Measures of Variation - 3.2 - Part 2

Measures of Variation - 3.2 - Part 2

In part 1 the concept of standard deviation was introduced. In this part, we will dig more into standard deviation, as well as looking at coefficients of variation.

Review: Standard deviation

How do we interpret standard deviation?

• Standard deviation is a measure of consistency.

• Specifically, standard deviation finds the average distance a data value falls from the mean of a data set.

• This means, the higher the standard deviation is, the greater the gap between data values and the mean is. In other words, the higher the standard deviation is, the more spread out a data set is.

• High standard deviation = low consistency

Other important formulas

Population Standard Deviation

Population Standard Deviation cont.

Population Standard Deviation cont.

Multiple Choice

Which is the formula for population standard deviation?

• $\sigma=\sqrt{\frac{\Sigma\left(x-\mu\right)^2}{N}}$

• $s=\sqrt{\frac{\Sigma\left(x-\overline{x}\right)^2}{n-1}}$

Multiple Choice

Which is the formula for population mean?

• $\overline{x}=\frac{\Sigma x}{n}$

• $\mu=\frac{\Sigma x}{N}$

Multiple Choice

Variance is...

• the population or sample mean squared

• the population or sample standard deviation squared

Coefficient of Variation

• Displays standard deviation as a percentage of the sample or the population mean.

• Why is this important? Because standard deviation depends on the units of measure (remember how the cartons of blueberries were in ounces and the blossom diameters were in inches). Therefore it can be difficult to compare standard deviations from various populations (i.e., what varied more, the blossoms from Hybrid A or the blueberries from supplier 2?).

Coefficient of Variation cont.

Example 1:

• The Trading Post on Grand Mesa is a small, family - run store in a remote part of Colorado. The Grand Mesa region contains many good fishing lakes, so the Trading Post sells spinners (a type of fishing lure). There is a very limited selection of spinners, however. The Trading Post has only eight different types of spinners for sale. The prices (in dollars) are below:

• 2.  1.95  2.6  2  1.85  2.25  2.15  2.25

• Since the store only sells 8 different types of spinners, the above values are considered the population.

Example 1 cont.

Example 1 cont.

Example 1 cont.

Example 2:

• Cabela’s in Sidney, Nebraska, is a very large outfitter that carries a broad selection of fishing tackle. It markets its products nationwide. A random sample of 10 spinners from Cabela’s extensive spring fishing products gave the following prices (in dollars):

• 1.69, 1.49, 3.09, 1.79, 1.39, 2.89, 1.49, 1.39, 1.49, 1.99

• Note that the values listed above are a sample of all spinners sold at that Cabela's

Example 2 cont.

Fill in the Blank

What is the sample mean of the data values in question 2:

1.69,  1.49,  3.09,   1.79,   1.39,  2.89,  1.49,  1.39,  1.49,  1.99

 $\overline{x}=$  

Fill in the Blank

What is the sample standard deviation of the data values in question 2:

1.69, 1.49, 3.09, 1.79, 1.39, 2.89, 1.49, 1.39, 1.49, 1.99

 $s=$  

Fill in the Blank

What is the coefficient of variation of the data values in question 2:

1.69, 1.49, 3.09, 1.79, 1.39, 2.89, 1.49, 1.39, 1.49, 1.99

CV $=\frac{s}{\overline{x}}\cdot100=$  

Interpreting Coefficient of Variation

The standard deviation makes up 33.16% of the mean for the sample of Cabela's spinners. This is around 3 times that of The Trading post. This means Cabela’s selection (the sample we examined) has a larger variety of spinner costs. This makes sense given that The Trading Post is a small shop, whereas Cabela’s is a national chain. Cabela’s average price is lower, relative variability is higher.