1. Honors Statistics Chapter 4 Part 4
Displaying and Summarizing Quantitative Data

2. Learning Goals
Know how to display the distribution of a quantitative variable with a histogram, a stem-and-leaf display, or a dotplot.
Know how to display the relative position of quantitative variable with a Cumulative Frequency Curve and analysis the Cumulative Frequency Curve.
Be able to describe the distribution of a quantitative variable in terms of its shape.
Be able to describe any anomalies or extraordinary features revealed by the display of a variable.

3. Learning Goals
Be able to determine the shape of the distribution of a variable by knowing something about the data.
Know the basic properties and how to compute the mean and median of a set of data.
Understand the properties of a skewed distribution.
Know the basic properties and how to compute the standard deviation and IQR of a set of data.

4. Learning Goals
Understand which measures of center and spread are resistant and which are not.
Be able to select a suitable measure of center and a suitable measure of spread for a variable based on information about its distribution.
Be able to describe the distribution of a quantitative variable in terms of its shape, center, and spread.

5. Learning Goal 3
Be able to describe the distribution of a quantitative variable in terms of its shape.

6. Learning Goal 3: What is the Shape of the Distribution?
Does the histogram have a single central peak or several separated peaks?
Is the histogram symmetric?
Do any unusual features stick out?
In any graph, look for the overall pattern and any striking deviations from that pattern.

7. Learning Goal 3: Shape, Center, and Spread
When describing a distribution, make sure to always talk about three things: shape, center, and spread…
Actually you should comment on four things when describing a distribution. The three above and any deviations from the shape.
These deviations from the shape are called ‘outliers’ and will be discussed later.

8. Learning Goal 3: Shape - Peaks
Does the histogram have a single central peak or several separated peaks?
Peaks in a histogram are also 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.

9. Learning Goal 3: Shape: Unimodal - Example
Unimodal – single peak.

10. Learning Goal 3: Shape: Bimodal - Example
Bimodal - two peaks.

11. Learning Goal 3: Shape: Multimodal - Example
Multimodal – three or more peaks.

12. Learning Goal 3: Shape: Bimodal or Multimodal
A bimodal or multimodal shape distribution might indicate that the data are from two or more different populations.

13. Learning Goal 3: Shape: Uniform
A histogram that doesn’t appear to have any mode and in which all the bars are approximately the same height is called uniform or rectangular.
A distribution in which every class has equal frequency, no mode.
A uniform distribution is symmetrical with the added property that the bars are the same height.

14. Learning Goal 3: Shape: Uniform - Example
Uniform – no mode, symmetrical.

15. Learning Goal 3: Shape: Modal Comparison

16. Learning Goal 3: Shape: Symmetrical
In a symmetrical distribution, the data values are evenly distributed on both sides of the mean.
If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, the histogram is symmetric.

17. Learning Goal 3: Shape: Symmetrical - Example
Symmetrical – The distribution’s shape is generally the same if folded down the middle.

18. Learning Goal 3: Shape: Skewed
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.

19. Learning Goal 3: Shape: Skewed Right - Example
In a skewed right distribution, most of the data values fall to the left, and the “tail” of the distribution is to the right.

20. Learning Goal 3: Shape: Skewed Left - Example
In a skewed left distribution, most of the data values fall to the right, and the “tail” of the distribution is to the left.

21. Learning Goal 3: Shape: Skewed - Comparison
A distribution is skewed to the left if the left tail is longer than the right tail
A distribution is skewed to the right if the right tail is longer than the left tail

22. Learning Goal 3: Shape: Other Common Terms
Hump – high bar
Valley – between 2 peaks
Gap – no data

23. Learning Goal 3: Shapes

24. Learning Goal 4
Be able to describe any anomalies or extraordinary features revealed by the display of a variable.

25. Learning Goal 4: Overall Pattern - Anything Unusual?
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.

26. Learning Goal 4: Deviations from the Overall Pattern
Outliers – An individual observation that falls outside the overall pattern of the distribution. Extreme Values – either high or low.
Causes:
Data Mistake
Special nature of some observations
Outliers

27. Learning Goal 4: Outliers
An Outlier falls far from the rest of the data.

28. Learning Goal 4: Outliers
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 two states clearly not belonging 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.
This is from the book. Imagine you are doing a study of health care in the 50 US states, and need to know how they differ in terms of their elderly population.
This is a histogram of the number of states grouped by the percentage of their residents that are 65 or over.
You can see there is one very small number and one very large number, with a gap between them and the rest of the distribution.
Values that fall outside of the overall pattern are called outliers. They might be interesting, they might be mistakes - I get those in my data from typos in entering RNA sequence data into the computer.
They might only indicate that you need more samples. Will be paying a lot of attention to them throughout class both for what we can learn about biology and also because they can cause trouble with your statistics.
Guess which states they are (florida and alaska).
Alaska
Florida

29. Learning Goal 5
Be able to determine the shape of the distribution of a variable by knowing something about the data.

30. Learning Goal 5: Determine the Shape of a Distribution - Example

31. Learning Goal 5: Determine the Shape of a Distribution - Example
It’s often a good idea to think about what the distribution of a data set might look like before we collect the data. What do you think the distribution of each of the following data sets will look like?
Number of Miles run by Saturday morning joggers at a park.
Roughly symmetric, slightly skewed right.
Hours spent by U.S. adults watching football on Thanksgiving Day.
Bimodal. Many people watch no football, others watch most of one or more games.
Amount of winnings of all people playing a particular state’s lottery last week.
Strongly skewed to the right, with almost everyone at $0, a few small prizes, with the winner an outlier.

32. Learning Goal 5: Determine the Shape of a Distribution – Your Turn
Consider a data set containing IQ scores for the general public. What shape would you expect a histogram of this data set to have?
Symmetric
Skewed to the left
Skewed to the right
Bimodal

33. Learning Goal 5: Determine the Shape of a Distribution – Your Turn
Consider a data set of the scores of students on a very easy exam in which most score very well but a few score very poorly. What shape would you expect a histogram of this data set to have?
Symmetric
Skewed to the left
Skewed to the right
Bimodal