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

4.2 Shapes of Distributions

Number of ModesOne 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?

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

Symmetry or SkewnessA 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 SkewnessA 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.

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, 7Mean is 4.Median is 4.Mode is 4.

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, 90The mean is 36.1.The median is 30.The mode is 20.

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.

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Which is a better measure of “average” (or of the center of the distribution) for a skewed distribution:the median or the mean? Why?

VariationVariation 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?

To summarize--

The general shape of a distribution can be discussed using:

1.The number of modes2.Symmetry or skewness3.Variation.