outside the overall pattern.

• Center – What number is the data centered around?

• Spread - Is the distribution approximately symmetric or is it skewed in one direction? Is it skewed to the right (right tail longer) or left?

Slide 4- 18

Think Before You Draw, Again

• Remember the “Make a picture” rule?

• Now that we have options for data displays, you need to Think carefully about which type of display to make.

• Before making a stem-and-leaf display, a histogram, or a dotplot, check the• Quantitative Data Condition: The data are values of a quantitative

variable whose units are known.

Slide 4- 19

Describing a distribution of dataWhen describing a distribution, make sure to always tell about four things:

1.) Shape

2.) Outliers

3.) Center

4.) Spread

Slide 4- 20

Humps and Bumps

1. Does the histogram have a single, central hump or several separated bumps?• Humps in a histogram are called modes.

• A histogram with one main peak is dubbed unimodal; histograms with two peaks are bimodal; histograms with three or more peaks are called multimodal.

Slide 4- 21

What is the Shape of the Distribution?

1. Does the histogram have a single, central hump or several separated bumps?

2. Is the histogram symmetric?

3. Do any unusual features stick out?

Slide 4- 22

Humps and Bumps (cont.)

• A bimodal histogram has two apparent peaks:

Slide 4- 23

Humps and Bumps (cont.)

• A histogram that doesn’t appear to have any mode and in which all the bars are approximately the same height is called uniform:

Slide 4- 24

Symmetry

2. Is the histogram symmetric?

• If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, the histogram is symmetric.

Slide 4- 25

Symmetry (cont.)• The (usually) thinner ends of a distribution are called the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail.

• In the figure below, the histogram on the left is said to be skewed left, while the histogram on the right is said to be skewed right.

Slide 4- 26

Where is the Center of the Distribution?

• If you had to pick a single number to describe all the data what would you pick?

• It’s easy to find the center when a histogram is unimodal and symmetric—it’s right in the middle.

• On the other hand, it’s not so easy to find the center of a skewed histogram or a histogram with more than one mode.

• For now, we will “eyeball” the center of the distribution. In the next chapter we will find the center numerically.

Slide 4- 27

Anything Unusual?

3. Do any unusual features stick out?• Sometimes it’s the unusual features that tell us something

interesting or exciting about the data.

• You should always mention any stragglers, or outliers, that stand off away from the body of the distribution.

• Are there any gaps in the distribution? If so, we might have data from more than one group.

Slide 4- 28

Anything Unusual? (cont.)

• The following histogram has outliers—there are three cities in the leftmost bar:

Slide 4- 29

Comparing Distributions (cont.)• Compare the following distributions of ages for female and male heart attack patients:

Slide 4- 30

Comparing Distributions

• Often we would like to compare two or more distributions instead of looking at one distribution by itself.

• When looking at two or more distributions, it is very important that the histograms have been put on the same scale. Otherwise, we cannot really compare the two distributions.

• When we compare distributions, we talk about the shape, center, and spread of each distribution.

Slide 4- 31

Timeplots: Order, Please!

• For some data sets, we are interested in how the data behave over time. In these cases, we construct timeplots of the data.

Slide 4- 32

*Re-expressing Skewed Data to Improve Symmetry

Figure 4.11

Histogram of the average age of U.S. Presidents at Inaugaration

• The purpose of an Exploratory Data Analysis is to organize data and identify patterns/departures.

• PLOT YOUR DATA - Choose an appropriate graph

• Look for overall pattern and departures from pattern

• Shape {mound, bimodal, skewed, uniform}

• Outliers {points clearly away from body of data}

• Center {What number “typifies” the data?}

• Spread {How “variable” are the data values?}