Topic: Normal distribution

Activity 17, exploration 2, p. 302-304 (Sevilla, A., & Somers, K., 2007).

The following tables provide information about the top women-owned businesses in the U.S. The first table gives year 2000 revenue in millions of dollars and number of employees for Pennsylvania women-owned companies with year 2000 revenues of $70 million or higher. The second table gives the same information for Michigan women-owned companies with year 2000 revenues of $70 million or higher.

Pennsylvania Companies
2000 Revenue ($ million) / Number of Employees
84 Lumber / 2000 / 4400
Charming Shoppes / 1600 / 12000
Rodale / 500 / 1300
Mothers Work / 366.3 / 3800
Harmelin Media / 200 / 105
McGettigan Partners / 175 / 400
Wetherill Associates / 135 / 500
Hanna Holdings / 73.7 / 1284
Michigan Companies
2000 Revenue ($ million) / Number of Employees
Ilitch Holdings / 800 / 700
Plastech Engineered Products / 420 / 3500
Jerome-Duncan Ford / 350 / 300
Elder Ford / 287.6 / 108
Patsy Lou Williamson Auto. Gp. / 221 / 240
Mexican Industries / 174 / 1463
Rush Trucking / 153 / 2000
Jaguar-Saab of Troy / 143 / 134
Manpower Metro Detroit / 118 / 250
Continental Plastics / 107 / 650
Leco / 82 / 800
Rodgers Chevrolet GEO / 75 / 80
Strategic Staffing Solutions / 75 / 600
Two Men & a Truck Internat'l / 75 / 40
Systrand Manufacturing / 72 / 230

1. Use a calculator or computer to compute the mean and standard deviation of the year 2000 revenue for the Pennsylvania companies in the first table.

Mean / 631.25
Standard Deviation / 741.681

2. Use a calculator or computer to compute the mean and standard deviation of the year 2000 revenue for the Michigan companies in the second table.

Mean / 210.173
Standard Deviation / 195.695

3. Explain what the values you calculated in parts a and b of this exploration tell you about the data sets.

The mean tells us about the typical revenue for each sample of countries. That is, it tells us where the center of the distribution is.

The standard deviation tells us about the variability of the points. For instance, if our points are normally distributed, we can expect 95% of our data points to fall within 2 standard deviations of the mean.

4. How would the mean and standard deviations change if the largest data value in each set were removed?

For Pennsylvania Companies:

Mean / 435.7142857
Standard Deviation / 533.7803557

For Michigan Companies:

Mean / 168.0428571
Standard Deviation / 112.1150283

As expected, the mean, in both cases, goes down. Because the mean is sensitive to extreme values, we expect that if the most extreme values are removed, that the mean will change. Similarly, the standard deviation is also sensitive to extreme values. As a result of removing extreme values, we would expect the standard deviation to decrease. Indeed, we see that this is what happens.

5. Find the mean and the standard deviation of the number of employees for the Pennsylvania companies in the first table.

Mean / 2973.625
Standard Deviation / 3978.875

6. Find the mean and the standard deviation of the number of employees for the Michigan companies in the second table and compare to your results in #5 of this exploration.

Mean / 739.667
Standard Deviation / 941.006

What we see is that the Michigan companies have far fewer workers, on average, than to Pennsylvania companies do. We also see, however, that the number of workers is far more variable in Pennsylvania. This means that on average companies are larger, but that there are more very large companies and more very small companies in Pennsylvania, whereas companies in Michigan tend to be more homogenous, with moderate size.