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Chapter 2: Describing Distributions with Numbers

Knowledge Prerequisites

Graphing and describing distributions

1. declarative knowledge (definitions)
   1. distribution
   2. histogram
   3. class width
   4. shape, center and spread of a distribution
   5. outlier of a distribution
   6. skewness of a dsitribution
   7. stem and leaf of a stemplot
2. procedural knowledge
   1. how to construct a histogram
   2. how to change class width and starting point for histogram
   3. how to construct a stemplot
3. conditional knowledge
   1. know which graph is more appropriate for given data
4. resources
   1. Chapter 1 of textbook

Learning Goals

1. declarative knowledge (definitions)
   1. mean, $\overline{x}$
   2. median, M
   3. quartiles, $Q_1$ and $Q_3$
   4. five-number summary
   5. boxplot
   6. standard deviation, $s$
   7. spread
   8. variance, $s^2$
   9. resistant measure
   10. 1.5 IQR rule (interquartile range)
2. procedural knowledge
   1. how to find the mean of a set of data
   2. how to find the median of a set of data
   3. how to find the five-number summary of a set of data
   4. how to find standard deviation of a set of data
   5. how to find variance of a set of data
   6. how to construct a boxplot from given data
3. conditional knowledge
   1. know the difference between mean and median and what they say about a distribution.
   2. know the difference between standard deviation and quartiles and what they say about a distribution.
   3. know the difference between resistant measures and non-resistant measures.
   4. why is the five-number summary a good description of a distribution.

Questions to Ponder

1. What is the best way to study/prepare for the concepts in this section? Are there are other methods to study these concepts?
2. What questions do you need to ask the instructor?

Purpose of this Section

1. To quantify measures of center and spread.

General Notes

Calculating the standard deviation (a measure of how far, on average, the observations are from the mean.)

The first column of the following table are test scores from a previous statistics class $(\overline{x}=70)$:

The variance is $s^2=\frac{2018}{11-1}=201.8$. The standard deviation is $s=\sqrt{s^2}=\sqrt{201.8}\cong 14.21$.

Properties of the standard deviation:

1. $s$ measures spread about the MEAN.
2. $s=0$ when ALL observations have the same value; otherwise $s>0$.
3. $s$ is larger when the observations are more spread out.
4. $s$ has the same units of measurement as the original data.
5. $s$ is NOT a resistant measure of the spread of a distribution.

Resources

• One Variable Statistical Calculator - http://bcs.whfreeman.com/ips4e/cat_010/applets/histogramIPS.html - This applet is organized as five "tabbed" display screens. Select a data file from the Data Set tab (and you can enter your own data). The applet provides summary statistics, histogram, stemplot.
• The Data Applet - http://www.stat.uiuc.edu/~stat100/java/DataApplet.html - This applet provides the summary statistics and various graphs of provided data sets (you can modify these data sets).
• The Six Characteristics of a Dataset - http://exploringdata.cqu.edu.au/six_char.htm - Once some data have been gathered, the first step in working with the data is to look at it in a variety of ways. This describes the use of six characteristics of a dataset in analysing the dataset.

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