The Range

Chapter 1 1.3 Data Analysis Learning Goal: To be able to describe the general shape of a distribution in terms of its number of modes, skewness, and variation.

Number of Modes One way to describe the shape of a distribution is by its number of peaks, or modes. Uniform distribution—has no mode because all data values have the same frequency.

Any peak is considered a mode, even if all peaks do not have the same height. A distribution with a single peak is called a single-peaked, or unimodal, distribution. A distribution with two peaks, even though not the same size, is a bimodal distribution. What is the following distribution?

Symmetry or Skewness A distribution is symmetric if its left half is a mirror image of its right half. A symmetric distribution with a single peak and a bell shape is known as a normal distribution.

Symmetry or Skewness A distribution is left-skewed (or negatively skewed) if the values are more spread out on the left, meaning that some low values are likely to be outliers. A distribution is right skewed or positively skewed if the values are more spread out on the right. It has a tail pulled toward the right.

Data can be "skewed", meaning it tends to have a long tail on one side or the other:

Example

Stem and Leaf Plot

5a

6 a,b

Box and Whisker Plot A box-and-whisker plot shows the distribution of data. The middle half of the data is represented by a “box” with a vertical line at the median. The box extends to the upper and lower quartiles. The upper quartile is the median of the upper half of the data. The lower quartile is the median of the lower half of the data. *It helps you to interpret and represent data. *It gives a visual representation of data.

The box extends to the upper and lower quartiles. A box-and-whisker plot shows the distribution of data. The middle half of the data is represented by a “box” with a vertical line at the median. The box extends to the upper and lower quartiles. The upper quartile is the median of the upper half of the data. The lower quartile is the median of the lower half of the data. Data set: 85,92,78,88,90,88,89 78 85 88 88 89 90 92 Lower quartile Median Upper quartile What is a box-and-whisker plot? What are the upper and lower quartiles?

Example Back to Back S & L Plot

The first step to creating any Stem and Leaf Plot is to write our values out from lowest to highest, and group them into Tens; eg. ones, tens, twenties, thirties, etc

Stem and Leaf Plot for the “Brand A” Phone.

Stem and Leaf Plot for the “Brand B” phone

Two Stem and Leaf Plots and the final steps we need to take to combine them into a single plot.

Back-to back Stem and Leaf Plots

Variation Variation describes how widely data are spread out about the center of a data set. How would you expect the variation to differ between times in a 5K city run and a 5K run in a state meet?

What is the relationship between mean, median and mode of a left-skewed distribution? Find the mean, median and mode of: 0, 5, 10, 20, 40, 45, 45, 50, 50, 50, 60, 60, 60, 60, 60, 60, 70, 70, 70, 70, 70, 70, 70, 70 The mean is 51.5. The median is 60. The mode is 70.

What is the relationship between mean, median and mode of a right-skewed distribution? Find the mean, median, and mode of: 20, 20, 20, 20, 20, 20, 20, 20, 30, 30, 30, 30, 30, 30, 45, 45, 45, 50, 50, 60, 70, 90 The mean is 36.1. The median is 30. The mode is 20.

House prices in the United States. For each of the following situations, state whether you expect the distributions to be symmetric, left-skewed or right-skewed. House prices in the United States. Weight in a sample of 30 year old men. The heights of all players in the NBA.

Copyright © 2009 Pearson Education, Inc. Which is a better measure of “average” (or of the center of the distribution) for a skewed distribution: the median or the mean? Why? Page 160 Copyright © 2009 Pearson Education, Inc.

The body temperature of 2000 randomly selected college students How many modes would you expect for each of the following distributions? Why? Make a rough sketch with clearly labeled axes? The body temperature of 2000 randomly selected college students The attendance at Disney World during a year The last digit of your phone number

To summarize-- The general shape of a distribution can be discussed using: The number of modes Symmetry or skewness Variation.

What is the relationship between mean, median and mode for a normal distribution? Find the mean median and mode of: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7 Mean is 4. Median is 4. Mode is 4.